MediaWiki:Gold-data.json

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id

1

pid

51

eid

"math.51.18"

title

"Bessel function"

formulae

id

"FORMULA_0f521573a47e7fd187dafed615b0ecce"

formula

"\begin{align}J_{-(m+\frac{1}{2})}(x) &= (-1)^{m+1} Y_{m+\frac{1}{2}}(x), \\Y_{-(m+\frac{1}{2})}(x) &= (-1)^m J_{m+\frac{1}{2}}(x).\end{align}"

semanticFormula

"\begin{align}\BesselJ{- (m + \frac{1}{2})}@{x} &= (- 1)^{m+1} \BesselY{m+\frac{1}{2}}@{x} , \\ \BesselY{- (m + \frac{1}{2})}@{x} &= (-1)^m \BesselJ{m+\frac{1}{2}}@{x} .\end{align}"

confidence

0.8803349492974287

translations

Mathematica

translation

"BesselJ[- (m +Divide[1,2]), x] == (- 1)^(m + 1)* BesselY[m +Divide[1,2], x] BesselY[- (m +Divide[1,2]), x] == (- 1)^(m)* BesselJ[m +Divide[1,2], x]"

translationInformation

subEquations

"BesselJ[- (m +Divide[1,2]), x] = (- 1)^(m + 1)* BesselY[m +Divide[1,2], x]"

"BesselY[- (m +Divide[1,2]), x] = (- 1)^(m)* BesselJ[m +Divide[1,2], x]"

freeVariables

"m"

"x"

constraints

Empty array

tokenTranslations

\pgcd	"Greatest common divisor; Example: \pgcd{m,n} Will be translated to: GCD[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/27.1#p2.t1.r3 Mathematica: https://reference.wolfram.com/language/ref/GCD.html"
\BesselY	"Bessel function second kind; Example: \BesselY{v}@{z} Will be translated to: BesselY[$0, $1] Branch Cuts: (-\infty, 0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/10.2#E3 Mathematica: https://"
\BesselJ	"Bessel function first kind; Example: \BesselJ{v}@{z} Will be translated to: BesselJ[$0, $1] Branch Cuts: if v \notin \Integers: (-\infty, 0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/10.2#E2 Mathematica: https://reference.wolfram.com/language/ref/BesselJ.html"

Maple

translation

"BesselJ(- (m +(1)/(2)), x) = (- 1)^(m + 1)* BesselY(m +(1)/(2), x); BesselY(- (m +(1)/(2)), x) = (- 1)^(m)* BesselJ(m +(1)/(2), x)"

translationInformation

subEquations

"BesselJ(- (m +(1)/(2)), x) = (- 1)^(m + 1)* BesselY(m +(1)/(2), x)"

"BesselY(- (m +(1)/(2)), x) = (- 1)^(m)* BesselJ(m +(1)/(2), x)"

freeVariables

"m"

"x"

constraints

Empty array

tokenTranslations

\pgcd	"Greatest common divisor; Example: \pgcd{m,n} Will be translated to: gcd($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/27.1#p2.t1.r3 Maple: https://www.maplesoft.com/support/help/Maple/view.aspx?path=gcd"
\BesselY	"Bessel function second kind; Example: \BesselY{v}@{z} Will be translated to: BesselY($0, $1) Branch Cuts: (-\infty, 0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/10.2#E3 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=Bessel"
\BesselJ	"Bessel function first kind; Example: \BesselJ{v}@{z} Will be translated to: BesselJ($0, $1) Branch Cuts: if v \notin \Integers: (-\infty, 0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/10.2#E2 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=Bessel"

positions

section	8
sentence	8
word	32

includes

"Y_{\alpha}"

"J_{-\alpha}(x)"

"J"

"J_{\alpha}(x)"

"Y_{n}"

"J_{n}(x)"

"m"

"Y_{\alpha}(x)"

"J_{\alpha}"

"x"

"(-1)^{m}"

"J_{n}"

"J_{\alpha}(z)"

"J_{\alpha}(k)"

"Y"

"J_{n + m}(x)"

isPartOf

Empty array

definiens

definition	"Bessel function first kind"
score	2

definition	"Bessel function second kind"
score	2

definition	"above relation"
score	0

definition	"spherical Bessel"
score	1

definition	"integer"
score	1

definition	"nonnegative integer"
score	1

definition	"relationship"
score	0

definition	"function"
score	1

definition	"recurrence relation"
score	1

definition	"Bessel"
score	1

definition	"large number of other known integral"
score	0

definition	"positive zero"
score	0

definition	"entire function of genus"
score	0

definition	"identity"
score	0

definition	"orthogonality relation"
score	0

definition	"Bessel function"
score	2

definition	"term"
score	0

definition	"real zero"
score	0

definition	"similar relation"
score	0

definition	"Hankel"
score	1

definition	"Bessel function of the second kind"
score	2

definition	"limit"
score	0

definition	"ordinary Bessel function"
score	1

definition	"case"
score	0

definition	"negative integer"
score	0

definition	"integral formula"
score	0

definition	"small argument"
score	0

definition	"average"
score	0

definition	"Bessel function of the first kind"
score	2

definition	"reference"
score	0

definition	"series expansion"
score	0

definition	"spherical Bessel function"
score	1

definition	"Abel 's identity"
score	0

definition	"important property of Bessel 's equation"
score	1

definition	"particular Bessel"
score	1

definition	"solution of Bessel 's equation"
score	0

definition	"Wronskian of the solution"
score	0

definition	"series"
score	0

definition	"closure equation"
score	0

id

2

pid

52

eid

"math.52.404"

title

"Ellipse"

formulae

id

"FORMULA_d3e28ddd096754fb8e1e52aaaa4f7770"

formula

"E(e) \,=\, \int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta"

semanticFormula

"\compellintEk@{e} = \int_0^{\cpi / 2} \sqrt{1 - e^2 \sin^2 \theta} \diff{\theta}"

confidence

0.8896531556938116

translations

Mathematica

translation

"EllipticE[(e)^2] == Integrate[Sqrt[1 - (e)^(2)*(Sin[\[Theta]])^(2)], {\[Theta], 0, Pi/2}, GenerateConditions->None]"

translationInformation

subEquations

"EllipticE[(e)^2] = Integrate[Sqrt[1 - (e)^(2)*(Sin[\[Theta]])^(2)], {\[Theta], 0, Pi/2}, GenerateConditions->None]"

freeVariables

Empty array

constraints

Empty array

tokenTranslations

\cpi	"Pi was translated to: Pi"
\expe	"Recognizes e with power as the exponential function. It was translated as a function."
\compellintEk	"Legendre's complete elliptic integral of the second kind; Example: \compellintEk@{k} Will be translated to: EllipticE[($0)^2] Relevant links to definitions: DLMF: http://dlmf.nist.gov/19.2#E8 Mathematica: https://"
\sin	"Sine; Example: \sin@@{z} Will be translated to: Sin[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E1 Mathematica: https://reference.wolfram.com/language/ref/Sin.html"

Maple

translation

"EllipticE(e) = int(sqrt(1 - (e)^(2)*(sin(theta))^(2)), theta = 0..Pi/2)"

translationInformation

subEquations

"EllipticE(e) = int(sqrt(1 - (e)^(2)*(sin(theta))^(2)), theta = 0..Pi/2)"

freeVariables

Empty array

constraints

Empty array

tokenTranslations

\cpi	"Pi was translated to: Pi"
\expe	"Recognizes e with power as the exponential function. It was translated as a function."
\compellintEk	"Legendre's complete elliptic integral of the second kind; Example: \compellintEk@{k} Will be translated to: EllipticE($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/19.2#E8 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=EllipticE"
\sin	"Sine; Example: \sin@@{z} Will be translated to: sin($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=sin"

positions

section	37
sentence	0
word	39

includes

"\theta"

"E"

"\pi a b"

"\pi"

"e"

"E(e)"

isPartOf

Empty array

definiens

definition	"complete elliptic integral of the second kind"
score	2

definition	"elementary function"
score	1

definition	"function"
score	1

definition	"length of the semi-major axis"
score	2

definition	"eccentricity"
score	2

definition	"circumference"
score	0

definition	"ellipse"
score	1

definition	"angle"
score	1

definition	"angular coordinate"
score	1

definition	"center"
score	0

definition	"formula"
score	0

definition	"rotation angle"
score	0

id

3

pid

53

eid

"math.53.6"

title

"Elliptic integral"

formulae

id

"FORMULA_04e9de23897a3b23dee1a9b7312ad99e"

formula

"F(x;k) = u"

semanticFormula

"\incellintFk@{\asin@{\Jacobiellsnk@@{u}{k}}}{k} = u"

confidence

0

translations

Mathematica

translation

"EllipticF[ArcSin[JacobiSN[u, (k)^2]], (k)^2] == u"

translationInformation

subEquations

"EllipticF[ArcSin[JacobiSN[u, (k)^2]], (k)^2] = u"

freeVariables

"k"

"u"

constraints

Empty array

tokenTranslations

Empty object

Maple

translation

"EllipticF(JacobiSN(u, k), k) = u"

translationInformation

subEquations

"EllipticF(JacobiSN(u, k), k) = u"

freeVariables

"k"

"u"

constraints

Empty array

tokenTranslations

Empty object

positions

section	2
sentence	6
word	5

includes

"u"

"F"

"x"

"k"

isPartOf

"F(\varphi,k) = F\left(\varphi \,|\, k^2\right) = F(\sin \varphi ; k) = \int_0^\varphi \frac {\mathrm{d}\theta}{\sqrt{1 - k^2 \sin^2 \theta}}"

"F(x ; k) = \int_{0}^{x} \frac{\mathrm{d}t}{\sqrt{\left(1 - t^2\right)\left(1 - k^2 t^2\right)}}"

"E(\varphi,k) = E\left(\varphi \,|\,k^2\right) = E(\sin\varphi;k) = \int_0^\varphi \sqrt{1-k^2 \sin^2\theta}\,\mathrm{d}\theta"

"E(x;k) = \int_0^x \frac{\sqrt{1-k^2 t^2} }{\sqrt{1-t^2}}\,\mathrm{d}t"

definiens

definition	"inverse to the elliptic integral"
score	1

definition	"Jacobian elliptic function"
score	2

definition	"Legendre"
score	1

definition	"normal form"
score	1

definition	"trigonometric form"
score	1

definition	"incomplete elliptic integral of the second kind"
score	0

definition	"incomplete elliptic integral of the first kind"
score	2

id

4

pid

54

eid

"math.54.195"

title

"Gamma function"

formulae

id

"FORMULA_19a0f00da77cc439ad679c579a295bd2"

formula

"\frac{1}{\Gamma(z)} = \frac{i}{2\pi}\int_C (-t)^{-z}e^{-t}\,dt"

semanticFormula

"\frac{1}{\EulerGamma@{z}} = \frac{\iunit}{2 \cpi} \int_C(- t)^{-z} \expe^{-t} \diff{t}"

confidence

0.8809245132365588

translations

Empty object

positions

section	11
sentence	10
word	9

includes

"C"

"\Gamma"

"\frac {1}{\Gamma (z)}"

"z"

"1"

"\Gamma(r)"

"t"

"\pi"

"\Gamma (z)"

"\Gamma(z)"

"\Pi\left(z\right)"

"\Gamma\left(z\right)"

"e^{-x}"

isPartOf

Empty array

definiens

definition	"related expression"
score	0

definition	"integer"
score	0

definition	"reflection formula"
score	1

definition	"end"
score	0

definition	"Hankel contour"
score	2

definition	"Riemann sphere"
score	1

definition	"Hankel 's formula for the gamma function"
score	2

definition	"gamma function"
score	2

definition	"reciprocal gamma function"
score	2

id

5

pid

55

eid

""

title

"Logarithm"

formulae

id

"FORMULA_579837194f2124b255d579031524a91c"

formula

"2^{4} = 2 \times2 \times 2 \times 2 = 16"

semanticFormula

"2^{4} = 2 \times2 \times 2 \times 2 = 16"

confidence

0

translations

Mathematica

translation

"(2)^(4) == 2 * 2 * 2 * 2 == 16"

translationInformation

subEquations

"(2)^(4) = 2 * 2 * 2 * 2"

"2 * 2 * 2 * 2 = 16"

freeVariables

Empty array

constraints

Empty array

tokenTranslations

\times	"was translated to: *"

Maple

translation

"(2)^(4) = 2 * 2 * 2 * 2 = 16"

translationInformation

subEquations

"(2)^(4) = 2 * 2 * 2 * 2"

"2 * 2 * 2 * 2 = 16"

freeVariables

Empty array

constraints

Empty array

tokenTranslations

\times	"was translated to: *"

positions

section	4
sentence	0
word	3

includes

"2"

"^{4}"

isPartOf

Empty array

definiens

definition	"example"
score	2

id

6

pid

56

eid

"math.56.40"

title

"Riemann zeta function"

formulae

id

"FORMULA_bd88ec58aa42c7a59bc2f4ff458a54cf"

formula

"\psi(x) := \sum_{n=1}^\infty e^{-n^2 \pi x}"

semanticFormula

"\psi(x) : = \sum_{n=1}^\infty \expe^{- n^2 \cpi x}"

confidence

0.9073333333333333

translations

Mathematica

translation	"\[Psi][x_] := Sum[Exp[-(n)^(2)Pix], {n, 1, Infinity}]"

Maple

translation	"psi := (x) -> sum(exp(-(n)^(2)Pix), n=1..infinity)"

positions

section	4
sentence	7
word	23

includes

"1"

"n"

"2"

"x"

"\psi"

isPartOf

Empty array

definiens

definition	"analytic continuation"
score	0

definition	"absolute convergence"
score	0

definition	"convenience"
score	0

definition	"inversion"
score	0

definition	"process"
score	0

definition	"stricter requirement"
score	0

definition	"series"
score	1

definition	"definition"
score	2

id

7

pid

57

eid

"math.57.2"

title

"Logarithmic integral function"

formulae

id

"FORMULA_36fb8f8330168b8f8acda0dc36851474"

formula

"\operatorname{li}(x) = \lim_{\varepsilon \to 0+} \left( \int_0^{1-\varepsilon} \frac{dt}{\ln t} + \int_{1+\varepsilon}^x \frac{dt}{\ln t} \right)"

semanticFormula

"\logint@{x} = \lim_{\varepsilon \to 0+}(\int_0^{1-\varepsilon} \frac{\diff{t}}{\ln t} + \int_{1+\varepsilon}^x \frac{\diff{t}}{\ln t})"

confidence

0.8728566391293461

translations

Mathematica

translation

"LogIntegral[x] == Limit[Integrate[Divide[1,Log[t]], {t, 0, 1 - \[CurlyEpsilon]}, GenerateConditions->None]+ Integrate[Divide[1,Log[t]], {t, 1 + \[CurlyEpsilon], x}, GenerateConditions->None], \[CurlyEpsilon] -> 0, Direction -> "FromAbove", GenerateConditions->None]"

translationInformation

subEquations

"LogIntegral[x] = Limit[Integrate[Divide[1,Log[t]], {t, 0, 1 - \[CurlyEpsilon]}, GenerateConditions->None]+ Integrate[Divide[1,Log[t]], {t, 1 + \[CurlyEpsilon], x}, GenerateConditions->None], \[CurlyEpsilon] -> 0, Direction -> "FromAbove", GenerateConditions->None]"

freeVariables

"x"

constraints

Empty array

tokenTranslations

\logint	"Logarithmic integral; Example: \logint@{x} Will be translated to: LogIntegral[$0] Constraints: x > 1 Mathematica uses other branch cuts: (-\inf, 1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/6.2#E8 Mathematica: https://reference.wolfram.com/language/ref/LogIntegral.html"
\ln	"Natural logarithm; Example: \ln@@{z} Will be translated to: Log[$0] Constraints: z != 0 Branch Cuts: (-\infty, 0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.2#E2 Mathematica: https://reference.wolfram.com/language/ref/Log.html"

Maple

translation

"Li(x) = limit(int((1)/(ln(t)), t = 0..1 - varepsilon)+ int((1)/(ln(t)), t = 1 + varepsilon..x), varepsilon = 0, right)"

translationInformation

subEquations

"Li(x) = limit(int((1)/(ln(t)), t = 0..1 - varepsilon)+ int((1)/(ln(t)), t = 1 + varepsilon..x), varepsilon = 0, right)"

freeVariables

"x"

constraints

Empty array

tokenTranslations

\logint	"Logarithmic integral; Example: \logint@{x} Will be translated to: Li($0) Constraints: x > 1 Relevant links to definitions: DLMF: http://dlmf.nist.gov/6.2#E8 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=Li"
\ln	"Natural logarithm; Example: \ln@@{z} Will be translated to: ln($0) Constraints: z != 0 Branch Cuts: (-\infty, 0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.2#E2 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=ln"

positions

section	1
sentence	2
word	22

includes

"x"

"x)"

isPartOf

Empty array

definiens

definition	"Cauchy principal value"
score	2

definition	"function"
score	1

definition	"singularity"
score	1

definition	"special function"
score	1

definition	"integral representation"
score	1

definition	"integral logarithm li"
score	2

definition	"logarithmic integral function"
score	2

definition	"logarithmic integral"
score	2

definition	"function li"
score	1

id

8

pid

58

eid

"math.58.61"

title

"Gaussian quadrature"

formulae

id

"FORMULA_8c49145544fca24efb8de07eb1275c09"

formula

"w_{i} = \frac{1}{p'_{n}(x_{i})}\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx"

semanticFormula

"w_{i} = \frac{1}{p'_{n}(x_{i})} \int_{a}^{b} \omega(x) \frac{p_{n}(x)}{x-x_{i}} \diff{x}"

confidence

0

translations

Mathematica

translation	"Subscript[w, i] = Divide[1, Subscript[p\[Prime], n][Subscript[x, i]]]Integrate[\[Omega][x]Divide[Subscript[p,n][x], x-Subscript[x,i]], {x, a, b}]"

positions

section	5
sentence	4
word	24

includes

"a"

"b"

"w_{i}"

"p_n(x)"

"p_{k}(x)"

"p_{n}"

"x_{i}"

"\omega(x)"

"p_{n}(x)"

"\omega"

"x_i"

"a_{n}"

"P_{n}"

"w_i"

"r(x_{i})"

"i"

"n"

"x"

"P_{n}(x)"

"\frac{p_{n}(x)}{x-x_{i}}"

"p'_{n}(x_{i})"

"p_{n}(x_{i})"

"x_{j}"

"p_r"

"p_s"

"\mathbf{e}_n"

"x_j"

"1"

isPartOf

Empty array

definiens

definition	"yield"
score	0

definition	"integral expression for the weight"
score	2

definition	"integrand"
score	1

definition	"L'Hôpital 's rule"
score	0

definition	"limit"
score	0

definition	"polynomial of degree"
score	0

id

9

pid

59

eid

"math.59.52"

title

"Lambert W function"

formulae

id

"FORMULA_fe13643d8449f601f150fd50c0751cf2"

formula

"\begin{align}x & =ue^u, \\[5pt]\frac{dx}{du} & =(u+1)e^u.\end{align}"

semanticFormula

"\begin{align}x & =\LambertW@{x}\expe^{\LambertW@{x}}, \\ \deriv{x}{\LambertW@{x}} &=(\LambertW@{x} + 1) \expe^{\LambertW@{x}} .\end{align}"

confidence

0

translations

Mathematica

translation

"x == ProductLog[x]*(E)^(ProductLog[x]) D[x,ProductLog[x]] = (ProductLog[x] + 1)*Exp[ProductLog[x]]"

translationInformation

subEquations

"x = ProductLog[x]*(E)^(ProductLog[x])"

"D[x,ProductLog[x]] = (ProductLog[x] + 1)*Exp[ProductLog[x]]"

freeVariables

"u"

"x"

constraints

Empty array

tokenTranslations

\expe	"Recognizes e with power as the exponential function. It was translated as a function."

Maple

translation

"x = LambertW(x)*exp(u); diff(x, [LambertW(x)$1]) = (LambertW(x) + 1)*exp(LambertW(x))"

translationInformation

subEquations

"x = LambertW(x)*exp(u)"

"diff(x, [LambertW(x)$1]) = (LambertW(x) + 1)*exp(LambertW(x))"

freeVariables

"u"

"x"

constraints

Empty array

tokenTranslations

\expe	"Recognizes e with power as the exponential function. It was translated as a function."

positions

section	12
sentence	1
word	14

includes

"e^{w}"

"x"

isPartOf

Empty array

definiens

definition	"substitution"
score	2

definition	"third identity"
score	0

definition	"second identity"
score	1

id

10

pid

60

eid

"math.60.57"

title

"Legendre polynomials"

formulae

id

"FORMULA_8646bd0d06e9454aaa39dfc506fe54f7"

formula

"\frac{1}{\left| \mathbf{x}-\mathbf{x}' \right|} = \frac{1}{\sqrt{r^2+{r'}^2-2r{r'}\cos\gamma}} = \sum_{\ell=0}^\infty \frac{{r'}^\ell}{r^{\ell+1}} P_\ell(\cos \gamma)"

semanticFormula

"\frac{1}{|\mathbf{x} - \mathbf{x} '|} = \frac{1}{\sqrt{r^2+{r'}^2-2r{r'}\cos\gamma}} = \sum_{\ell=0}^\infty \frac{{r'}^\ell}{r^{\ell+1}} \LegendrepolyP{\ell}@{\cos \gamma}"

confidence

0.808438593520797

translations

Mathematica	"Divide[1, Abs[x - x\[Prime]]] == Divide[1, Sqrt[r^2+(r\[Prime])^(2)-2rr\[Prime] Cos[\[Gamma]]]] == Sum[Divide[(r\[Prime])^(\[ScriptL]), r^(\[ScriptL]+1)]*LegendreP[\[ScriptL], Cos[\[Gamma]]], {\[ScriptL], 0, Infinity}]"

positions

section	6
sentence	0
word	21

includes

"P_n(x)"

"P_n"

"P_n(\cos\theta)"

"P_{n}(x)"

"P_m"

"r"

"r{'}"

"\mathbf{x}"

"\mathbf{x}{'}"

"\gamma"

"P"

isPartOf

Empty array

definiens

definition	"expansion"
score	2

definition	"Adrien-Marie Legendre as the coefficient"
score	0

definition	"angle"
score	1

definition	"Legendre polynomial"
score	2

definition	"length of the vector"
score	1

definition	"vector"
score	1

definition	"polynomial"
score	1

id

11

pid

61

eid

"math.61.27"

title

"Error function"

formulae

id

"FORMULA_523ec091d0929f0fa69ae7e0d563a72b"

formula

"\operatorname{erf}^{(k)}(z) = \frac{2 (-1)^{k-1}}{\sqrt{\pi}} \mathit{H}_{k-1}(z) e^{-z^2} = \frac{2}{\sqrt{\pi}} \frac{d^{k-1}}{dz^{k-1}} \left(e^{-z^2}\right),\qquad k=1, 2, \dots"

semanticFormula

"\erf@@{(z)}^{(k)} = \frac{2 (-1)^{k-1}}{\sqrt{\cpi}} \HermitepolyH{k-1}@{z} \expe^{-z^2} = \frac{2}{\sqrt{\cpi}} \deriv [{k-1}]{ }{z}(\expe^{-z^2}) , \qquad k = 1 , 2 , \dots"

confidence

0.82607945540953

translations

Mathematica

translation

"D[Erf[z], {z, k}] == Divide[2*(- 1)^(k - 1),Sqrt[Pi]]*HermiteH[k - 1, z]*Exp[- (z)^(2)] == Divide[2,Sqrt[Pi]]*D[Exp[- (z)^(2)], {z, k - 1}]"

translationInformation

subEquations

"D[Erf[z], {z, k}] = Divide[2*(- 1)^(k - 1),Sqrt[Pi]]*HermiteH[k - 1, z]*Exp[- (z)^(2)]"

"Divide[2*(- 1)^(k - 1),Sqrt[Pi]]*HermiteH[k - 1, z]*Exp[- (z)^(2)] = Divide[2,Sqrt[Pi]]*D[Exp[- (z)^(2)], {z, k - 1}]"

freeVariables

"k"

"z"

constraints

"k == 1 , 2 , \[Ellipsis]"

tokenTranslations

\deriv1	"Derivative; Example: \deriv[n]{f}{x} Will be translated to: D[$1, {$2, $0}] Relevant links to definitions: DLMF: http://dlmf.nist.gov/1.4#E4 Mathematica: https://"
\cpi	"Pi was translated to: Pi"
\HermitepolyH	"Hermite polynomial; Example: \HermitepolyH{n}@{x} Will be translated to: HermiteH[$0, $1] Relevant links to definitions: DLMF: http://dlmf.nist.gov/18.3#T1.t1.r13 Mathematica: https://"
\expe	"Recognizes e with power as the exponential function. It was translated as a function."
\erf	"Error function; Example: \erf@@{z} Will be translated to: Erf[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/7.2#E1 Mathematica: https://reference.wolfram.com/language/ref/Erf.html"

Maple

translation

"diff(erf(z), [z$k]) = (2*(- 1)^(k - 1))/(sqrt(Pi))*HermiteH(k - 1, z)*exp(- (z)^(2)) = (2)/(sqrt(Pi))*diff(exp(- (z)^(2)), [z$(k - 1)])"

translationInformation

subEquations

"diff(erf(z), [z$k]) = (2*(- 1)^(k - 1))/(sqrt(Pi))*HermiteH(k - 1, z)*exp(- (z)^(2))"

"(2*(- 1)^(k - 1))/(sqrt(Pi))*HermiteH(k - 1, z)*exp(- (z)^(2)) = (2)/(sqrt(Pi))*diff(exp(- (z)^(2)), [z$(k - 1)])"

freeVariables

"k"

"z"

constraints

"k = 1 , 2 , .."

tokenTranslations

\deriv1	"Derivative; Example: \deriv[n]{f}{x} Will be translated to: diff($1, [$2$($0)]) Relevant links to definitions: DLMF: http://dlmf.nist.gov/1.4#E4 Maple: https://www.maplesoft.com/support/help/Maple/view.aspx?path=diff"
\cpi	"Pi was translated to: Pi"
\HermitepolyH	"Hermite polynomial; Example: \HermitepolyH{n}@{x} Will be translated to: HermiteH($0, $1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/18.3#T1.t1.r13 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=HermiteH"
\expe	"Recognizes e with power as the exponential function. It was translated as a function."
\erf	"Error function; Example: \erf@@{z} Will be translated to: erf($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/7.2#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=erf"

positions

section	5
sentence	4
word	6

includes

"erf"

"e^{-t^2}"

"-1"

"z"

"z)"

"e"

"\mathit{H}"

"z^{\bar{n}}"

isPartOf

Empty array

definiens

definition	"Higher order derivative"
score	2

definition	"physicists ' Hermite polynomial"
score	1

definition	"name error function"
score	1

definition	"erfc"
score	1

definition	"error function"
score	2

definition	"erf"
score	1

id

12

pid

62

eid

"math.62.44"

title

"Chebyshev polynomials"

formulae

id

"FORMULA_d9eb68704833b0f525c4ca81a749d9ca"

formula

"x_k = \cos\left(\frac{\pi(k+1/2)}{n}\right),\quad k=0,\ldots,n-1"

semanticFormula

"x_k = \cos(\frac{\cpi(k + 1 / 2)}{n}) , \quad k = 0 , \ldots , n - 1"

confidence

0

translations

Mathematica

translation

"Subscript[x, k] == Cos[Divide[Pi*(k + 1/2),n]]"

translationInformation

subEquations

"Subscript[x, k] = Cos[Divide[Pi*(k + 1/2),n]]"

freeVariables

"Subscript[x, k]"

"k"

"n"

constraints

"k == 0 , \[Ellipsis], n - 1"

tokenTranslations

\cos	"Cosine; Example: \cos@@{z} Will be translated to: Cos[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E2 Mathematica: https://reference.wolfram.com/language/ref/Cos.html"
\cpi	"Pi was translated to: Pi"

Maple

translation

"x[k] = cos((Pi*(k + 1/2))/(n))"

translationInformation

subEquations

"x[k] = cos((Pi*(k + 1/2))/(n))"

freeVariables

"k"

"n"

"x[k]"

constraints

"k = 0 , .. , n - 1"

tokenTranslations

\cos	"Cosine; Example: \cos@@{z} Will be translated to: cos($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E2 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=cos"
\cpi	"Pi was translated to: Pi"

positions

section	8
sentence	2
word	18

includes

"n"

"x"

"n x"

"-1"

"k = 0"

"x_{k}"

isPartOf

Empty array

definiens

definition	"root"
score	2

definition	"one"
score	0

definition	"trigonometric definition"
score	0

definition	"fact"
score	0

definition	"different simple root"
score	1

definition	"Chebyshev polynomial of the first kind"
score	1

definition	"Chebyshev polynomial"
score	1

id

13

pid

63

eid

"math.63.109"

title

"Hermite polynomials"

formulae

id

"FORMULA_249043719eb4dd70350b460363255e11"

formula

"E(x, y; u) := \sum_{n=0}^\infty u^n \, \psi_n (x) \, \psi_n (y) = \frac{1}{\sqrt{\pi (1 - u^2)}} \, \exp\left(-\frac{1 - u}{1 + u} \, \frac{(x + y)^2}{4} - \frac{1 + u}{1 - u} \, \frac{(x - y)^2}{4}\right)"

semanticFormula

"E(x , y ; u) : = \sum_{n=0}^\infty u^n \psi_n(x) \psi_n(y) = \frac{1}{\sqrt{\cpi(1 - u^2)}} \exp(- \frac{1 - u}{1 + u} \frac{(x + y)^2}{4} - \frac{1 + u}{1 - u} \frac{(x - y)^2}{4})"

confidence

0

translations

Mathematica

translation	"\[CapitalEpsilon][x_, y_, u_] := Sum[(u)^(n)* Subscript[\[Psi], n][x]* Subscript[\[Psi], n][y], {n, 0, Infinity}, GenerateConditions->None] == Divide[1,Sqrt[Pi(1 - (u)^(2))]]Exp[-Divide[1 - u,1 + u]Divide[(x + y)^(2),4]-Divide[1 + u,1 - u]Divide[(x - y)^(2),4]]"

Maple

translation	"Epsilon := (x, y, u) -> sum((u)^(n)* psi[n](x)* psi[n](y), n = 0..infinity) = (1)/(sqrt(Pi(1 - (u)^(2))))exp(-(1 - u)/(1 + u)((x + y)^(2))/(4)-(1 + u)/(1 - u)((x - y)^(2))/(4))"

positions

section	25
sentence	2
word	16

includes

"u"

"\psi_{n}"

"H_{n}(x)"

"\psi_{n}(x)"

"x^{n}"

"n"

"x"

"H_{n}"

"He_{n}(x)"

"He_{n}"

"D_{n}(z)"

"E(x,y;u)"

"H_{n}(y)"

isPartOf

Empty array

definiens

definition	"distributional identity"
score	1

definition	"separable kernel"
score	1

definition	"Mehler 's formula"
score	2

definition	"Hermite polynomial"
score	1

definition	"Hermite function"
score	2

definition	"Hermite"
score	1

definition	"bivariate Gaussian probability density"
score	1

definition	"Gaussian probability density"
score	1

definition	"Gaussian probability"
score	1

id

14

pid

64

eid

"math.64.8"

title

"Legendre function"

formulae

id

"FORMULA_06f9b7b1d3f141742ad1c582b55056ba"

formula

"x = \pm 1"

semanticFormula

"x = \pm 1"

confidence

0

translations

Mathematica

translation

"x == \[PlusMinus]1"

translationInformation

subEquations

"x = + 1"

"x = - 1"

freeVariables

"x"

constraints

Empty array

tokenTranslations

\pm	"was translated to: \[PlusMinus]"

Maple

translation

"x = &+- 1"

translationInformation

subEquations

"x = + 1"

"x = - 1"

freeVariables

"x"

constraints

Empty array

tokenTranslations

\pm	"was translated to: &+-"

positions

section	3
sentence	1
word	11

includes

Empty array

isPartOf

Empty array

definiens

definition	"value"
score	2

id

15

pid

65

eid

"math.65.27"

title

"Bernoulli polynomials"

formulae

id

"FORMULA_a7fcf738c676932d58f39ff9f7df22ae"

formula

"E_n=2^nE_n(\tfrac{1}{2})"

semanticFormula

"\EulernumberE{n} = 2^n\EulerpolyE{n}@{\tfrac{1}{2}}"

confidence

0.8953028732079359

translations

Mathematica

translation	"EulerE[n] == (2)^(n)* EulerE[n, Divide[1,2]]"

Maple

translation	"euler(n) = (2)^(n)* euler(n, (1)/(2))"

positions

section	8
sentence	4
word	6

includes

"B_{n}"

"n"

"E_{k}"

isPartOf

Empty array

definiens

definition	"Euler number"
score	2

id

16

pid

66

eid

"math.66.8"

title

"Trigonometric integral"

formulae

id

"FORMULA_0feb8031b89a9707b164163ec50265f0"

formula

"\operatorname{Si}(ix) = i\operatorname{Shi}(x)"

semanticFormula

"\sinint@{\iunit x} = \iunit \sinhint@{x}"

confidence

0.8811682126384021

translations

Mathematica

translation

"SinIntegral[I*x] == I*SinhIntegral[x]"

translationInformation

subEquations

"SinIntegral[I*x] == I*SinhIntegral[x]"

freeVariables

"x"

constraints

Empty array

tokenTranslations

Shi	"Was interpreted as a function call because of a leading \operatorname."
\iunit	"Imaginary unit was translated to: I"
\sinint	"Sine integral; Example: \sinint@{z} Will be translated to: SinIntegral[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/6.2#E9 Mathematica: https://reference.wolfram.com/language/ref/SinIntegral.html"

Maple

translation

"Si(I*x) = I*Shi(x)"

translationInformation

subEquations

"Si(I*x) = I*Shi(x)"

freeVariables

"x"

constraints

Empty array

tokenTranslations

Shi	"Was interpreted as a function call because of a leading \operatorname."
\iunit	"Imaginary unit was translated to: I"
\sinint	"Sine integral; Example: \sinint@{z} Will be translated to: Si($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/6.2#E9 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=Si"

positions

section	3
sentence	1
word	9

includes

"Si"

"Si(x)"

"x"

isPartOf

Empty array

definiens

definition	"ordinary sine"
score	1

definition	"Trigonometric integral"
score	2

definition	"hyperbolic sine integral"
score	2

id

17

pid

67

eid

"math.67.29"

title

"Beta function"

formulae

id

"FORMULA_5f59825d73d63a9990498edca7222261"

formula

"f(z)=\frac{1}{\Beta(x,y)}"

semanticFormula

"f(x, y) = \frac{1}{\EulerBeta@{x}{y}}"

confidence

0.8953028732079359

translations

Mathematica

translation	"f[x_, y_] := Divide[1,Beta[x, y]]"

Maple

translation	"f := (x,y) -> (1)/(Beta(x, y))"

positions

section	6
sentence	0
word	12

includes

"x, y"

"\Beta"

"y"

"x"

isPartOf

Empty array

definiens

definition	"function about the form"
score	0

definition	"reciprocal beta function"
score	2

definition	"definite integral of trigonometric function"
score	1

definition	"integral representation"
score	0

definition	"product"
score	0

definition	"power"
score	0

definition	"multiple-angle"
score	0

definition	"beta function"
score	2

id

18

pid

68

eid

"math.68.51"

title

"Fresnel integral"

formulae

id

"FORMULA_b7dae135f3b04317078f86b595fe7dae"

formula

"\begin{align}\int x^m e^{ix^n}\,dx & =\frac{x^{m+1}}{m+1}\,_1F_1\left(\begin{array}{c} \frac{m+1}{n}\\1+\frac{m+1}{n}\end{array}\mid ix^n\right) \\[6px]& =\frac{1}{n} i^\frac{m+1}{n}\gamma\left(\frac{m+1}{n},-ix^n\right),\end{align}"

semanticFormula

"\begin{align}\int x^m \exp(\iunit x^n) \diff{x} &= \frac{x^{m+1}}{m+1}\genhyperF{1}{1}@{\frac{m+1}{n}}{1+\frac{m+1}{n}}{\iunit x^n}\\ &=\frac{1}{n} \iunit^{(m+1)/n} \incgamma@{\frac{m+1}{n}}{-\iunit x^n}\end{align}"

confidence

0.869061849326977

translations

Mathematica

translation	"Integrate[(x)^(m)* Exp[I(x)^(n)], x, GenerateConditions->None] == Divide[(x)^(m + 1),m + 1]HypergeometricPFQ[{Divide[m + 1,n]}, {1 +Divide[m + 1,n]}, I(x)^(n)] == Divide[1,n](I)^((m + 1)/n)* Gamma[Divide[m + 1,n], 0, - I*(x)^(n)]"

Maple

translation	"int((x)^(m)* exp(I(x)^(n)), x) = ((x)^(m + 1))/(m + 1)hypergeom([(m + 1)/(n)], [1 +(m + 1)/(n)], I(x)^(n)) = (1)/(n)(I)^((m + 1)/n)* GAMMA((m + 1)/(n))-GAMMA((m + 1)/(n), - I*(x)^(n))"

positions

section	5
sentence	0
word	14

includes

"dx"

"x"

isPartOf

Empty array

definiens

definition	"incomplete gamma function"
score	2

definition	"confluent hypergeometric function"
score	2

definition	"Fresnel integral"
score	1

definition	"imaginary part"
score	1

id

19

pid

69

eid

"math.69.117"

title

"Classical orthogonal polynomials"

formulae

id

"FORMULA_725c6b6b645d425d3b385ac2c002da77"

formula

"T_n(x) = \frac{\Gamma(1/2)\sqrt{1-x^2}}{(-2)^n\,\Gamma(n+1/2)} \ \frac{d^n}{dx^n}\left([1-x^2]^{n-1/2}\right)"

semanticFormula

"\ChebyshevpolyT{n}@{x} = \frac{\EulerGamma{1/2}\sqrt{1-x^2}}{(-2)^n\EulerGamma{n+1/2}} \deriv [n]{ }{x}([1 - x^2]^{n-1/2})"

confidence

0

translations

Mathematica

translation	"ChebyshevT[n, x] == Divide[Gamma[1/2]Sqrt[1 - (x)^(2)],(- 2)^(n) Gamma[n + 1/2]]*D[(1 - (x)^(2))^(n - 1/2), {x, n}]"

Maple

translation	"ChebyshevT(n, x) = (GAMMA(1/2)sqrt(1 - (x)^(2)))/((- 2)^(n) GAMMA(n + 1/2))*diff((1 - (x)^(2))^(n - 1/2), [x$(n)])"

positions

section	18
sentence	3
word	4

includes

"\ L_n"

"H_n"

"P_{n}"

"n-r"

"n"

"P_{n}(x)"

"-1/2"

"e_{n}"

"P_n"

"\lambda_{n}"

"-1"

"+1"

"U_n"

isPartOf

Empty array

definiens

definition	"Rodrigues ' formula"
score	2

definition	"orthogonal polynomial"
score	1

definition	"Chebyshev polynomials of the second kind"
score	1

definition	"classical orthogonal polynomial"
score	1

definition	"Chebyshev polynomial"
score	2

definition	"Gamma function"
score	2

id

20

pid

70

eid

"math.70.58"

title

"Generalized hypergeometric function"

formulae

id

"FORMULA_699b5f465d21dd6af7212cd8414f60c6"

formula

"{}_1F_0(1;;z) = \sum_{n \geqslant 0} z^n = (1-z)^{-1}"

semanticFormula

"\genhyperF{1}{0}@{1}{}{z} = \sum_{n \geqslant 0} z^n = (1-z)^{-1}"

confidence

0

translations

Mathematica

translation	"HypergeometricPFQ[{1}, {}, z] == Sum[(z)^(n), {n, 0, Infinity}, GenerateConditions->None] == (1 - z)^(- 1)"

Maple

translation	"hypergeom([1], [], z) = sum((z)^(n), n = 0..infinity) = (1 - z)^(- 1)"

positions

section	17
sentence	2
word	0

includes

"z"

"n"

"z)"

"_{p}F_{q}"

"^{n}"

isPartOf

Empty array

definiens

definition	"geometric series with ratio"
score	2

definition	"coefficient"
score	0

definition	"hypergeometric function"
score	2

id

21

pid

71

eid

"math.71.1-1"

title

"Dirichlet L-function"

formulae

id

"FORMULA_dcb9beab8f504cfc907c3165d24e5ad3"

formula

"\chi(-1) = 1"

semanticFormula

"\Dirichletchar@@{- 1}{k} = 1"

confidence

0.746792096089683

translations

Mathematica

translation	"DirichletCharacter[1, 1, -1] == 1"

positions

section	1
sentence	0
word	7

includes

"\chi"

isPartOf

"a=\begin{cases}0;&\mbox{if }\chi(-1)=1, \\ 1;&\mbox{if }\chi(-1)=-1,\end{cases}"

definiens

definition	"primitive character"
score	2

definition	"integer"
score	1

definition	"only zero"
score	0

definition	"Gamma function"
score	0

definition	"symbol"
score	0

definition	"functional equation"
score	0

definition	"Gauss sum"
score	0

definition	"Dirichlet character"
score	2

id

22

pid

72

eid

"math.72.15"

title

"Airy function"

formulae

id

"FORMULA_3b2520d05d324290456841271e8d565b"

formula

"\operatorname{Bi}'(z)\sim \frac{z^{\frac{1}{4}}e^{\frac{2}{3}z^{\frac{3}{2}}}}{\sqrt\pi\,}\left[ \sum_{n=0}^{\infty}\frac{1+6n}{1-6n} \dfrac{ \Gamma(n+\frac{5}{6})\Gamma(n+\frac{1}{6})\left(\frac{3}{4}\right)^n}{2\pi n! z^{3n/2}} \right]"

semanticFormula

"\AiryBi'@{z} \sim \frac{z^{\frac{1}{4}} \expe^{\frac{2}{3}z^{\frac{3}{2}}}}{\sqrt{\cpi}} [\sum_{n=0}^{\infty} \frac{1+6n}{1-6n} \frac{\EulerGamma@{n + \frac{5}{6}} \EulerGamma@{n + \frac{1}{6}}(\frac{3}{4})^n{2 \cpi n! z^{3n/2}}}]"

confidence

0.6525418663370697

translations

Empty object

positions

section	3
sentence	9
word	9

includes

"z"

"z)"

"= 0"

isPartOf

Empty array

definiens

definition	"z"
score	0

definition	"asymptotic formula for Ai"
score	1

definition	"Bi"
score	1

definition	"asymptotic behaviour of the Airy function"
score	1

definition	"Ai"
score	1

definition	"cosine"
score	2

definition	"definition of the Airy function"
score	1

definition	"Airy function"
score	2

definition	"Gamma function"
score	2

id

23

pid

73

eid

"math.73.41"

title

"Dawson function"

formulae

id

"FORMULA_f6b555bd8ce626d90119ab5eafdaeff2"

formula

"F'(y)=1-2yF(y)"

semanticFormula

"\DawsonsintF'@{y}=1-2y\DawsonsintF@{y}"

confidence

0

translations

Mathematica

translation	"D[DawsonF[y], {y, 1}] == 1 - 2yDawsonF[y]"

Maple

translation	"diff( dawson(y), y$(1) ) = 1 - 2ydawson(y)"

positions

section	2
sentence	9
word	1

includes

"y"

isPartOf

Empty array

definiens

definition	"polynomial"
score	0

definition	"Dawson function"
score	2

id

24

pid

74

eid

"math.74.0-1"

title

"Hurwitz zeta function"

formulae

id

"FORMULA_80a3608d4c2aae63f082861007c16c38"

formula

"s\not =1"

semanticFormula

"s \neq 1"

confidence

0

translations

Mathematica

translation	"s \[NotEqual] 1"

positions

section	0
sentence	2
word	24

includes

"s"

"1"

"\not = 1"

isPartOf

Empty array

definiens

definition	"value"
score	2

id

25

pid

75

eid

"math.75.6-1"

title

"Theta function"

formulae

id

"FORMULA_bfba6c35dbbcd8b89c6a29b1ffd6f517"

formula

"q = e^{i\pi\tau}"

semanticFormula

"q = \expe^{\iunit \cpi \tau}"

confidence

0

translations

Mathematica

translation	"q == Exp[IPi\[Tau]]"

Maple

translation	"q = exp(IPitau)"

positions

section	2
sentence	0
word	57

includes

"\tau"

"q"

"w = e^{\pi iz}"

"q = e^{\pi i\tau}"

isPartOf

"q = e^{\pi i\tau}"

"q = e^{2\pi i\tau}"

"\theta_F (z)= \sum_{m\in \Z^n} e^{2\pi izF(m)}"

"\hat{\theta}_F (z) = \sum_{k=0}^\infty R_F(k) e^{2\pi ikz}"

definiens

definition	"term of the nome"
score	2

definition	"nome"
score	2

id

26

pid

76

eid

"math.76.155"

title

"Jacobi elliptic functions"

formulae

id

"FORMULA_b54c03865b3efa9ea9112567cd66f59d"

formula

"\frac{\mathrm{d}}{\mathrm{d}z} \operatorname{dn}(z) = - k^2 \operatorname{sn}(z) \operatorname{cn}(z)"

semanticFormula

"\deriv [1]{ }{z} \Jacobielldnk@@{(z)}{k} = - k^2 \Jacobiellsnk@@{(z)}{k} \Jacobiellcnk@@{(z)}{k}"

confidence

0.6954186066124032

translations

Mathematica

translation	"D[JacobiDN[z, (k)^2], {z, 1}] == - (k)^(2)* JacobiSN[z, (k)^2]*JacobiCN[z, (k)^2]"

Maple

translation	"diff(JacobiDN(z, k), [z$(1)]) = - (k)^(2)* JacobiSN(z, k)*JacobiCN(z, k)"

positions

section	16
sentence	0
word	15

includes

"k"

"^{2}"

isPartOf

Empty array

definiens

definition	"derivative"
score	2

definition	"elliptic function"
score	2

definition	"basic Jacobi"
score	0

definition	"sn"
score	2

definition	"dn"
score	2

definition	"cn"
score	2

id

27

pid

77

eid

"math.77.118"

title

"Incomplete gamma function"

formulae

id

"FORMULA_c82b4ceebacd2b4a03b2eff406834e61"

formula

"\int_{-\infty}^\infty \frac {\gamma\left(\frac s 2, z^2 \pi \right)} {(z^2 \pi)^\frac s 2} e^{-2 \pi i k z} \mathrm d z = \frac {\Gamma\left(\frac {1-s} 2, k^2 \pi \right)} {(k^2 \pi)^\frac {1-s} 2}"

semanticFormula

"\int_{-\infty}^\infty \frac{\incgamma@{\frac s 2}{z^2 \cpi}}{(z^2 \cpi)^\frac s 2} \expe^{- 2 \cpi \iunit k z} \diff{z} = \frac{\incGamma@{\frac {1-s} 2}{k^2 \cpi}}{(k^2 \cpi)^\frac {1-s} 2}}"

confidence

0.8121295595054496

translations

Mathematica

translation	"Integrate[Divide[Gamma[Divide[s,2], 0, (z)^(2)* Pi],((z)^(2)* Pi)^(Divide[s,2])]Exp[- 2PiIkz], {z, - Infinity, Infinity}, GenerateConditions->None] == Divide[Gamma[Divide[1 - s,2], (k)^(2) Pi],((k)^(2)* Pi)^(Divide[1 - s,2])]"

Maple

translation	"int((GAMMA((s)/(2))-GAMMA((s)/(2), (z)^(2)* Pi))/(((z)^(2)* Pi)^((s)/(2)))exp(- 2PiIkz), z = - infinity..infinity) = (GAMMA((1 - s)/(2), (k)^(2) Pi))/(((k)^(2)* Pi)^((1 - s)/(2)))"

positions

section	25
sentence	1
word	15

includes

"\gamma(s, z)"

"\gamma"

"z^s"

"\Gamma"

"\gamma(s,z)"

"k"

"z"

"z="

"2\pi"

"\gamma(u,v)"

"\gamma(s,x)"

"s"

"z^{s}"

"e^{-x}"

"\gamma(a,x)"

isPartOf

Empty array

definiens

definition	"Fourier"
score	1

definition	"upper incomplete Gamma function"
score	2

definition	"lower incomplete Gamma function"
score	2

id

28

pid

78

eid

"math.78.0-1"

title

"Polylogarithm"

formulae

id

"FORMULA_e939f30d07578c2fb0d8cb5201db3c79"

formula

"_{1}(z) ="

semanticFormula

"\polylog{1}@{z} = -\ln@{1-z}"

confidence

0

translations

Mathematica

translation	"PolyLog[1, z] = -Log[1 - z]"

Maple

translation	"polylog(1, z) = -ln(1 - z)"

positions

section	0
sentence	7
word	11

includes

"_{1}"

"z"

"z) ="

"z)"

"1"

isPartOf

"\operatorname{Li}_{1}(z) = -\ln(1-z)"

"\operatorname{Ti}_0(z) = {z \over 1+z^2}, \quad \operatorname{Ti}_1(z) = \arctan z, \quad \operatorname{Ti}_2(z) = \int_0^z {\arctan t \over t} dt, \quad \ldots\quad \operatorname{Ti}_{n+1}(z) = \int_0^z \frac{\operatorname{Ti}_n(t)}{t} dt"

definiens

definition	"natural logarithm"
score	2

definition	"logarithm"
score	2

definition	"polylogarithm function"
score	2

definition	"dilogarithm"
score	1

definition	"trilogarithm"
score	1

id

29

pid

79

eid

"math.79.11"

title

"Sinc function"

formulae

id

"FORMULA_6340f4a043f912a3557e084aaf03792a"

formula

"\int_{-\infty}^\infty \operatorname{sinc}(t) \, e^{-i 2 \pi f t}\,dt = \operatorname{rect}(f)"

semanticFormula

"\int_{-\infty}^\infty \operatorname{sinc}(t) \expe^{- \iunit 2 \cpi f t} \diff{t} = \operatorname{rect}(f)"

confidence

0

translations

Mathematica

translation	"Integrate[sinc[(t)]Exp[- I2Pif*t], {t, - Infinity, Infinity}, GenerateConditions->None] == rect[f]"

Maple

translation	"int(sinc((t))exp(- I2Pif*t), t = - infinity..infinity) = rect(f)"

positions

section	1
sentence	9
word	16

includes

"\pi"

"\infty"

"sinc"

isPartOf

Empty array

definiens

definition	"argument"
score	0

definition	"continuous Fourier"
score	2

definition	"rectangular function"
score	2

definition	"sinc"
score	2

definition	"ordinary frequency"
score	1

id

30

pid

80

eid

"math.80.26"

title

"Exponential integral"

formulae

id

"FORMULA_a9a738ef9d4e46360dd9b87b39c691bf"

formula

"N=1"

semanticFormula

"N=1"

confidence

0

translations

Mathematica

translation	"N == 1"

Maple

translation	"N = 1"

positions

section	4
sentence	3
word	30

includes

"N"

isPartOf

Empty array

definiens

definition	"large value"
score	2

definition	"value"
score	2

id

31

pid

81

eid

"math.81.84"

title

"Laguerre polynomials"

formulae

id

"FORMULA_f179a85d8102cbedb67cf60b188a68b7"

formula

"\sum_{n=0}^\infty \frac{n!\,\Gamma\left(\alpha + 1\right)}{\Gamma\left(n+\alpha+1\right)}L_n^{(\alpha)}(x)L_n^{(\alpha)}(y)t^n=\frac{1}{(1-t)^{\alpha + 1}}e^{-(x+y)t/(1-t)}\,_0F_1\left(;\alpha + 1;\frac{xyt}{(1-t)^2}\right)"

semanticFormula

"\sum_{n=0}^\infty \frac{n! \EulerGamma@{\alpha + 1}}{\EulerGamma@{n + \alpha + 1}} \LaguerrepolyL[\alpha]{n}@{x} \LaguerrepolyL[\alpha]{n}@{x} t^n = \frac{1}{(1-t)^{\alpha + 1}} \expe^{-(x+y)t/(1-t)} \genhyperF{0}{1}@{}{\alpha + 1}{\frac{xyt}{(1-t)^2}}"

confidence

0.8953028732079359

translations

Mathematica

translation	"Sum[Divide[(n)!Gamma[\[Alpha]+ 1],Gamma[n + \[Alpha]+ 1]]LaguerreL[n, \[Alpha], x]LaguerreL[n, \[Alpha], x](t)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,(1 - t)^(\[Alpha]+ 1)]Exp[-(x + y)t/(1 - t)]HypergeometricPFQ[{}, {\[Alpha]+ 1}, Divide[xy*t,(1 - t)^(2)]]"

Maple

translation	"sum((factorial(n)GAMMA(alpha + 1))/(GAMMA(n + alpha + 1))LaguerreL(n, alpha, x)LaguerreL(n, alpha, x)(t)^(n), n = 0..infinity) = (1)/((1 - t)^(alpha + 1))exp(-(x + y)t/(1 - t))hypergeom([], [alpha + 1], (xy*t)/((1 - t)^(2)))"

positions

section	15
sentence	0
word	10

includes

"\alpha"

"L_{n}^{(\alpha)}"

"L_n^{(\alpha)}(x)"

"n"

isPartOf

Empty array

definiens

definition	"Hille formula"
score	2

definition	"Laguerre polynomial"
score	2

definition	"series on the left converge"
score	0

definition	"generalized Laguerre polynomial"
score	2

definition	"confluent hypergeometric function"
score	2

id

32

pid

82

eid

"math.82.8"

title

"Associated Legendre polynomials"

formulae

id

"FORMULA_6f29e15c07089506a70db1b3f54b27a5"

formula

"c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!}"

semanticFormula

"c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!}"

confidence

0

translations

Mathematica

translation	"Subscript[c, l, m] == (- 1)^(m)*Divide[(\[ScriptL]- m)!,(\[ScriptL]+ m)!]"

Maple

translation	"c[l, m] = (- 1)^(m)*(factorial(ell - m))/(factorial(ell + m))"

positions

section	1
sentence	7
word	26

includes

"m"

"(-1)^{m}"

"- 1"

isPartOf

Empty array

definiens

definition	"proportionality constant"
score	2

id

33

pid

83

eid

"math.83.3"

title

"Scorer's function"

formulae

id

"FORMULA_c8116180276232704ca3e9f67f207565"

formula

"\mathrm{Gi}(x) = \frac{1}{\pi} \int_0^\infty \sin\left(\frac{t^3}{3} + xt\right)\, dt"

semanticFormula

"\ScorerGi@{x} = \frac{1}{\cpi} \int_0^\infty \sin(\frac{t^3}{3} + xt) \diff{t}"

confidence

0.7929614010341081

translations

Mathematica

translation

"ScorerGi[x] == Divide[1,Pi]*Integrate[Sin[Divide[(t)^(3),3]+ x*t], {t, 0, Infinity}, GenerateConditions->None]"

translationInformation

subEquations

"ScorerGi[x] = Divide[1,Pi]*Integrate[Sin[Divide[(t)^(3),3]+ x*t], {t, 0, Infinity}, GenerateConditions->None]"

freeVariables

"x"

constraints

Empty array

tokenTranslations

\ScorerGi	"Scorer function Gi; Example: \ScorerGi@{z} Will be translated to: ScorerGi[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/9.12#i Mathematica: https://"
\cpi	"Pi was translated to: Pi"
\sin	"Sine; Example: \sin@@{z} Will be translated to: Sin[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E1 Mathematica: https://reference.wolfram.com/language/ref/Sin.html"

Maple

translation

"AiryBi(x)*(int(AiryAi(t), t = (x) .. infinity))+AiryAi(x)*(int(AiryBi(t), t = 0 .. (x))) = (1)/(Pi)*int(sin(((t)^(3))/(3)+ x*t), t = 0..infinity)"

translationInformation

subEquations

"AiryBi(x)*(int(AiryAi(t), t = (x) .. infinity))+AiryAi(x)*(int(AiryBi(t), t = 0 .. (x))) = (1)/(Pi)*int(sin(((t)^(3))/(3)+ x*t), t = 0..infinity)"

freeVariables

"x"

constraints

Empty array

tokenTranslations

\ScorerGi	"Scorer function Gi; Example: \ScorerGi@{z} Will be translated to: Alternative translations: [AiryBi($0)(int(AiryAi(t), t = ($0) .. infinity))+AiryAi($0)(int(AiryBi(t), t = 0 .. ($0)))]Relevant links to definitions: DLMF: http://dlmf.nist.gov/9.12#i Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=Airy"
\cpi	"Pi was translated to: Pi"
\sin	"Sine; Example: \sin@@{z} Will be translated to: sin($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=sin"

positions

section	0
sentence	1
word	15

includes

"x"

"x)"

isPartOf

Empty array

definiens

definition	"Scorer 's function"
score	2

definition	"special function"
score	1

id

34

pid

84

eid

"math.84.31"

title

"Voigt profile"

formulae

id

"FORMULA_e663d20df3cca1ae5dec645d320cd511"

formula

"\frac{\partial^2}{\partial x^2} V(x;\sigma,\gamma)= \frac{x^2-\gamma^2-\sigma^2}{\sigma^4} \frac{\operatorname{Re}[w(z)]}{\sigma\sqrt{2 \pi}}-\frac{2 x \gamma}{\sigma^4} \frac{\operatorname{Im}[w(z)]}{\sigma\sqrt{2 \pi}}+\frac{\gamma}{\sigma^4}\frac{1}{\pi}"

semanticFormula

"\deriv[2]{}{x} V(x ; \sigma , \gamma) = \frac{x^2-\gamma^2-\sigma^2}{\sigma^4} \frac{\realpart [\Faddeevaw@{z}]}{\sigma \sqrt{2 \cpi}} - \frac{2 x \gamma}{\sigma^4} \frac{\imagpart [\Faddeevaw@{z}]}{\sigma \sqrt{2 \cpi}} + \frac{\gamma}{\sigma^4} \frac{1}{\cpi}"

confidence

0.8620216359266987

translations

Mathematica

translation

"D[PDF[VoigtDistribution[\[Gamma], \[Sigma]], x], {x, 2}] == Divide[x^2 - \[Gamma]^2 - \[Sigma]^2, \[Sigma]^4] * Divide[ Re[ Exp[-(Divide[x+I*y,\[Sigma]*Sqrt[2]])^2]*Erfc[-I*(Divide[x+I*y,\[Sigma]*Sqrt[2]])] ], \[Sigma]*Sqrt[2*Pi]] - Divide[2*x*y, \[Sigma]^4] * Divide[Im[Exp[-(Divide[x+I*y,\[Sigma]*Sqrt[2]])^2]*Erfc[-I*(Divide[x+I*y,\[Sigma]*Sqrt[2]])]], \[Sigma]*Sqrt[2*Pi]] + Divide[\[Gamma],\[Sigma]^4]*Divide[1,Pi]"

positions

section	6
sentence	0
word	20

includes

"w(z)]"

"z"

"V(x;\sigma,\gamma)"

"x"

"w(z)"

isPartOf

Empty array

definiens

definition	"term of the Faddeeva function"
score	2

definition	"second derivative profile"
score	2

definition	"real part of the Faddeeva function"
score	2

definition	"Faddeeva function"
score	2

definition	"Voigt function"
score	2

definition	"Voigt profile"
score	2

id

35

pid

85

eid

"math.85.57"

title

"Lerch zeta function"

formulae

id

"FORMULA_a0cc62efe3cabac6d8bebe5b8b94b5fa"

formula

"\Phi(z,s,a) = \frac{1}{1-z} \frac{1}{a^{s}} + \sum_{n=1}^{N-1} \frac{(-1)^{n} \mathrm{Li}_{-n}(z)}{n!} \frac{(s)_{n}}{a^{n+s}} +O(a^{-N-s})"

semanticFormula

"\Phi(z , s , a) = \frac{1}{1-z} \frac{1}{a^{s}} + \sum_{n=1}^{N-1} \frac{(-1)^{n} \polylog{-n}@{z}}{n!} \frac{\Pochhammersym{s}{n}}{a^{n+s}} + \bigO{a^{-N-s}}"

confidence

0.8662724998444776

translations

Mathematica

translation	"\[CapitalPhi][z, s, a] == Divide[1,1 - z]Divide[1,(a)^(s)]+ Sum[Divide[(- 1)^(n) PolyLog[-n, z],(n)!]*Divide[Pochhammer[s, n],(a)^(n + s)], {n, 1, N - 1}, GenerateConditions->None]+ O[a]^(- N - s)"

positions

section	6
sentence	1
word	23

includes

"a"

"\Phi(z,s,a)"

"z"

"s"

isPartOf

Empty array

definiens

definition	"asymptotic expansion"
score	2

definition	"Pochhammer symbol"
score	1

definition	"Lerch transcendent"
score	2

definition	"polylogarithm"
score	2

definition	"polylogarithm function"
score	2

definition	"Pochhammer symbol"
score	2

id

36

pid

86

eid

"math.86.44"

title

"Confluent hypergeometric function"

formulae

id

"FORMULA_d83a3ce5244b566d8f71edb7f81afa43"

formula

"M(1,2,z)=(e^z-1)/z,\ \ M(1,3,z)=2!(e^z-1-z)/z^2"

semanticFormula

"\KummerconfhyperM@{1}{2}{z} = (\expe^z - 1) / z , \KummerconfhyperM@{1}{3}{z} = 2! (\expe^z - 1 - z) / z^2"

confidence

0.912945064646862

translations

Mathematica

translation	"Hypergeometric1F1[1, 2, z] == (Exp[z]- 1)/z Hypergeometric1F1[1, 3, z] == (2)!*(Exp[z]- 1 - z)/(z)^(2)"

Maple

translation	"KummerM(1, 2, z) = (exp(z)- 1)/z; KummerM(1, 3, z) = factorial(2)*(exp(z)- 1 - z)/(z)^(2)"

positions

section	10
sentence	4
word	0

includes

"M"

"U(a, b, z)"

"z"

"U(n,c,z)"

"\Phi(a, b, z)"

"M(n,b,z)"

"M(a, b, z)"

isPartOf

Empty array

definiens

definition	"etc"
score	0

definition	"Kummer 's function of the first kind"
score	2

definition	"confluent hypergeometric function"
score	1

definition	"hypergeometric function"
score	1

id

37

pid

87

eid

"math.87.54"

title

"Mathieu function"

formulae

id

"FORMULA_f694135eafc20195a9d96ca3ce8af674"

formula

"\sigma = \pm 1"

semanticFormula

"\sigma = \pm 1"

confidence

0

translations

Mathematica

translation

"\[Sigma] == \[PlusMinus]1"

translationInformation

subEquations

"\[Sigma] = + 1"

"\[Sigma] = - 1"

freeVariables

"\[Sigma]"

constraints

Empty array

tokenTranslations

\pm	"was translated to: \[PlusMinus]"

Maple

translation

"sigma = &+- 1"

translationInformation

subEquations

"sigma = + 1"

"sigma = - 1"

freeVariables

"sigma"

constraints

Empty array

tokenTranslations

\pm	"was translated to: &+-"

positions

section	4
sentence	1
word	27

includes

Empty array

isPartOf

Empty array

definiens

definition	"value"
score	2

id

38

pid

88

eid

"math.88.0"

title

"Parabolic cylinder function"

formulae

id

"FORMULA_bec6388631b20f2af14e375b13e1533f"

formula

"\frac{d^2f}{dz^2} + \left(\tilde{a}z^2+\tilde{b}z+\tilde{c}\right)f=0"

semanticFormula

"\deriv [2]{f}{z} +(\tilde{a} z^2 + \tilde{b} z + \tilde{c}) f = 0"

confidence

0

translations

Mathematica

translation	"D[f[z], {z, 2}] + (az^2 + bz + c)*f[z] == 0"

positions

section	0
sentence	0
word	19

includes

"z"

isPartOf

Empty array

definiens

definition	"solution to the differential equation"
score	2

definition	"special function"
score	1

definition	"mathematics"
score	0

definition	"parabolic cylinder function"
score	1

id

39

pid

89

eid

"math.89.23"

title

"Painlevé transcendents"

formulae

id

"FORMULA_0a306ab913684a1ba3935715d3dd8ad8"

formula

"c=\infty"

semanticFormula

"c=\infty"

confidence

0

translations

Mathematica

translation	"c == Infinity"

Maple

translation	"c = infinity"

positions

section	9
sentence	5
word	23

includes

"c"

isPartOf

Empty array

definiens

definition	"central charge of the Virasoro algebra"
score	2

definition	"combination of conformal block"
score	1

definition	"Painlevé VI equation"
score	1

definition	"two-dimensional conformal field theory"
score	1

id

40

pid

90

eid

"math.90.7"

title

"Hypergeometric function"

formulae

id

"FORMULA_aaffb0ad8dea17d68491d9fb6ebcfbe3"

formula

"c = a + 1"

semanticFormula

"c = a + 1"

confidence

0

translations

Mathematica

translation	"c == a + 1"

Maple

translation	"c := a + 1"

positions

section	3
sentence	0
word	20

includes

"a"

"c"

isPartOf

Empty array

definiens

definition	"value"
score	2

id

41

pid

91

eid

"math.91.47"

title

"Barnes G-function"

formulae

id

"FORMULA_6bc0d742c4d25c1abb61158150489676"

formula

"\frac{1}{\Gamma(z)}= z e^{\gamma z} \prod_{k=1}^\infty \left\{ \left(1+\frac{z}{k}\right)e^{-z/k} \right\}"

semanticFormula

"\frac{1}{\EulerGamma@{z}} = z \expe^{\EulerConstant z} \prod_{k=1}^\infty \{(1 + \frac{z}{k}) \expe^{-z/k} \}"

confidence

0.8614665289982916

translations

Mathematica

translation

"Divide[1,Gamma[z]] == z*Exp[EulerGamma*z]*Product[(1 +Divide[z,k])*Exp[- z/k], {k, 1, Infinity}, GenerateConditions->None]"

translationInformation

subEquations

"Divide[1,Gamma[z]] = z*Exp[EulerGamma*z]*Product[(1 +Divide[z,k])*Exp[- z/k], {k, 1, Infinity}, GenerateConditions->None]"

freeVariables

"z"

constraints

Empty array

tokenTranslations

\expe	"Recognizes e with power as the exponential function. It was translated as a function."
\EulerConstant	"Euler-Mascheroni constant was translated to: EulerGamma"
\EulerGamma	"Euler Gamma function; Example: \EulerGamma@{z} Will be translated to: Gamma[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/5.2#E1 Mathematica: https://reference.wolfram.com/language/ref/Gamma.html"

Maple

translation

"(1)/(GAMMA(z)) = z*exp(gamma*z)*product((1 +(z)/(k))*exp(- z/k), k = 1..infinity)"

translationInformation

subEquations

"(1)/(GAMMA(z)) = z*exp(gamma*z)*product((1 +(z)/(k))*exp(- z/k), k = 1..infinity)"

freeVariables

"z"

constraints

Empty array

tokenTranslations

\expe	"Recognizes e with power as the exponential function. It was translated as a function."
\EulerConstant	"Euler-Mascheroni constant was translated to: gamma"
\EulerGamma	"Euler Gamma function; Example: \EulerGamma@{z} Will be translated to: GAMMA($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/5.2#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=GAMMA"

positions

section	8
sentence	0
word	55

includes

"\,\Gamma(x)"

"\, \gamma"

"z"

"\,\gamma"

isPartOf

Empty array

definiens

definition	"Euler"
score	1

definition	"Mascheroni"
score	1

definition	"gamma function"
score	2

id

42

pid

92

eid

"math.92.1-1"

title

"Heun function"

formulae

id

"FORMULA_8c78ef87048e61947a6d7d4b5e06aa63"

formula

"192/24 = 8 = 2 \times 4"

semanticFormula

"192/24 = 8 = 2 \times 4"

confidence

0

translations

Mathematica

translation

"192/24 == 8 == 2 * 4"

translationInformation

subEquations

"192/24 = 8"

"8 = 2 * 4"

freeVariables

Empty array

constraints

Empty array

tokenTranslations

\times	"was translated to: *"

numericResults

overallResult

"SUCCESS"

numberOfTests

2

numberOfFailedTests

0

numberOfSuccessfulTests

2

numberOfSkippedTests

0

numberOfErrorTests

0

wasAborted

false

crashed

false

testCalculationsGroups

lhs

"192/24"

rhs

"8"

testExpression

"(192/24)-(8)"

activeConstraints

Empty array

testCalculations

result

"SUCCESS"

resultExpression

"0."

testValues

Empty object

lhs

"8"

rhs

"2 * 4"

testExpression

"(8)-(2 * 4)"

activeConstraints

Empty array

testCalculations

result

"SUCCESS"

resultExpression

"0."

testValues

Empty object

symbolicResults

overallResult

"SUCCESS"

numberOfTests

2

numberOfFailedTests

0

numberOfSuccessfulTests

2

numberOfSkippedTests

0

numberOfErrorTests

0

crashed

false

testCalculationsGroup

lhs

"192/24"

rhs

"8"

testExpression

"(192/24)-(8)"

testCalculations

result	"SUCCESS"
testTitle	"Simple"
testExpression	"FullSimplify[(192/24)-(8)]"
resultExpression	"0"
wasAborted	false
conditionallySuccessful	false

lhs

"8"

rhs

"2 * 4"

testExpression

"(8)-(2 * 4)"

testCalculations

result	"SUCCESS"
testTitle	"Simple"
testExpression	"FullSimplify[(8)-(2 * 4)]"
resultExpression	"0"
wasAborted	false
conditionallySuccessful	false

SymPy

translation

"192/24 == 8 == 2 * 4"

translationInformation

subEquations

"192/24 = 8"

"8 = 2 * 4"

freeVariables

Empty array

constraints

Empty array

tokenTranslations

\times	"was translated to: *"

Maple

translation

"192/24 = 8 = 2 * 4"

translationInformation

subEquations

"192/24 = 8"

"8 = 2 * 4"

freeVariables

Empty array

constraints

Empty array

tokenTranslations

\times	"was translated to: *"

numericResults

overallResult

"SUCCESS"

numberOfTests

2

numberOfFailedTests

0

numberOfSuccessfulTests

2

numberOfSkippedTests

0

numberOfErrorTests

0

wasAborted

false

crashed

false

testCalculationsGroups

lhs

"192/24"

rhs

"8"

testExpression

"evalf((192/24)-(8))"

activeConstraints

Empty array

testCalculations

result

"SUCCESS"

resultExpression

"0."

testValues

Empty object

lhs

"8"

rhs

"2 * 4"

testExpression

"evalf((8)-(2 * 4))"

activeConstraints

Empty array

testCalculations

result

"SUCCESS"

resultExpression

"0."

testValues

Empty object

symbolicResults

overallResult

"SUCCESS"

numberOfTests

2

numberOfFailedTests

0

numberOfSuccessfulTests

2

numberOfSkippedTests

0

numberOfErrorTests

0

crashed

false

testCalculationsGroup

lhs

"192/24"

rhs

"8"

testExpression

"(192/24)-(8)"

testCalculations

result	"SUCCESS"
testTitle	"Simple"
testExpression	"simplify((192/24)-(8))"
resultExpression	"0"
wasAborted	false
conditionallySuccessful	false

lhs

"8"

rhs

"2 * 4"

testExpression

"(8)-(2 * 4)"

testCalculations

result	"SUCCESS"
testTitle	"Simple"
testExpression	"simplify((8)-(2 * 4))"
resultExpression	"0"
wasAborted	false
conditionallySuccessful	false

positions

section	3
sentence	1
word	25

includes

Empty array

isPartOf

Empty array

definiens

Empty array

id

43

pid

93

eid

"math.93.0-1"

title

"Gegenbauer polynomials"

formulae

id

"FORMULA_34d9d355f0c0e28d91465c3b575fb0a1"

formula

"=2"

semanticFormula

"\alpha = 2"

confidence

0

translations

Mathematica

translation	"\[Alpha] = 2"

Maple

translation	"alpha = 2"

positions

section	1
sentence	0
word	17

includes

Empty array

isPartOf

"\begin{align}C_0^\alpha(x) & = 1 \\C_1^\alpha(x) & = 2 \alpha x \\C_n^\alpha(x) & = \frac{1}{n}[2x(n+\alpha-1)C_{n-1}^\alpha(x) - (n+2\alpha-2)C_{n-2}^\alpha(x)].\end{align}"

definiens

definition	"value"
score	2

id

44

pid

94

eid

"math.94.4"

title

"Basic hypergeometric series"

formulae

id

"FORMULA_33e3b57bb75d5ea3b5b8ddcceef38430"

formula

"\lim_{q\to 1}\;_{j}\phi_k \left[\begin{matrix} q^{a_1} & q^{a_2} & \ldots & q^{a_j} \\ q^{b_1} & q^{b_2} & \ldots & q^{b_k} \end{matrix} ; q,(q-1)^{1+k-j} z \right]=\;_{j}F_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_j \\ b_1 & b_2 & \ldots & b_k \end{matrix} ;z \right]"

semanticFormula

"\lim_{q\to 1} \qgenhyperphi{j}{k}@{q^{a_1} , q^{a_2} , \ldots , q^{a_j}}{q^{b_1} , q^{b_2} , \ldots , q^{b_k}}{q}{(q - 1)^{1+k-j} z} = \genhyperF{j}{k}@{a_1 , a_2 , \ldots , a_j}{b_1 , b_2 , \ldots , b_k}{z}"

confidence

0

translations

Empty object

positions

section	1
sentence	5
word	13

includes

"q^{n}"

"q"

"b"

"a"

"z"

isPartOf

Empty array

definiens

definition	"q-analog of the hypergeometric series"
score	2

definition	"unilateral basic hypergeometric series"
score	2

definition	"basic hypergeometric series"
score	2

id

45

pid

95

eid

"math.95.0"

title

"Whittaker function"

formulae

id

"FORMULA_16ec3a3583ee2b4621d316bf839c1725"

formula

"\frac{d^2w}{dz^2}+\left(-\frac{1}{4}+\frac{\kappa}{z}+\frac{1/4-\mu^2}{z^2}\right)w=0"

semanticFormula

"\deriv [2]{w}{z} +(- \frac{1}{4} + \frac{\kappa}{z} + \frac{1/4-\mu^2}{z^2}) w = 0"

confidence

0

translations

Mathematica

translation	"D[w, {z, 2}]+(-Divide[1,4]+Divide[\[Kappa],z]+Divide[1/4 - \[Mu]^(2),(z)^(2)])*w == 0"

Maple

translation	"diff(w, [z$(2)])+(-(1)/(4)+(kappa)/(z)+(1/4 - (mu)^(2))/((z)^(2)))*w = 0"

positions

section	0
sentence	2
word	4

includes

"\mu"

"\kappa"

"z"

isPartOf

Empty array

definiens

definition	"Whittaker 's equation"
score	2

definition	"Whittaker function"
score	1

id

46

pid

96

eid

"math.96.1"

title

"Lemniscatic elliptic function"

formulae

id

"FORMULA_24137d79f0a282f42fdf9ea93576e998"

formula

"e_1=\tfrac12,\qquad e_2=0,\qquad e_3=-\tfrac12"

semanticFormula

"e_1=\tfrac12,\qquad e_2=0,\qquad e_3=-\tfrac12"

confidence

0

translations

Mathematica

translation	"Subscript[e, 1] == Divide[1,2] Subscript[e, 2] = 0 Subscript[e, 3] = -Divide[1,2]"

Maple

translation	"e[1] := (1)/(2); e[2] := 0; e[3] := -(1)/(2)"

positions

section	0
sentence	5
word	11

includes

"e_{1}"

"e_{2}"

"e_{3}"

isPartOf

Empty array

definiens

definition	"constant"
score	2

id

47

pid

98

eid

"math.98.53-1"

title

"Meijer G-function"

formulae

id

"FORMULA_028eb01ef675c90ea0f74fcdd93fc78c"

formula

"\gamma> 0,n-p=m-q> 0"

semanticFormula

"\gamma> 0,n-p=m-q> 0"

confidence

0

translations

Mathematica

translation	"\[Gamma] > 0 n - p == m - q > 0"

Maple

translation	"gamma > 0; n - p = m - q > 0"

positions

section	12
sentence	0
word	17

includes

"m"

"q"

"p=q> 0"

"n"

"p=q"

"\gamma>"

isPartOf

Empty array

definiens

definition	"constraint"
score	2

id

48

pid

99

eid

"math.99.30"

title

"3-j symbol"

formulae

id

"FORMULA_3f987b881a59a03904ff9a79476faae0"

formula

"\begin{pmatrix} j \\ m \quad m'\end{pmatrix}:= \sqrt{2 j + 1}\begin{pmatrix} j & 0 & j \\ m & 0 & m'\end{pmatrix}= (-1)^{j - m'} \delta_{m, -m'}"

semanticFormula

"\begin{pmatrix} j \\ m \quad m'\end{pmatrix}:= \sqrt{2 j + 1}\begin{pmatrix} j & 0 & j \\ m & 0 & m'\end{pmatrix}= (-1)^{j - m'} \delta_{m, -m'}"

confidence

0

translations

Mathematica

translation	"Wigner[j_, m_, m\[Prime]_] := Sqrt[2j+1] {{j, 0, j}, {m, 0, m\[Prime]}} = (-1)^(j-m\[Prime])*Subscript[\[Delta], m, -m\[Prime]]"

positions

section	10
sentence	0
word	23

includes

"j"

"m"

isPartOf

Empty array

definiens

definition	"Wigner 1-jm symbol"
score	2

definition	"metric tensor in angular-momentum theory"
score	2

definition	"quantity"
score	0

id

49

pid

100

eid

"math.100.14"

title

"6-j symbol"

formulae

id

"FORMULA_21d6ec52b25bb130bf068c4857bbcb93"

formula

"\begin{Bmatrix} i & j & \ell\\ k & m & n \end{Bmatrix}= (\Phi_{i,j}^{k,m})_{\ell,n}"

semanticFormula

"\Wignersixjsym{i}{j}{\ell}{k}{m}{n} = (\Phi_{i,j}^{k,m})_{\ell,n}"

confidence

0.8624533614429312

translations

Empty object

positions

section	5
sentence	4
word	23

includes

"j"

isPartOf

Empty array

definiens

definition	"6j symbol"
score	2

definition	"associativity isomorphism"
score	2

definition	"symbol"
score	1

definition	"vector space isomorphism"
score	2

definition	"Wigner"
score	1

definition	"Wigner 's 6 - j symbol"
score	2

id

50

pid

101

eid

"math.101.32"

title

"9-j symbol"

formulae

id

"FORMULA_08d08037d9e64d85aa3645470ce645af"

formula

"\sum_{j_7 j_8} (2j_7+1)(2j_8+1) \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} \begin{Bmatrix} j_1 & j_2 & j_3'\\ j_4 & j_5 & j_6'\\ j_7 & j_8 & j_9 \end{Bmatrix} = \frac{\delta_{j_3j_3'}\delta_{j_6j_6'} \begin{Bmatrix} j_1 & j_2 & j_3 \end{Bmatrix} \begin{Bmatrix} j_4 & j_5 & j_6\end{Bmatrix} \begin{Bmatrix} j_3 & j_6 & j_9 \end{Bmatrix}} {(2j_3+1)(2j_6+1)}"

semanticFormula

"\sum_{j_7 j_8} (2j_7+1)(2j_8+1) \Wignerninejsym{j_1}{j_2}{j_3}{j_4}{j_5}{j_6}{j_7}{j_8}{j_9} \Wignerninejsym{j_1}{j_2}{j_3'}{j_4}{j_5}{j_6'}{j_7}{j_8}{j_9} = \frac{\delta_{j_3j_3'}\delta_{j_6j_6'} \begin{Bmatrix} j_1 & j_2 & j_3 \end{Bmatrix} \begin{Bmatrix} j_4 & j_5 & j_6\end{Bmatrix} \begin{Bmatrix} j_3 & j_6 & j_9 \end{Bmatrix}}{(2j_3+1)(2j_6+1)}"

confidence

0

translations

Empty object

positions

section	5
sentence	0
word	10

includes

"j"

"_{4}"

isPartOf

Empty array

definiens

definition	"orthogonality relation"
score	1

definition	"triangular delta"
score	2

definition	"symbol"
score	1

definition	"Wigner 's 9 - j symbol"
score	2

id

51

pid

102

eid

"math.102.5"

title

"Kravchuk polynomials"

formulae

id

"FORMULA_6b7eb62a3e02e45fb1365dd2f07a5bbc"

formula

"\mathcal{K}_k(x; n,q) = \sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \binom {n-j}{k-j} \binom{x}{j}"

semanticFormula

"\KrawtchoukpolyK{k}@{x}{n}{q} = \sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \binom {n-j}{k-j} \binom{x}{j}"

confidence

0

translations

Mathematica

translation	"K[k_, x_, n_, q_] := Sum[(- q)^(j)(q - 1)^(k - j)Binomial[n - j,k - j]*Binomial[x,j], {j, 0, k}, GenerateConditions->None]"

positions

section	2
sentence	0
word	9

includes

"q"

"n"

isPartOf

Empty array

definiens

definition	"following alternative expression"
score	0

definition	"Kravchuk polynomial"
score	2

id

52

pid

103

eid

"math.103.8"

title

"Kelvin functions"

formulae

id

"FORMULA_07453e6baf8f216467f9b664de795bfc"

formula

"g_1(x) = \sum_{k \geq 1} \frac{\sin(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2"

semanticFormula

"g_1(x) = \sum_{k \geq 1} \frac{\sin(k \cpi / 4)}{k! (8x)^k} \prod_{l = 1}^k(2 l - 1)^2"

confidence

0

translations

Mathematica

translation	"Subscript[g, 1][x_] := Sum[Divide[Sin[kPi/4],(k)!(8x)^(k)]Product[(2*l - 1)^(2), {l, 1, k}, GenerateConditions->None], {k, 1, Infinity}, GenerateConditions->None]"

Maple

translation	"g[1] := (x) -> sum((sin(kPi/4))/(factorial(k)(8x)^(k))product((2*l - 1)^(2), l = 1..k), k = 1..infinity)"

positions

section	1
sentence	1
word	28

includes

"x"

"x)"

"g_1(x)"

isPartOf

Empty array

definiens

definition	"series expansion"
score	1

definition	"special case"
score	0

definition	"asymptotic series"
score	1

definition	"definition"
score	2

id

53

pid

104

eid

"math.104.2"

title

"Lommel function"

formulae

id

"FORMULA_03f5cb50caaedb9f0a4ada231fd61c58"

formula

"S_{\mu,\nu}(z) = s_{\mu,\nu}(z) + 2^{\mu-1} \Gamma\left(\frac{\mu + \nu + 1}{2}\right) \Gamma\left(\frac{\mu - \nu + 1}{2}\right)\left(\sin \left[(\mu - \nu)\frac{\pi}{2}\right] J_\nu(z) - \cos \left[(\mu - \nu)\frac{\pi}{2}\right] Y_\nu(z)\right)"

semanticFormula

"\LommelS{\mu}{\nu}@{z} = \Lommels{\mu}{\nu}@{z} + 2^{\mu-1} \EulerGamma@{\frac{\mu + \nu + 1}{2}} \EulerGamma@{\frac{\mu - \nu + 1}{2}}(\sin [(\mu - \nu) \frac{\cpi}{2}] \BesselJ{\nu}@{z} - \cos [(\mu - \nu) \frac{\cpi}{2}] \BesselY{\nu}@{z})"

confidence

0.8775479393290169

translations

Mathematica

translation

"S[\[Mu]_, \[Nu]_, z_] := Divide[Pi,2]*(BesselY[\[Nu], z]*Integrate[(x)^\[Mu]* BesselJ[\[Nu], x], {x, 0, z}, GenerateConditions->None]- BesselJ[\[Nu], z]*Integrate[(x)^\[Mu]* BesselY[\[Nu], x], {x, 0, z}, GenerateConditions->None]) + (2)^(\[Mu]- 1)* Gamma[Divide[\[Mu]+ \[Nu]+ 1,2]]*Gamma[Divide[\[Mu]- \[Nu]+ 1,2]]*(Sin[((\[Mu]- \[Nu])*Divide[Pi,2])*]*BesselJ[\[Nu], z]- Cos[((\[Mu]- \[Nu])*Divide[Pi,2])*]*BesselY[\[Nu], z])"

Maple

translation	"LommelS1(mu, nu, z) = (Pi)/(2)(BesselY(nu, z)int((x)^(mu)* BesselJ(nu, x), x = 0..z)- BesselJ(nu, z)int((x)^(mu) BesselY(nu, x), x = 0..z))"

positions

section	0
sentence	1
word	18

includes

"s_{\mu,\nu}(z)"

"S_{\mu,\nu}(z)"

"J_{\nu}(z)"

"Y_{\nu}(z)"

isPartOf

Empty array

definiens

definition	"Lommel function"
score	2

definition	"Bessel function of the first kind"
score	2

definition	"Bessel function of the second kind"
score	2

id

54

pid

105

eid

"math.105.18"

title

"Struve function"

formulae

id

"FORMULA_6dc2da7f595d2f199fbc15768167f006"

formula

"\mathbf{H}_{\alpha}(z) = \frac{z^{\alpha+1}}{2^{\alpha}\sqrt{\pi} \Gamma \left (\alpha+\tfrac{3}{2} \right )} {}_1F_2 \left (1,\tfrac{3}{2}, \alpha+\tfrac{3}{2},-\tfrac{z^2}{4} \right )"

semanticFormula

"\StruveH{\alpha}@{z} = \frac{z^{\alpha+1}}{2^{\alpha} \sqrt{\cpi} \EulerGamma@{\alpha + \tfrac{3}{2}}} \genhyperF{1}{2}@{1}{\tfrac{3}{2}, \alpha + \tfrac{3}{2}}{- \tfrac{z^2}{4}}"

confidence

0.8740850655136605

translations

Mathematica

translation	"StruveH[\[Alpha], z] == Divide[(z)^(\[Alpha]+ 1),(2)^\[Alpha]Sqrt[Pi]Gamma[\[Alpha]+Divide[3,2]]]*HypergeometricPFQ[{1}, {Divide[3,2], \[Alpha]+Divide[3,2]}, -Divide[(z)^(2),4]]"

Maple

translation	"StruveH(alpha, z) = ((z)^(alpha + 1))/((2)^(alpha)sqrt(Pi)GAMMA(alpha +(3)/(2)))*hypergeom([1], [(3)/(2), alpha +(3)/(2)], -((z)^(2))/(4))"

positions

section	6
sentence	2
word	31

includes

"_{1}F_{2}"

"\mathbf{K}_\alpha(x)"

"\alpha"

"\Gamma(z)"

"\mathbf{H}_{\alpha}(x)"

"\mathbf{L}_{\alpha}(x)"

"\mathbf{H}_{\alpha}(z)"

"Y_{\alpha}(x)"

"\mathbf{M}_\alpha(x)"

isPartOf

Empty array

definiens

definition	"hypergeometric function"
score	2

definition	"Struve"
score	2

definition	"Struve function"
score	2

definition	"gamma function"
score	2

id

55

pid

106

eid

"math.106.7"

title

"Hill differential equation"

formulae

id

"FORMULA_3a6745862e8f6ef2b93c343ad82b40c0"

formula

"f(t+p) = f(t)"

semanticFormula

"f(t+p) = f(t)"

confidence

0

translations

Mathematica

translation	"f[t + p] == f[t]"

Maple

translation	"f(t + p) = f(t)"

positions

section	0
sentence	1
word	21

includes

"f(t)"

"t"

"p"

"f(t+\pi)=f(t)"

isPartOf

"f(t+\pi)=f(t)"

definiens

definition	"function"
score	2

definition	"periodic function by minimal period"
score	2

id

56

pid

108

eid

"math.108.3"

title

"Anger function"

formulae

id

"FORMULA_014efde25f995ccd08168a36ec7ef86d"

formula

"\mathbf{J}_\nu(z)=\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}+\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}"

semanticFormula

"\AngerJ{\nu}@{z} = \cos \frac{\cpi\nu}{2} \sum_{k=0}^\infty \frac{(-1)^k z^{2k}}{4^k\EulerGamma@{k+\frac{\nu}{2}+1}\EulerGamma@{k-\frac{\nu}{2}+1}}+\sin\frac{\cpi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^k z^{2k+1}}{2^{2k+1}\EulerGamma@{k+\frac{\nu}{2}+\frac{3}{2}}\EulerGamma@{k-\frac{\nu}{2}+\frac{3}{2}}}"

confidence

0.8648813564530858

translations

Mathematica

translation	"AngerJ[\[Nu], z] == Cos[Divide[Pi\[Nu],2]]Sum[Divide[(- 1)^(k)* (z)^(2k),(4)^(k) Gamma[k +Divide[\[Nu],2]+ 1]Gamma[k -Divide[\[Nu],2]+ 1]], {k, 0, Infinity}, GenerateConditions->None]+ Sin[Divide[Pi\[Nu],2]]Sum[Divide[(- 1)^(k) (z)^(2k + 1),(2)^(2k + 1)* Gamma[k +Divide[\[Nu],2]+Divide[3,2]]*Gamma[k -Divide[\[Nu],2]+Divide[3,2]]], {k, 0, Infinity}, GenerateConditions->None]"

Maple

translation	"AngerJ(nu, z) = cos((Pinu)/(2))sum(((- 1)^(k)* (z)^(2k))/((4)^(k) GAMMA(k +(nu)/(2)+ 1)GAMMA(k -(nu)/(2)+ 1)), k = 0..infinity)+ sin((Pinu)/(2))sum(((- 1)^(k) (z)^(2k + 1))/((2)^(2k + 1)* GAMMA(k +(nu)/(2)+(3)/(2))*GAMMA(k -(nu)/(2)+(3)/(2))), k = 0..infinity)"

positions

section	2
sentence	0
word	8

includes

"J_{\nu}"

"\mathbf{J}_{\nu}"

"\nu"

isPartOf

Empty array

definiens

definition	"power series expansion"
score	2

definition	"Anger function"
score	2

definition	"Gamma function"
score	2

id

57

pid

109

eid

"math.109.27"

title

"Lamé function"

formulae

id

"FORMULA_7d20395e75eeb74df48a681897d9d727"

formula

"(\operatorname{Ec})^'_{2K} = (\operatorname{Ec})^'_0 = 0, \;\; (\operatorname{Es})^'_{2K} = (\operatorname{Es})^'_0 = 0"

semanticFormula

"(\operatorname{Ec})_{2K}^' =(\operatorname{Ec})_0^' = 0 ,(\operatorname{Es})_{2K}^' =(\operatorname{Es})_0^' = 0"

confidence

0

translations

Empty object

positions

section	2
sentence	3
word	31

includes

"\operatorname{Ec}"

"\operatorname{Es}"

isPartOf

Empty array

definiens

definition	"boundary condition"
score	2

definition	"ellipsoidal wave"
score	2

id

58

pid

110

eid

"math.110.1"

title

"Gauss–Hermite quadrature"

formulae

id

"FORMULA_cdf8d887d4b5ad1a7724773d8eef8fd2"

formula

"\int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)"

semanticFormula

"\int_{-\infty}^{+\infty} \expe^{-x^2} f(x) \diff{x} \approx \sum_{i=1}^n w_i f(x_i)"

confidence

0

translations

Empty object

positions

section	0
sentence	1
word	3

includes

"\int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx"

"n"

"x_{i}"

"w_{i}"

isPartOf

Empty array

definiens

definition	"value of integral"
score	2

definition	"form of Gaussian quadrature"
score	2

definition	"Gauss -- Hermite quadrature"
score	2

definition	"Hermite polynomial"
score	1

definition	"associated weight"
score	2

id

59

pid

111

eid

"math.111.0"

title

"Askey–Wilson polynomials"

formulae

id

"FORMULA_cfe946a0547913234ac79d398f269607"

formula

"p_n(x;a,b,c,d|q) =(ab,ac,ad;q)_na^{-n}\;_{4}\phi_3 \left[\begin{matrix} q^{-n}&abcdq^{n-1}&ae^{i\theta}&ae^{-i\theta} \\ ab&ac&ad \end{matrix} ; q,q \right]"

semanticFormula

"\AskeyWilsonpolyp{n}@{x}{a}{b}{c}{d}{q} = \qmultiPochhammersym{ab , ac , ad}{q}{n} a^{-n} \qgenhyperphi{4}{3}@{q^{-n} , abcdq^{n-1} , a\expe^{\iunit\theta} , a\expe^{-\iunit\theta}}{ab , ac , ad}{q}{q}"

confidence

0

translations

Mathematica

translation	"p[n_, x_, a_, b_, c_, d_, q_] := Product[QPochhammer[Part[{ab , ac , ad},i],q,n],{i,1,Length[{ab , ac , ad}]}](a)^(- n) QHypergeometricPFQ[{(q)^(- n), abcd(q)^(n - 1), aExp[I\[Theta]], aExp[- I\[Theta]]},{ab , ac , a*d},q,q]"

positions

section	0
sentence	3
word	4

includes

"\phi"

"_{n}"

"n"

isPartOf

Empty array

definiens

definition	"basic hypergeometric function"
score	2

definition	"q-Pochhammer symbol"
score	2

definition	"Askey–Wilson polynomials"
score	2

id

60

pid

112

eid

"math.112.0"

title

"Hahn polynomials"

formulae

id

"FORMULA_777007203448847310455e0b0eaaeb2c"

formula

"Q_n(x;\alpha,\beta,N)= {}_3F_2(-n,-x,n+\alpha+\beta+1;\alpha+1,-N+1;1)."

semanticFormula

"\HahnpolyQ{n}@{x}{\alpha}{\beta}{N} = \genhyperF{3}{2}@{- n , - x , n + \alpha + \beta + 1}{\alpha + 1 , - N + 1}{1}"

confidence

0.8953028732079359

translations

Mathematica

translation	"Q[n_, x_, \[Alpha]_, \[Beta]_, N_] := HypergeometricPFQ[{- n , - x , n + \[Alpha]+ \[Beta]+ 1}, {\[Alpha]+ 1 , - N + 1}, 1]"

positions

section	0
sentence	3
word	11

includes

"R_{n}(x;\gamma,\delta,N)"

"S_{n}(x;a,b,c)"

isPartOf

Empty array

definiens

definition	"Hahn polynomial"
score	2

definition	"basic hypergeometric function"
score	2

definition	"hypergeometric function"
score	2

id

61

pid

113

eid

"math.113.2"

title

"Charlier polynomials"

formulae

id

"FORMULA_b76bcf7237b989f6b5d90082fafa53f1"

formula

"\sum_{x=0}^\infty \frac{\mu^x}{x!} C_n(x; \mu)C_m(x; \mu)=\mu^{-n} e^\mu n! \delta_{nm}, \quad \mu>0"

semanticFormula

"\sum_{x=0}^\infty \frac{\mu^x}{x!} \CharlierpolyC{n}@{x}{\mu} \CharlierpolyC{m}@{x}{\mu} = \mu^{-n} \expe^\mu n! \delta_{nm} , \quad \mu > 0"

confidence

0

translations

Mathematica

translation	"Sum[Divide[\[Mu]^x, x!] * HypergeometricPFQ[{-n, -x}, {}, -Divide[1,\[Mu]]] * HypergeometricPFQ[{-m, -x}, {}, -Divide[1,\[Mu]]], {x, 0, Infinity}] == \[Mu]^(-n)Exp[\[Mu]]n!*Subscript[\[Delta], n, m]"

positions

section	0
sentence	2
word	5

includes

Empty array

isPartOf

Empty array

definiens

definition	"orthogonality relation"
score	2

definition	"Charlier polynomial"
score	2

id

62

pid

114

eid

"math.114.0"

title

"Q-Racah polynomials"

formulae

id

"FORMULA_51c23bddc19530680328afbf28235b90"

formula

"p_n(q^{-x}+q^{x+1}cd;a,b,c,d;q) = {}_4\phi_3\left[\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&q^{x+1}cd\\aq&bdq&cq\\ \end{matrix};q;q\right]"

semanticFormula

"\qRacahpolyR{n}@{q^{-x} + q^{x+1} cd}{a}{b}{c}{d}{q} = \qgenhyperphi{4}{3}@{q^{-n}, abq^{n+1}, q^{-x}, q^{x+1}cd}{aq , bdq , cq}{q}{q}"

confidence

0

translations

Empty object

positions

section	1
sentence	0
word	15

includes

Empty array

isPartOf

Empty array

definiens

definition	"term of basic hypergeometric function"
score	2

id

63

pid

115

eid

"math.115.0"

title

"Q-Charlier polynomials"

formulae

id

"FORMULA_925d68ff3ddf733a69ec9936dfede5d6"

formula

"\displaystyle c_n(q^{-x};a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a)"

semanticFormula

"c_n(q^{-x} ; a ; q) = \qgenhyperphi{2}{1}@{q^{-n} , q^{-x}}{0}{q}{- q^{n+1} / a}"

confidence

0.5776294951318733

translations

Empty object

positions

section	1
sentence	0
word	15

includes

Empty array

isPartOf

Empty array

definiens

definition	"q-Charlier polynomial"
score	2

definition	"term of the basic hypergeometric function"
score	2

id

64

pid

116

eid

"math.116.0"

title

"Meixner polynomials"

formulae

id

"FORMULA_29a1f82de004c5721c8dfc5dd1dc5b98"

formula

"M_n(x,\beta,\gamma) = \sum_{k=0}^n (-1)^k{n \choose k}{x\choose k}k!(x+\beta)_{n-k}\gamma^{-k}"

semanticFormula

"\MeixnerpolyM{n}@{x}{\beta}{\gamma} = \sum_{k=0}^n(- 1)^k{n \choose k}{x\choose k} k! \Pochhammersym{x + \beta}{n-k} \gamma^{-k}"

confidence

0.8953028732079359

translations

Mathematica

translation	"M[n_, x_, \[Beta]_, \[Gamma]_] := Sum[(- 1)^(k)Binomial[n,k]Binomial[x,k](k)!Pochhammer[x + \[Beta], n - k]*\[Gamma]^(- k), {k, 0, n}, GenerateConditions->None]"

positions

section	0
sentence	1
word	16

includes

Empty array

isPartOf

Empty array

definiens

definition	"Meixner polynomial"
score	2

definition	"Pochhammer symbol"
score	1

definition	"term of binomial coefficient"
score	1

id

65

pid

117

eid

"math.117.19"

title

"Appell series"

formulae

id

"FORMULA_85014aaf0c7c1f4fe433115e796a03db"

formula

"x(1-x) \frac {\partial^2F_1(x,y)} {\partial x^2} + y(1-x) \frac {\partial^2F_1(x,y)} {\partial x \partial y} + [c - (a+b_1+1) x] \frac {\partial F_1(x,y)} {\partial x} - b_1 y \frac {\partial F_1(x,y)} {\partial y} - a b_1 F_1(x,y) = 0"

semanticFormula

"x(1-x) \deriv[2]{\AppellF{1}@{a}{b_1}{b_2}{\gamma}{x}{y}}{x} + y(1-x) \frac{\pdiff[2]{\AppellF{1}@{a}{b_1}{b_2}{\gamma}{x}{y}}}{\pdiff{x}\pdiff{y}} + [c - (a+b_1+1) x] \deriv[1]{\AppellF{1}@{a}{b_1}{b_2}{\gamma}{x}{y}}{x} - b_1 y \deriv[1]{\AppellF{1}@{a}{b_1}{b_2}{\gamma}{x}{y}}{y} - a b_1 \AppellF{1}@{a}{b_1}{b_2}{\gamma}{x}{y} = 0"

confidence

0

translations

Mathematica

translation

"x*(1-x) * D[AppellF[a, Subscript[b, 1], Subscript[b, 2], \[Gamma], x, y], {x,2}] + y*(1-x) * D[AppellF[a, Subscript[b, 1], Subscript[b, 2], \[Gamma], x, y], x, y] + (c - (a+Subscript[b, 1]+1)*x) * D[AppellF[a, Subscript[b, 1], Subscript[b, 2], \[Gamma], x, y], x] - Subscript[b,1] * y * D[AppellF[a, Subscript[b, 1], Subscript[b, 2], \[Gamma], x, y], y] - a*Subscript[b,1]*AppellF[a, Subscript[b, 1], Subscript[b, 2], \[Gamma], x, y] == 0"

positions

section	3
sentence	0
word	39

includes

"y"

"x"

"F_{1}"

"F"

"_{1}F_{1}"

isPartOf

Empty array

definiens

definition	"Appell"
score	2

definition	"partial differential equation"
score	2

definition	"system of differential equation"
score	1

definition	"system of second-order differential equation"
score	2

definition	"Appell series"
score	2

id

66

pid

118

eid

"math.118.0"

title

"Theta function of a lattice"

formulae

id

"FORMULA_39f4baaa3543f22706b6f7701518f3eb"

formula

"\Theta_\Lambda(\tau) = \sum_{x\in\Lambda}e^{i\pi\tau\|x\|^2}\qquad\mathrm{Im}\,\tau > 0"

semanticFormula

"\Theta_\Lambda(\tau) = \sum_{x\in\Lambda} \expe^{\iunit \cpi \tau \|x \|^2} \qquad \imagpart \tau > 0"

confidence

0

translations

Mathematica

translation	"\[CapitalTheta][\[CapitalLambda]_, \[Tau]_] := Sum[Exp[IPi\[Tau]*(Norm[x])^(2)], {x, \[CapitalLambda]}]"

positions

section	1
sentence	0
word	15

includes

"\Lambda"

isPartOf

Empty array

definiens

definition	"theta function"
score	2

definition	"lattice"
score	1

definition	"Theta function of a lattice"
score	1

id

67

pid

119

eid

"math.119.0"

title

"Heine–Stieltjes polynomials"

formulae

id

"FORMULA_d673cd2334542e8f83f099798c4027b3"

formula

"\frac{d^2 S}{dz^2}+\left(\sum _{j=1}^N \frac{\gamma _j}{z - a_j} \right) \frac{dS}{dz} + \frac{V(z)}{\prod _{j=1}^N (z - a_j)}S = 0"

semanticFormula

"\deriv [2]{S}{z} +(\sum_{j=1}^N \frac{\gamma _j}{z - a_j}) \deriv[]{S}{z} + \frac{V(z)}{\prod _{j=1}^N (z - a_j)} S = 0"

confidence

0

translations

Empty object

positions

section	0
sentence	1
word	6

includes

"V(z)"

"S"

"V"

isPartOf

Empty array

definiens

definition	"form"
score	0

definition	"Fuchsian equation"
score	2

definition	"polynomial"
score	1

definition	"degree"
score	0

definition	"Edward Burr Van Vleck"
score	0

definition	"Heine"
score	1

definition	"polynomial solution"
score	1

definition	"Stieltjes polynomial"
score	1

definition	"Van Vleck polynomial"
score	1

id

68

pid

120

eid

"math.120.0"

title

"Stieltjes–Wigert polynomials"

formulae

id

"FORMULA_583d3b9e00bbd73091b01f368d1a82c7"

formula

"w(x) = \frac{k}{\sqrt{\pi}} x^{-1/2} \exp(-k^2\log^2 x)"

semanticFormula

"w(x) = \frac{k}{\sqrt{\cpi}} x^{-1/2} \exp(- k^2 \log^2 x)"

confidence

0

translations

Mathematica

translation	"w[x_] := Divide[k,Sqrt[Pi]](x)^(- 1/2) Exp[- (k)^(2)* (Log[x])^(2)]"

Maple

translation	"w := (x) -> (k)/(sqrt(Pi))(x)^(- 1/2) exp(- (k)^(2)* (log(x))^(2))"

positions

section	0
sentence	0
word	38

includes

"\frac{k}{\sqrt{\pi}} x^{-1/2} \exp \left(-k^2 \log^2 x \right)"

isPartOf

Empty array

definiens

definition	"weight function"
score	2

definition	"positive real line"
score	0

definition	"basic Askey scheme"
score	1

definition	"family of basic hypergeometric orthogonal polynomial"
score	1

definition	"mathematics"
score	0

definition	"Stieltjes -- Wigert polynomial"
score	2

definition	"Thomas Jan Stieltjes"
score	0

definition	"Carl Severin Wigert"
score	0

definition	"example of such weight function"
score	0

id

69

pid

121

eid

"math.121.23"

title

"Modular lambda function"

formulae

id

"FORMULA_4e5334aa6f5fa551b0718a2372816061"

formula

"y^2=x(x-1)(x-\lambda)"

semanticFormula

"y^2=x(x-1)(x-\lambda)"

confidence

0

translations

Mathematica

translation	"(y)^(2) == x(x - 1)(x - \[Lambda])"

Maple

translation	"(y)^(2) = x(x - 1)(x - lambda)"

positions

section	2
sentence	5
word	13

includes

"\lambda"

isPartOf

Empty array

definiens

definition	"elliptic curve of Legendre form"
score	2

definition	"relation to the j-invariant"
score	1

definition	"relation to the j-invariant"
score	1

id

70

pid

122

eid

"math.122.3"

title

"Meixner–Pollaczek polynomials"

formulae

id

"FORMULA_96d19b4b504f801548c69064d662043b"

formula

"P_1^{(\lambda)}(x;\phi)=2(\lambda\cos\phi + x\sin\phi)"

semanticFormula

"\MeixnerPollaczekpolyP{\lambda}{1}@{x}{\phi} = 2(\lambda \cos \phi + x \sin \phi)"

confidence

0.8953028732079359

translations

Empty object

positions

section	1
sentence	0
word	9

includes

"P_{m}^{(\lambda)}(x;\varphi)"

isPartOf

Empty array

definiens

definition	"first few Meixner -- Pollaczek polynomial"
score	2

id

71

pid

123

eid

"math.123.0"

title

"Jacobi polynomials"

formulae

id

"FORMULA_c8b5b9184e45bca39744427c45693115"

formula

"P_n^{(\alpha,\beta)}(z)=\frac{(\alpha+1)_n}{n!}\,{}_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\tfrac{1}{2}(1-z)\right)"

semanticFormula

"\JacobipolyP{\alpha}{\beta}{n}@{z} = \frac{\Pochhammersym{\alpha + 1}{n}}{n!} \genhyperF{2}{1}@{- n , 1 + \alpha + \beta + n}{\alpha + 1}{\tfrac{1}{2}(1 - z)}"

confidence

0.7595006538205181

translations

Mathematica

translation	"JacobiP[n, \[Alpha], \[Beta], z] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]HypergeometricPFQ[{- n , 1 + \[Alpha]+ \[Beta]+ n}, {\[Alpha]+ 1}, Divide[1,2](1 - z)]"

Maple

translation	"JacobiP(n, alpha, beta, z) = (pochhammer(alpha + 1, n))/(factorial(n))hypergeom([- n , 1 + alpha + beta + n], [alpha + 1], (1)/(2)(1 - z))"

positions

section	1
sentence	0
word	12

includes

"P_{n}^{(\alpha, \beta)}(x)"

"(\alpha+1)_n"

"n"

"n + \alpha + \beta"

"P_{n}^{(\alpha, \beta)}"

"\alpha,\beta"

"z"

isPartOf

Empty array

definiens

definition	"Pochhammer 's symbol"
score	2

definition	"hypergeometric function"
score	2

definition	"Jacobi polynomial"
score	2

id

72

pid

124

eid

"math.124.0"

title

"Continuous dual Hahn polynomials"

formulae

id

"FORMULA_b0d448ba925dc6b2bf2ce32a1253dee4"

formula

"S_n(x^2;a,b,c)= {}_3F_2(-n,a+ix,a-ix;a+b,a+c;1)."

semanticFormula

"\contdualHahnpolyS{n}@{x^2}{a}{b}{c} = \genhyperF{3}{2}@{- n , a + \iunit x , a - \iunit x}{a + b , a + c}{1}"

confidence

0.7132263353695951

translations

Empty object

positions

section	0
sentence	1
word	10

includes

"R_{n}(x;\gamma,\delta,N)"

isPartOf

Empty array

definiens

definition	"hypergeometric function"
score	1

definition	"dual Hahn polynomial"
score	1

definition	"continuous Hahn polynomial"
score	1

definition	"continuous dual Hahn polynomial"
score	2

id

73

pid

125

eid

"math.125.15"

title

"Continuous Hahn polynomials"

formulae

id

"FORMULA_ff971744100fef3b34b2c93b6adc3efb"

formula

"P_n^{(\alpha,\beta)}=\lim_{t\to\infty}t^{-n}p_n\left(\tfrac12xt; \tfrac12(\alpha+1-it), \tfrac12(\beta+1+it), \tfrac12(\alpha+1+it), \tfrac12(\beta+1-it)\right)"

semanticFormula

"\JacobipolyP{\alpha}{\beta}{n}@{x} = \lim_{t\to\infty} t^{-n} \contHahnpolyp{n}@{\tfrac12 xt}{\tfrac12(\alpha + 1 - \iunit t)}{\tfrac12(\beta + 1 + \iunit t)}{\tfrac12(\alpha + 1 + \iunit t)}{\tfrac12(\beta + 1 - \iunit t)}"

confidence

0.9041995034970904

translations

Mathematica

translation

"JacobiP[n, \[Alpha], \[Beta], x] == Limit[(t)^(- n)* I^(n)*Divide[Pochhammer[Divide[1,2]*(\[Alpha]+ 1 - I*t) + Divide[1,2]*(\[Alpha]+ 1 + I*t), n]*Pochhammer[Divide[1,2]*(\[Alpha]+ 1 - I*t) + Divide[1,2]*(\[Beta]+ 1 - I*t), n], (n)!] * HypergeometricPFQ[{-(n), n + 2*Re[Divide[1,2]*(\[Alpha]+ 1 - I*t) + Divide[1,2]*(\[Beta]+ 1 + I*t)] - 1, Divide[1,2]*(\[Alpha]+ 1 - I*t) + I*(Divide[1,2]*x*t)}, {Divide[1,2]*(\[Alpha]+ 1 - I*t) + Divide[1,2]*(\[Alpha]+ 1 + I*t), Divide[1,2]*(\[Alpha]+ 1 - I*t) + Divide[1,2]*(\[Beta]+ 1 - I*t)}, 1], t -> Infinity, GenerateConditions->None]"

positions

section	5
sentence	2
word	20

includes

"p_{n}(x;a,b,c,d)"

"F_{n}"

"P_{n}^{(\alpha,\beta)}"

isPartOf

Empty array

definiens

definition	"case of the continuous Hahn polynomial"
score	1

definition	"Jacobi polynomial"
score	2

definition	"continuous Hahn polynomial"
score	2

id

74

pid

126

eid

"math.126.7"

title

"Dual Hahn polynomials"

formulae

id

"FORMULA_657ec9a2e460e61adc6857260291be56"

formula

"\sum^{b-1}_{s=a}w_n^{(c)}(s,a,b)w_m^{(c)}(s,a,b)\rho(s)[\Delta x(s-\frac{1}{2}) ]=\delta_{nm}d_n^2"

semanticFormula

"\sum_{s=a}^{b-1} \dualHahnpolyR{n}@{c}{s}{a}{b} \dualHahnpolyR{m}@{c}{s}{a}{b} \rho(s) [\Delta x(s - \frac{1}{2})] = \delta_{nm} d_n^2"

confidence

0

translations

Empty object

positions

section	1
sentence	0
word	8

includes

"n"

isPartOf

Empty array

definiens

definition	"Dual Hahn polynomial"
score	2

id

75

pid

127

eid

"math.127.0"

title

"Continuous q-Hahn polynomials"

formulae

id

"FORMULA_67e28846328978f4e08bb6b69fe6c549"

formula

"p_n(x;a,b,c|q)=a^{-n}e^{-inu}(abe^{2iu},ac,ad;q)_n*_4\Phi_3(q^{-n},abcdq^{n-1},ae^{i{(t+2u)}},ae^{-it};abe^{2iu},ac,ad;q;q)"

semanticFormula

"p_n(x ; a , b , c|q) = a^{-n} \expe^{-\iunit nu} \qmultiPochhammersym{ab\expe^{2\iunit u} , ac , ad}{q}{n} * \qgenhyperphi{4}{3}@{q^{-n} , abcdq^{n-1} , a\expe^{\iunit{(t+2u)}} , a\expe^{-\iunit t}}{ab\expe^{2\iunit u} , ac , ad}{q}{q}"

confidence

0.8662724998444776

translations

Mathematica

translation	"p[n_, x_, a_, b_, c_, q_] := (a)^(- n)* Exp[- I\[Nu]]Product[QPochhammer[Part[{abExp[2Iu], ac , ad},i],q,n],{i,1,Length[{abExp[2Iu], ac , ad}]}]* QHypergeometricPFQ[{(q)^(- n), abcd(q)^(n - 1), aExp[I(t + 2u)], aExp[- It]},{abExp[2Iu], ac , a*d},q,q]"

positions

section	1
sentence	0
word	15

includes

"q"

isPartOf

Empty array

definiens

definition	"polynomial"
score	1

definition	"term of basic hypergeometric function"
score	1

definition	"Pochhammer symbol"
score	1

definition	"continuous FORMULA_7694f4a66316e53c8cdd9d9954bd611d - Hahn polynomial"
score	2

definition	"q - Pochhammer symbol"
score	2

id

76

pid

128

eid

"math.128.0"

title

"Continuous dual q-Hahn polynomials"

formulae

id

"FORMULA_95daf919f18506606090e49a38d1c1a6"

formula

"p_n(x;a,b,c\mid q)=\frac{(ab,ac;q)_n}{a^n}\cdot {_3\Phi_2}(q^-n,ae^{i\theta},ae^{-i\theta}; ab, ac \mid q;q)"

semanticFormula

"p_n(x ; a , b , c \mid q) = \frac{\qmultiPochhammersym{ab , ac}{q}{n}}{a^n} \cdot \qgenhyperphi{3}{2}@{q^- n , ae^{\iunit \theta} , ae^{- \iunit \theta}}{ab , ac}{q}{q}"

confidence

0.8662724998444776

translations

Mathematica

translation	"p[n_, x_, a_, b_, c_, q_] := Divide[Product[QPochhammer[Part[{ab , ac},i],q,n],{i,1,Length[{ab , ac}]}],(a)^(n)] * QHypergeometricPFQ[{(q)^(-)* n , a(e)^(I\[Theta]), a(e)^(- I\[Theta])},{ab , ac},q,q]"

positions

section	1
sentence	0
word	15

includes

"q"

isPartOf

Empty array

definiens

definition	"polynomial"
score	1

definition	"term of basic hypergeometric function"
score	2

definition	"Pochhammer symbol"
score	1

definition	"continuous dual FORMULA_7694f4a66316e53c8cdd9d9954bd611d - Hahn polynomial"
score	2

id

77

pid

129

eid

"math.129.0"

title

"Q-Hahn polynomials"

formulae

id

"FORMULA_b3a9ac90714e1e705d2a88b30e79cca0"

formula

"Q_n(x;a,b,N;q)=\;_{3}\phi_2\left[\begin{matrix} q^-n & abq^n+1 & x \\ aq & q^-N \end{matrix} ; q,q \right]"

semanticFormula

"\qHahnpolyQ{n}@{x}{a}{b}{N}{q} = \qgenhyperphi{3}{2}@{q^-n , abq^n+1 , x}{aq , q^-N}{q}{q}"

confidence

0

translations

Mathematica

translation	"Q[n_, x_, a_, b_, N_, q_] := QHypergeometricPFQ[{(q)^(-)* n , ab(q)^(n)+ 1 , x},{aq , (q)^(-) N},q,q]"

positions

section	1
sentence	0
word	15

includes

Empty array

isPartOf

Empty array

definiens

definition	"q - Hahn polynomial"
score	2

definition	"polynomial"
score	1

definition	"term of basic hypergeometric function"
score	2

definition	"Pochhammer symbol"
score	0

id

78

pid

131

eid

"math.131.0"

title

"Al-Salam–Chihara polynomials"

formulae

id

"FORMULA_52a07ce46212cbc2298415c5fca6e075"

formula

"x="

semanticFormula

"x=\cos@{\theta}"

confidence

0

translations

Mathematica

translation	"x = Cos[\[Theta]]"

Maple

translation	"x = cos(theta)"

positions

section	1
sentence	0
word	20

includes

Empty array

isPartOf

Empty array

definiens

definition	"cosine function"
score	2

definition	"substitution"
score	2

id

79

pid

132

eid

"math.132.7"

title

"Orthogonal polynomials on the unit circle"

formulae

id

"FORMULA_f2d41903301a99a3fade5f2f49450694"

formula

"\Phi_n^*(z)=z^n\overline{\Phi_n(1/\overline{z})}"

semanticFormula

"\Phi_n^*(z) = z^n{\conj{\Phi_n(1 / \conj{z})}}"

confidence

0.7579553437219001

translations

Mathematica

translation	"\[CapitalPhi]\[Prima][n_, z_] := z^n*Conjugate[\[CapitalPhi][n, Divide[1, Conjugate[z]]]]"

positions

section	2
sentence	0
word	8

includes

"\Phi_n(z)"

"z^n"

"\alpha_n"

isPartOf

Empty array

definiens

definition	"polynomial"
score	2

id

80

pid

133

eid

"math.133.8"

title

"Orthogonal polynomials"

formulae

id

"FORMULA_c0641714ec593f58211623652c4a34f0"

formula

"P_n(x) = c_n \, \det \begin{bmatrix}m_0 & m_1 & m_2 &\cdots & m_n \\m_1 & m_2 & m_3 &\cdots & m_{n+1} \\&&\vdots&& \vdots \\m_{n-1} &m_n& m_{n+1} &\cdots &m_{2n-1}\\1 & x & x^2 & \cdots & x^n\end{bmatrix}"

semanticFormula

"P_n(x) = c_n \det \begin{bmatrix}m_0 & m_1 & m_2 &\cdots & m_n \\m_1 & m_2 & m_3 &\cdots & m_{n+1} \\&&\vdots&& \vdots \\m_{n-1} &m_n& m_{n+1} &\cdots &m_{2n-1}\\1 & x & x^2 & \cdots & x^n\end{bmatrix}"

confidence

0

translations

Empty object

positions

section	5
sentence	0
word	16

includes

"P_{n}"

"c_{n}"

"P_{m}"

isPartOf

Empty array

definiens

definition	"constant"
score	0

definition	"normalisation"
score	0

definition	"orthogonal polynomial"
score	2

definition	"term of the moment"
score	0

id

81

pid

134

eid

"math.134.0"

title

"Little q-Jacobi polynomials"

formulae

id

"FORMULA_c492265e4cd4beeeb776dad843dc1f73"

formula

"\displaystyle p_n(x;a,b;q) = {}_2\phi_1(q^{-n},abq^{n+1};aq;q,xq)"

semanticFormula

"\littleqJacobipolyp{n}@{x}{a}{b}{q} = \qgenhyperphi{2}{1}@{q^{-n} , abq^{n+1}}{aq}{q}{xq}"

confidence

0.7229065246531701

translations

Mathematica

translation	"p[n_, x_, a_, b_, q_] := QHypergeometricPFQ[{(q)^(- n), ab(q)^(n + 1)},{aq},q,xq]"

positions

section	1
sentence	0
word	19

includes

"q"

"p_{n}(x;a,b;q)"

isPartOf

Empty array

definiens

definition	"Jacobi polynomial"
score	1

definition	"term of basic hypergeometric function"
score	2

definition	"Pochhammer symbol"
score	0

definition	"little q - Jacobi polynomial"
score	2

id

82

pid

135

eid

"math.135.0"

title

"Big q-Jacobi polynomials"

formulae

id

"FORMULA_0680f701a101288f89487a7a3fabefb1"

formula

"\displaystyle P_n(x;a,b,c;q)={}_3\phi_2(q^{-n},abq^{n+1},x;aq,cq;q,q)"

semanticFormula

"\bigqJacobipolyP{n}@{x}{a}{b}{c}{q} = \qgenhyperphi{3}{2}@{q^{-n} , abq^{n+1} , x}{aq , cq}{q}{q}"

confidence

0.7424814142326033

translations

Mathematica

translation	"p[n_, x_, a_, b_, c_, q_] := QHypergeometricPFQ[{(q)^(- n), ab(q)^(n + 1), x},{aq , cq},q,q]"

positions

section	1
sentence	0
word	11

includes

"P_{n}(x;a,b,c;q)"

"q"

isPartOf

Empty array

definiens

definition	"polynomial"
score	1

definition	"term of basic hypergeometric function"
score	2

definition	"big q - Jacobi polynomial"
score	2

id

83

pid

137

eid

"math.137.0"

title

"Big q-Laguerre polynomials"

formulae

id

"FORMULA_aa5a6972c7e8327e316eddc8fd8e9b08"

formula

"P_n(x;a,b;q)=\frac{1}{(b^{-1}*q^{-n};q,n)}*_2\Phi_1(q^{-n},aqx^{-1};aq|q;\frac{x}{b})"

semanticFormula

"P_n(x;a,b;q) =\frac{1}{\qmultiPochhammersym{b^{-1}*q^{-n}}{q}{n}} * \qgenhyperphi{2}{1}@{q^{-n},aqx^{-1}}{aq}{q}{\frac{x}{b}}"

confidence

0

translations

Mathematica

translation	"P[n_, x_, a_, b_, q_] := Divide[1,Product[QPochhammer[Part[{(b)^(- 1)* (q)^(- n)},i],q,n],{i,1,Length[{(b)^(- 1)* (q)^(- n)}]}]]* QHypergeometricPFQ[{(q)^(- n), aq(x)^(- 1)},{a*q},q,Divide[x,b]]"

positions

section	1
sentence	0
word	15

includes

"q"

isPartOf

Empty array

definiens

definition	"polynomial"
score	1

definition	"term of basic hypergeometric function"
score	1

definition	"Pochhammer symbol"
score	1

definition	"q - Pochhammer symbol"
score	1

definition	"big q - Laguerre polynomial"
score	2

id

84

pid

138

eid

"math.138.0"

title

"Dual q-Krawtchouk polynomials"

formulae

id

"FORMULA_9221dfda453868628eb8bbcd2d414fdf"

formula

"K_n(\lambda(x);c,N|q)=_3\Phi_2(q^{-n},q^{-x},cq^{x-N};q^{-N},0|q;q)"

semanticFormula

"K_n(\lambda(x);c,N|q) = \qgenhyperphi{3}{2}@{q^{-n},q^{-x},cq^{x-N}}{q^{-N},0}{q}{q}"

confidence

0

translations

Empty object

positions

section	1
sentence	0
word	15

includes

"q"

isPartOf

Empty array

definiens

definition	"polynomial"
score	1

definition	"term of basic hypergeometric function"
score	2

definition	"Pochhammer symbol"
score	0

definition	"dual q - Krawtchouk polynomial"
score	2

id

85

pid

139

eid

"math.139.0"

title

"Continuous q-Laguerre polynomials"

formulae

id

"FORMULA_8c9e3af3c57272f3a6ddabba68ab4d3e"

formula

"P_{n}^{(\alpha)}(x|q)=\frac{(q^\alpha+1;q)_{n}}{(q;q)_{n}}"

semanticFormula

"P_{n}^{(\alpha)}(x|q) = \frac{\qmultiPochhammersym{q^\alpha+1}{q}{n}}{\qPochhammer{q}{q}{n}} \qgenhyperphi{3}{2}@{q^{-n},q^{\alpha/2+1/4}\expe^{\iunit\theta},q^{\alpha/2+1/4}*\expe^{-\iunit\theta}}{q^{\alpha+1},0}{q}{q}"

confidence

0

translations

Mathematica

translation	"P[n_, \[Alpha]_, x_, q_] := Divide[Product[QPochhammer[Part[{(q)^\[Alpha]+ 1},i],q,n],{i,1,Length[{(q)^\[Alpha]+ 1}]}],QPochhammer[q, q, n]]QHypergeometricPFQ[{(q)^(- n), (q)^(\[Alpha]/2 + 1/4) Exp[I\[Theta]], (q)^(\[Alpha]/2 + 1/4) Exp[- I*\[Theta]]},{(q)^(\[Alpha]+ 1), 0},q,q]"

positions

section	1
sentence	1
word	0

includes

"q"

isPartOf

Empty array

definiens

definition	"continuous q - Laguerre polynomial"
score	2

definition	"family of basic hypergeometric orthogonal polynomial"
score	2

definition	"Pochhammer symbol"
score	2

id

86

pid

142

eid

"math.142.0"

title

"Little q-Laguerre polynomials"

formulae

id

"FORMULA_4e548bca196e13d5af0eaadf2ea725d1"

formula

"\displaystyle p_n(x;a|q) = {}_2\phi_1(q^{-n},0;aq;q,qx) = \frac{1}{(a^{-1}q^{-n};q)_n}{}_2\phi_0(q^{-n},x^{-1};;q,x/a)"

semanticFormula

"p_n(x ; a|q) = \qgenhyperphi{2}{1}@{q^{-n} , 0}{aq}{q}{qx} = \frac{1}{\qmultiPochhammersym{a^{-1} q^{-n}}{q}{n}} \qgenhyperphi{2}{0}@{q^{-n} , x^{-1}}{}{q}{x/a}"

confidence

0.7219509974881755

translations

Mathematica	"p[n_, x_, a_, q_] := QHypergeometricPFQ[{(q)^(- n), 0},{aq},q,qx] == Divide[1,Product[QPochhammer[Part[{(a)^(- 1)* (q)^(- n)},i],q,n],{i,1,Length[{(a)^(- 1)* (q)^(- n)}]}]]*QHypergeometricPFQ[{(q)^(- n), (x)^(- 1)},{},q,x/a]"

positions

section	1
sentence	0
word	15

includes

"q"

"p_{n}(x;a|q)"

isPartOf

Empty array

definiens

definition	"polynomial"
score	1

definition	"term of basic hypergeometric function"
score	2

definition	"Pochhammer symbol"
score	1

definition	"little q - Laguerre polynomial"
score	2

definition	"q - Pochhammer symbol"
score	2

id

87

pid

143

eid

"math.143.0"

title

"Q-Bessel polynomials"

formulae

id

"FORMULA_c89da2fda6f9f6411ed4292f6d845f52"

formula

"y_{n}(x;a;q)=\;_{2}\phi_1 \left(\begin{matrix} q^{-N} & -aq^{n} \\ 0 \end{matrix} ; q,qx \right)"

semanticFormula

"y_{n}(x;a;q) = \qgenhyperphi{2}{1}@{q^{-N} , -aq^{n}}{0}{q}{qx}"

confidence

0.6264217257193126

translations

Mathematica	"y[n_, x_, a_, q_] := QHypergeometricPFQ[{(q)^(- N), - a(q)^(n)},{0},q,qx]"

positions

section	1
sentence	0
word	16

includes

Empty array

isPartOf

Empty array

definiens

definition	"polynomial"
score	1

definition	"term of basic hypergeometric function"
score	2

definition	"Pochhammer symbol"
score	0

definition	"q - Bessel polynomial"
score	1

id

88

pid

144

eid

"math.144.2"

title

"Discrete q-Hermite polynomials"

formulae

id

"FORMULA_b9974285610b7a82c94b6a504726df8c"

formula

"h_n(ix;q^{-1}) = i^n\hat h_n(x;q)"

semanticFormula

"\discqHermitepolyhI{n}@{\iunit x}{q^{-1}} = \iunit^n \discqHermitepolyhII{n}@{x}{q}"

confidence

0.8429359579302446

translations

Empty object

positions

section	1
sentence	0
word	27

includes

"\hat{h}_{n}(x;q)"

"q"

"h_{n}(x;q)"

isPartOf

Empty array

definiens

definition	"Hermite polynomial"
score	2

definition	"term of basic hypergeometric function"
score	1

definition	"Carlitz polynomial"
score	1

definition	"Al-Salam"
score	1

definition	"discrete q - Hermite polynomial"
score	2

id

89

pid

145

eid

"math.145.0"

title

"Q-Meixner–Pollaczek polynomials"

formulae

id

"FORMULA_fa6650cad7aed4d975716018ef03068f"

formula

"P_{n}(x;a\mid q) = a^{-n} e^{in\phi} \frac{a^2;q_n}{(q;q)_n} {_3}\Phi_2(q^-n, ae^{i(\theta+2\phi)}, ae^{-i\theta}; a^2, 0 \mid q; q)"

semanticFormula

"P_{n}(x ; a \mid q) = a^{-n} \expe^{\iunit n\phi} \frac{\qmultiPochhammersym{a^2}{q}{n}}{\qmultiPochhammersym{q}{q}{n}} \qgenhyperphi{3}{2}@{q^- n , a\expe^{\iunit(\theta + 2 \phi)} , a\expe^{- \iunit \theta}}{a^2, 0}{q}{q}"

confidence

0.8662724998444776

translations

Mathematica

translation	"P[n_, x_, a_, q_] := (a)^(- n)* Exp[In\[Phi]]Divide[Product[QPochhammer[Part[{(a)^(2)},i],q,n],{i,1,Length[{(a)^(2)}]}],Product[QPochhammer[Part[{q},i],q,n],{i,1,Length[{q}]}]]QHypergeometricPFQ[{(q)^(-)* n , aExp[I(\[Theta]+ 2\[Phi])], aExp[- I*\[Theta]]},{(a)^(2), 0},q,q]"

positions

section	1
sentence	0
word	16

includes

Empty array

isPartOf

Empty array

definiens

definition	"polynomial"
score	1

definition	"term of basic hypergeometric function"
score	2

definition	"Pochhammer symbol"
score	1

definition	"q - Pochhammer symbol"
score	2

definition	"Q Meixner – Pollaczek polynomials"
score	2

id

90

pid

149

eid

"math.149.0"

title

"Q-Laguerre polynomials"

formulae

id

"FORMULA_dea0af895f73964b98741e71bc0635cb"

formula

"\displaystyle L_n^{(\alpha)}(x;q) = \frac{(q^{\alpha+1};q)_n}{(q;q)_n} {}_1\phi_1(q^{-n};q^{\alpha+1};q,-q^{n+\alpha+1}x)"

semanticFormula

"\qLaguerrepolyL{\alpha}{n}@{x}{q} = \frac{\qmultiPochhammersym{q^{\alpha+1}}{q}{n}}{\qmultiPochhammersym{q}{q}{n}} \qgenhyperphi{1}{1}@{q^{-n}}{q^{\alpha+1}}{q}{- q^{n+\alpha+1} x}"

confidence

0.779734956061429

translations

Mathematica

translation	"L[n_, \[Alpha]_, x_, q_] := Divide[Product[QPochhammer[Part[{(q)^(\[Alpha]+ 1)},i],q,n],{i,1,Length[{(q)^(\[Alpha]+ 1)}]}],Product[QPochhammer[Part[{q},i],q,n],{i,1,Length[{q}]}]]QHypergeometricPFQ[{(q)^(- n)},{(q)^(\[Alpha]+ 1)},q,- (q)^(n + \[Alpha]+ 1) x]"

positions

section	1
sentence	0
word	18

includes

"q"

"P_{n}^{(\alpha)}(x;q)"

isPartOf

Empty array

definiens

definition	"Laguerre polynomial"
score	2

definition	"q - Laguerre polynomial"
score	2

definition	"term of basic hypergeometric function"
score	2

definition	"Pochhammer symbol"
score	1

definition	"q - Pochhammer symbol"
score	2

id

91

pid

150

eid

"math.150.3"

title

"Continuous q-Hermite polynomials"

formulae

id

"FORMULA_a10dc9de9b2b618ad2f2e96dc9eb0207"

formula

"\sum_{n=0}^\infty H_n(x \mid q) \frac{t^n}{(q;q)_n} = \frac{1}{\left( t e^{i \theta},t e^{-i \theta};q \right)_\infty}"

semanticFormula

"\sum_{n=0}^\infty \contqHermitepolyH{n}@{x}{q} \frac{t^n}{\qmultiPochhammersym{q}{q}{n}} = \frac{1}{\qmultiPochhammersym{t \expe^{\iunit \theta} , t \expe^{- \iunit \theta}}{q}{\infty}}"

confidence

0.7796357038819148

translations

Empty object

positions

section	3
sentence	0
word	0

includes

"q"

isPartOf

Empty array

definiens

definition	"continuous q - Hermite polynomial"
score	2

definition	"q - Pochhammer symbol"
score	2

id

92

pid

151

eid

"math.151.0"

title

"Ince equation"

formulae

id

"FORMULA_ce9ed9f979f486263028e3d86b63ac60"

formula

"w^{\prime\prime}+\xi\sin(2z)w^{\prime}+(\eta-p\xi\cos(2z))w=0. "

semanticFormula

"w^{\prime\prime}+\xi\sin(2z)w^{\prime}+(\eta-p\xi\cos(2z))w=0"

confidence

0

translations

Mathematica

translation	"D[w[z], {z, 2}] + \[Xi]Sin[2z]D[w[z], {z, 1}] + (\[Eta]-p\[Xi]Cos[2z])*w[z] == 0"

positions

section	0
sentence	0
word	19

includes

"p"

isPartOf

Empty array

definiens

definition	"differential equation"
score	2

definition	"Ince equation"
score	2

definition	"mathematics"
score	0

definition	"non-negative integer"
score	0

definition	"Edward Lindsay Ince"
score	0

definition	"polynomial solution"
score	0

definition	"Ince polynomial"
score	1

id

93

pid

152

eid

"math.152.1"

title

"Ferrers function"

formulae

id

"FORMULA_b5ab87b9cd2da05be00884345889d9e3"

formula

"Q_v^\mu(x)= \cos(\mu\pi)\left(\frac{1+x}{1-x}\right)^{\mu/2}\frac{F(v+1,-v;1-\mu;1/2-2/x)} {\Gamma(1-\mu ) }"

semanticFormula

"\FerrersQ[\mu]{v}@{x} = \cos(\mu \cpi)(\frac{1+x}{1-x})^{\mu/2} \frac{\hyperF@{v+1}{-v}{1-\mu}{1/2-2/x}}{\EulerGamma@{1-\mu}}"

confidence

0.8133162393162393

translations

Mathematica

translation	"LegendreQ[v, \[Mu], x] == Cos[(\[Mu]Pi)](Divide[1 + x,1 - x])^(\[Mu]/2)Divide[Hypergeometric2F1[v + 1, - v, 1 - \[Mu], 1/2 - 2/x],Gamma[1 - \[Mu]]]"

Maple

translation	"LegendreQ(v, mu, x) = cos((muPi))((1 + x)/(1 - x))^(mu/2)(hypergeom([v + 1, - v], [1 - mu], 1/2 - 2/x))/(GAMMA(1 - mu))"

positions

section	1
sentence	0
word	13

includes

Empty array

isPartOf

Empty array

definiens

definition	"Ferrers function of the second kind"
score	2

definition	"Ferrers function of the first kind"
score	1

definition	"Gamma function"
score	2

definition	"hypergeometric function"
score	2

id

94

pid

153

eid

"math.153.27"

title

"Incomplete Bessel functions"

formulae

id

"FORMULA_35ab66efafff0de40d98c0778ebb63c3"

formula

"H_{-v}^{(1)}(z,w)=e^{v\pi i}H_v^{(1)}(z,w)"

semanticFormula

"H_{-v}^{(1)}(z,w) = \expe^{v \cpi \iunit} H_v^{(1)}(z , w)"

confidence

0

translations

Empty object

positions

section	2
sentence	0
word	16

includes

"v"

"w"

"H_v^{(1)}(z,w)"

isPartOf

Empty array

definiens

definition	"incomplete Bessel function"
score	2

id

95

pid

154

eid

"math.154.0"

title

"Incomplete Bessel K function/generalized incomplete gamma function"

formulae

id

"FORMULA_c333a7966510ed0b8f4de3147eabe47a"

formula

"K_v(x,y)=\int_1^\infty\frac{e^{-xt-\frac{y}{t}}}{t^{v+1}}dt"

semanticFormula

"K_v(x , y) = \int_1^\infty \frac{\expe^{-xt-\frac{y}{t}}}{t^{v+1}} \diff{t}"

confidence

0

translations

Mathematica

translation	"K[v_, x_, y_] := Integrate[Divide[Exp[- x*t -Divide[y,t]],(t)^(v + 1)], {t, 1, Infinity}, GenerateConditions->None]"

positions

section	0
sentence	0
word	18

includes

"K_v(x,y)"

isPartOf

Empty array

definiens

definition	"mathematician"
score	0

definition	"type incomplete-version of Bessel function"
score	2

definition	"type generalized-version of incomplete gamma function"
score	0

MediaWiki:Gold-data.json

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