LaTeX to CAS translator

This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.

The demo-application converts LaTeX functions which directly translate to CAS counterparts.

Functions without explicit CAS support are available for translation via a DRMF package (under development).

The following LaTeX input ...

{\displaystyle C_n^{(\alpha)}(x) = \frac{(-1)^n}{2^n n!}\frac{\Gamma(\alpha+\frac{1}{2})\Gamma(n+2\alpha)}{\Gamma(2\alpha)\Gamma(\alpha+n+\frac{1}{2})}(1-x^2)^{-\alpha+1/2}\frac{d^n}{dx^n}\left[(1-x^2)^{n+\alpha-1/2}\right].}

${\displaystyle C_{n}^{(\alpha )}(x)={\frac {(-1)^{n}}{2^{n}n!}}{\frac {\Gamma (\alpha +{\frac {1}{2}})\Gamma (n+2\alpha )}{\Gamma (2\alpha )\Gamma (\alpha +n+{\frac {1}{2}})}}(1-x^{2})^{-\alpha +1/2}{\frac {d^{n}}{dx^{n}}}\left[(1-x^{2})^{n+\alpha -1/2}\right].}$

... is translated to the CAS output ...

Semantic latex: \ultrasphpoly{\alpha}{n}@{x} = \frac{(-1)^n}{2^n n!} \frac{\Gamma(\alpha+\frac{1}{2})\Gamma(n+2\alpha)}{\Gamma(2\alpha)\Gamma(\alpha+n+\frac{1}{2})}(1 - x^2)^{-\alpha+1/2} \deriv [n]{ }{x} [(1 - x^2)^{n+\alpha-1/2}]

Confidence: 0.6805

Mathematica

Translation: GegenbauerC[n, \[Alpha], x] == Divide[(- 1)^(n),(2)^(n)* (n)!]*Divide[\[CapitalGamma][\[Alpha]+Divide[1,2]]* \[CapitalGamma][n + 2*\[Alpha]],\[CapitalGamma][2*\[Alpha]]* \[CapitalGamma][\[Alpha]+ n +Divide[1,2]]]*(1 - (x)^(2))^(- \[Alpha]+ 1/2)* D[(1 - (x)^(2))^(n + \[Alpha]- 1/2), {x, n}]

Information

Sub Equations

GegenbauerC[n, \[Alpha], x] = Divide[(- 1)^(n),(2)^(n)* (n)!]*Divide[\[CapitalGamma][\[Alpha]+Divide[1,2]]* \[CapitalGamma][n + 2*\[Alpha]],\[CapitalGamma][2*\[Alpha]]* \[CapitalGamma][\[Alpha]+ n +Divide[1,2]]]*(1 - (x)^(2))^(- \[Alpha]+ 1/2)* D[(1 - (x)^(2))^(n + \[Alpha]- 1/2), {x, n}]

Free variables

\[Alpha]

\[CapitalGamma]

n

x

Symbol info

Ultraspherical Gegenbauer polynomial; Example: \ultrasphpoly{\lambda}{n}@{x}

Will be translated to: GegenbauerC[$1, $0, $2] Relevant links to definitions: DLMF: http://dlmf.nist.gov/18.3#T1.t1.r3 Mathematica: https://reference.wolfram.com/language/ref/GegenbauerC.html

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://reference.wolfram.com/language/ref/D.html

Could be the second Feigenbaum constant.

But this system doesn't know how to translate it as a constant. It was translated as a general letter.

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Test expression: (GegenbauerC[n, \[Alpha], x])-(Divide[(- 1)^(n),(2)^(n)* (n)!]*Divide[\[CapitalGamma]*(\[Alpha]+Divide[1,2])*\[CapitalGamma]*(n + 2*\[Alpha]),\[CapitalGamma]*(2*\[Alpha])*\[CapitalGamma]*(\[Alpha]+ n +Divide[1,2])]*(1 - (x)^(2))^(- \[Alpha]+ 1/2)* D[(1 - (x)^(2))^(n + \[Alpha]- 1/2), {x, n}])

ERROR:

{
    "result": "ERROR",
    "testTitle": "Simple",
    "testExpression": null,
    "resultExpression": null,
    "wasAborted": false,
    "conditionallySuccessful": false
}

Numeric

SymPy

Translation:

Information

Symbol info

(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \ultrasphpoly [\ultrasphpoly]

Tests

Symbolic

Numeric

Maple

Translation: GegenbauerC(n, alpha, x) = ((- 1)^(n))/((2)^(n)* factorial(n))*(Gamma(alpha +(1)/(2))* Gamma(n + 2*alpha))/(Gamma(2*alpha)* Gamma(alpha + n +(1)/(2)))*(1 - (x)^(2))^(- alpha + 1/2)* diff((1 - (x)^(2))^(n + alpha - 1/2), [x$(n)])

Information

Sub Equations

GegenbauerC(n, alpha, x) = ((- 1)^(n))/((2)^(n)* factorial(n))*(Gamma(alpha +(1)/(2))* Gamma(n + 2*alpha))/(Gamma(2*alpha)* Gamma(alpha + n +(1)/(2)))*(1 - (x)^(2))^(- alpha + 1/2)* diff((1 - (x)^(2))^(n + alpha - 1/2), [x$(n)])

Free variables

Gamma

alpha

n

x

Symbol info

Ultraspherical Gegenbauer polynomial; Example: \ultrasphpoly{\lambda}{n}@{x}

Will be translated to: GegenbauerC($1, $0, $2) Relevant links to definitions: DLMF: http://dlmf.nist.gov/18.3#T1.t1.r3 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=GegenbauerC

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

Could be the second Feigenbaum constant.

But this system doesn't know how to translate it as a constant. It was translated as a general letter.

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$C_{n}^{(\alpha )}(x)$

$(1-x^{2})^{\alpha -1/2}$

$\alpha$

$n$

$\mathbf {R} ^{n}$

Description

Rodrigues formula

Complete translation information:

{
  "id" : "FORMULA_e99ad97ac29599b3a2d06d92cd1f0d85",
  "formula" : "C_n^{(\\alpha)}(x) = \\frac{(-1)^n}{2^n n!}\\frac{\\Gamma(\\alpha+\\frac{1}{2})\\Gamma(n+2\\alpha)}{\\Gamma(2\\alpha)\\Gamma(\\alpha+n+\\frac{1}{2})}(1-x^2)^{-\\alpha+1/2}\\frac{d^n}{dx^n}\\left[(1-x^2)^{n+\\alpha-1/2}\\right]",
  "semanticFormula" : "\\ultrasphpoly{\\alpha}{n}@{x} = \\frac{(-1)^n}{2^n n!} \\frac{\\Gamma(\\alpha+\\frac{1}{2})\\Gamma(n+2\\alpha)}{\\Gamma(2\\alpha)\\Gamma(\\alpha+n+\\frac{1}{2})}(1 - x^2)^{-\\alpha+1/2} \\deriv [n]{ }{x} [(1 - x^2)^{n+\\alpha-1/2}]",
  "confidence" : 0.6805,
  "translations" : {
    "Mathematica" : {
      "translation" : "GegenbauerC[n, \\[Alpha], x] == Divide[(- 1)^(n),(2)^(n)* (n)!]*Divide[\\[CapitalGamma][\\[Alpha]+Divide[1,2]]* \\[CapitalGamma][n + 2*\\[Alpha]],\\[CapitalGamma][2*\\[Alpha]]* \\[CapitalGamma][\\[Alpha]+ n +Divide[1,2]]]*(1 - (x)^(2))^(- \\[Alpha]+ 1/2)* D[(1 - (x)^(2))^(n + \\[Alpha]- 1/2), {x, n}]",
      "translationInformation" : {
        "subEquations" : [ "GegenbauerC[n, \\[Alpha], x] = Divide[(- 1)^(n),(2)^(n)* (n)!]*Divide[\\[CapitalGamma][\\[Alpha]+Divide[1,2]]* \\[CapitalGamma][n + 2*\\[Alpha]],\\[CapitalGamma][2*\\[Alpha]]* \\[CapitalGamma][\\[Alpha]+ n +Divide[1,2]]]*(1 - (x)^(2))^(- \\[Alpha]+ 1/2)* D[(1 - (x)^(2))^(n + \\[Alpha]- 1/2), {x, n}]" ],
        "freeVariables" : [ "\\[Alpha]", "\\[CapitalGamma]", "n", "x" ],
        "tokenTranslations" : {
          "\\ultrasphpoly" : "Ultraspherical Gegenbauer polynomial; Example: \\ultrasphpoly{\\lambda}{n}@{x}\nWill be translated to: GegenbauerC[$1, $0, $2]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/18.3#T1.t1.r3\nMathematica:  https://reference.wolfram.com/language/ref/GegenbauerC.html",
          "\\deriv1" : "Derivative; Example: \\deriv[n]{f}{x}\nWill be translated to: D[$1, {$2, $0}]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/1.4#E4\nMathematica:  https://reference.wolfram.com/language/ref/D.html",
          "\\alpha" : "Could be the second Feigenbaum constant.\nBut this system doesn't know how to translate it as a constant. It was translated as a general letter.\n",
          "\\Gamma" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "ERROR",
        "numberOfTests" : 1,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 1,
        "crashed" : false,
        "testCalculationsGroup" : [ {
          "lhs" : "GegenbauerC[n, \\[Alpha], x]",
          "rhs" : "Divide[(- 1)^(n),(2)^(n)* (n)!]*Divide[\\[CapitalGamma]*(\\[Alpha]+Divide[1,2])*\\[CapitalGamma]*(n + 2*\\[Alpha]),\\[CapitalGamma]*(2*\\[Alpha])*\\[CapitalGamma]*(\\[Alpha]+ n +Divide[1,2])]*(1 - (x)^(2))^(- \\[Alpha]+ 1/2)* D[(1 - (x)^(2))^(n + \\[Alpha]- 1/2), {x, n}]",
          "testExpression" : "(GegenbauerC[n, \\[Alpha], x])-(Divide[(- 1)^(n),(2)^(n)* (n)!]*Divide[\\[CapitalGamma]*(\\[Alpha]+Divide[1,2])*\\[CapitalGamma]*(n + 2*\\[Alpha]),\\[CapitalGamma]*(2*\\[Alpha])*\\[CapitalGamma]*(\\[Alpha]+ n +Divide[1,2])]*(1 - (x)^(2))^(- \\[Alpha]+ 1/2)* D[(1 - (x)^(2))^(n + \\[Alpha]- 1/2), {x, n}])",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        } ]
      }
    },
    "SymPy" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \\ultrasphpoly [\\ultrasphpoly]"
        }
      }
    },
    "Maple" : {
      "translation" : "GegenbauerC(n, alpha, x) = ((- 1)^(n))/((2)^(n)* factorial(n))*(Gamma(alpha +(1)/(2))* Gamma(n + 2*alpha))/(Gamma(2*alpha)* Gamma(alpha + n +(1)/(2)))*(1 - (x)^(2))^(- alpha + 1/2)* diff((1 - (x)^(2))^(n + alpha - 1/2), [x$(n)])",
      "translationInformation" : {
        "subEquations" : [ "GegenbauerC(n, alpha, x) = ((- 1)^(n))/((2)^(n)* factorial(n))*(Gamma(alpha +(1)/(2))* Gamma(n + 2*alpha))/(Gamma(2*alpha)* Gamma(alpha + n +(1)/(2)))*(1 - (x)^(2))^(- alpha + 1/2)* diff((1 - (x)^(2))^(n + alpha - 1/2), [x$(n)])" ],
        "freeVariables" : [ "Gamma", "alpha", "n", "x" ],
        "tokenTranslations" : {
          "\\ultrasphpoly" : "Ultraspherical Gegenbauer polynomial; Example: \\ultrasphpoly{\\lambda}{n}@{x}\nWill be translated to: GegenbauerC($1, $0, $2)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/18.3#T1.t1.r3\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=GegenbauerC",
          "\\deriv1" : "Derivative; Example: \\deriv[n]{f}{x}\nWill be translated to: diff($1, [$2$($0)])\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/1.4#E4\nMaple: https://www.maplesoft.com/support/help/Maple/view.aspx?path=diff",
          "\\alpha" : "Could be the second Feigenbaum constant.\nBut this system doesn't know how to translate it as a constant. It was translated as a general letter.\n",
          "\\Gamma" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      }
    }
  },
  "positions" : [ {
    "section" : 1,
    "sentence" : 11,
    "word" : 7
  } ],
  "includes" : [ "C_{n}^{(\\alpha)}(x)", "(1 -x^{2})^{\\alpha-1/2}", "\\alpha", "n", "\\mathbf{R}^{n}" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "Rodrigues formula",
    "score" : 0.722
  } ]
}

Specify your own input

Formula input
Wikitext context	{{Use American English\|date = March 2019}} {{Short description\|Polynomial sequence}} In [[mathematics]], '''Gegenbauer polynomials''' or '''ultraspherical polynomials''' ''C''{{su\|p=(α)\|b=''n''}}(''x'') are [[orthogonal polynomials]] on the interval [−1,1] with respect to the [[weight function]] (1 − ''x''<sup>2</sup>)<sup>''α''–1/2</sup>. They generalize [[Legendre polynomials]] and [[Chebyshev polynomials]], and are special cases of [[Jacobi polynomials]]. They are named after [[Leopold Gegenbauer]]. ==Characterizations== <gallery widths="300" heights="200" class="float-right"> Mplwp gegenbauer Cn05a1.svg\|Gegenbauer polynomials with ''α''=1 Mplwp gegenbauer Cn05a2.svg\|Gegenbauer polynomials with ''α''=2 Mplwp gegenbauer Cn05a3.svg\|Gegenbauer polynomials with ''α''=3 Gegenbauer polynomials.gif\|An animation showing the polynomials on the ''xα''-plane for the first 4 values of ''n''. </gallery> A variety of characterizations of the Gegenbauer polynomials are available. * The polynomials can be defined in terms of their [[generating function]] {{harv\|Stein\|Weiss\|1971\|loc=§IV.2}}: ::<math>\frac{1}{(1-2xt+t^2)^\alpha}=\sum_{n=0}^\infty C_n^{(\alpha)}(x) t^n.</math> * The polynomials satisfy the [[recurrence relation]] {{harv\|Suetin\|2001}}: ::<math> \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} </math> * Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation {{harv\|Suetin\|2001}}: ::<math>(1-x^{2})y''-(2\alpha+1)xy'+n(n+2\alpha)y=0.\,</math> :When ''α'' = 1/2, the equation reduces to the Legendre equation, and the Gegenbauer polynomials reduce to the [[Legendre polynomials]]. :When ''α'' = 1, the equation reduces to the Chebyshev differential equation, and the Gegenbauer polynomials reduce to the [[Chebyshev polynomials]] of the second kind.<ref>Arfken, Weber, and Harris (2013) "Mathematical Methods for Physicists", 7th edition; ch. 18.4</ref> * They are given as [[Gaussian hypergeometric series]] in certain cases where the series is in fact finite: ::<math>C_n^{(\alpha)}(z)=\frac{(2\alpha)_n}{n!} \,_2F_1\left(-n,2\alpha+n;\alpha+\frac{1}{2};\frac{1-z}{2}\right).</math> :(Abramowitz & Stegun [http://www.math.sfu.ca/~cbm/aands/page_561.htm p. 561]). Here (2α)<sub>''n''</sub> is the [[rising factorial]]. Explicitly, ::<math> C_n^{(\alpha)}(z)=\sum_{k=0}^{\lfloor n/2\rfloor} (-1)^k\frac{\Gamma(n-k+\alpha)}{\Gamma(\alpha)k!(n-2k)!}(2z)^{n-2k}. </math> * They are special cases of the [[Jacobi polynomials]] {{harv\|Suetin\|2001}}: ::<math>C_n^{(\alpha)}(x) = \frac{(2\alpha)_n}{(\alpha+\frac{1}{2})_{n}}P_n^{(\alpha-1/2,\alpha-1/2)}(x).</math> :in which <math>(\theta)_n</math> represents the [[rising factorial]] of <math>\theta</math>. :One therefore also has the [[Rodrigues formula]] ::<math>C_n^{(\alpha)}(x) = \frac{(-1)^n}{2^n n!}\frac{\Gamma(\alpha+\frac{1}{2})\Gamma(n+2\alpha)}{\Gamma(2\alpha)\Gamma(\alpha+n+\frac{1}{2})}(1-x^2)^{-\alpha+1/2}\frac{d^n}{dx^n}\left[(1-x^2)^{n+\alpha-1/2}\right].</math> ==Orthogonality and normalization== For a fixed ''α'', the polynomials are orthogonal on [−1, 1] with respect to the weighting function (Abramowitz & Stegun [http://www.math.sfu.ca/~cbm/aands/page_774.htm p. 774]) :<math> w(z) = \left(1-z^2\right)^{\alpha-\frac{1}{2}}.</math> To wit, for ''n'' ≠ ''m'', :<math>\int_{-1}^1 C_n^{(\alpha)}(x)C_m^{(\alpha)}(x)(1-x^2)^{\alpha-\frac{1}{2}}\,dx = 0.</math> They are normalized by :<math>\int_{-1}^1 \left[C_n^{(\alpha)}(x)\right]^2(1-x^2)^{\alpha-\frac{1}{2}}\,dx = \frac{\pi 2^{1-2\alpha}\Gamma(n+2\alpha)}{n!(n+\alpha)[\Gamma(\alpha)]^2}.</math> ==Applications== The Gegenbauer polynomials appear naturally as extensions of Legendre polynomials in the context of [[potential theory]] and [[harmonic analysis]]. The [[Newtonian potential]] in '''R'''<sup>''n''</sup> has the expansion, valid with α = (''n'' − 2)/2, :<math>\frac{1}{\|\mathbf{x}-\mathbf{y}\|^{n-2}} = \sum_{k=0}^\infty \frac{\|\mathbf{x}\|^k}{\|\mathbf{y}\|^{k+n-2}}C_k^{(\alpha)}(\mathbf{x}\cdot \mathbf{y}).</math> When ''n'' = 3, this gives the Legendre polynomial expansion of the [[gravitational potential]]. Similar expressions are available for the expansion of the [[Poisson kernel]] in a ball {{harv\|Stein\|Weiss\|1971}}. It follows that the quantities <math>C^{((n-2)/2)}_k(\mathbf{x}\cdot\mathbf{y})</math> are [[spherical harmonics]], when regarded as a function of '''x''' only. They are, in fact, exactly the [[zonal spherical harmonic]]s, up to a normalizing constant. Gegenbauer polynomials also appear in the theory of [[Positive-definite function]]s. The [[Askey–Gasper inequality]] reads :<math>\sum_{j=0}^n\frac{C_j^\alpha(x)}{{2\alpha+j-1\choose j}}\ge 0\qquad (x\ge-1,\, \alpha\ge 1/4).</math> == See also== * [[Rogers polynomials]], the ''q''-analogue of Gegenbauer polynomials * [[Chebyshev polynomials]] * [[Romanovski polynomials]] == References == * {{Abramowitz_Stegun_ref\|22\|773}}{{dlmf\|id=18\|title=Orthogonal Polynomials\|first=Tom H. \|last=Koornwinder\|first2=Roderick S. C.\|last2= Wong\|first3=Roelof \|last3=Koekoek\|\|first4=René F. \|last4=Swarttouw}} {{citation\|last1=Stein\|first1=Elias\|authorlink1=Elias Stein\|first2=Guido\|last2=Weiss\|authorlink2=Guido Weiss\|title=Introduction to Fourier Analysis on Euclidean Spaces\|publisher=Princeton University Press\|year=1971\|isbn=978-0-691-08078-9\|location=Princeton, N.J.\|url-access=registration\|url=https://archive.org/details/introductiontofo0000stei}}. * {{springer\|title=Ultraspherical polynomials\|id=U/u095030\|first=P.K.\|last=Suetin}}. ;Specific <references /> [[Category:Orthogonal polynomials]] [[Category:Special hypergeometric functions]]
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LaTeX to CAS translator

Mathematica

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SymPy

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Maple

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