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 \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).}

${\displaystyle \sum _{j=0}^{n}{\frac {C_{j}^{\alpha }(x)}{2\alpha +j-1 \choose j}}\geq 0\qquad (x\geq -1,\,\alpha \geq 1/4).}$

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

Semantic latex: \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)

Confidence: 0

Mathematica

Translation: Sum[Divide[(Subscript[C, j])^\[Alpha][x],Binomial[2*\[Alpha]+ j - 1,j]], {j, 0, n}, GenerateConditions->None] >= 0

Information

Sub Equations

Sum[Divide[(Subscript[C, j])^\[Alpha][x],Binomial[2*\[Alpha]+ j - 1,j]], {j, 0, n}, GenerateConditions->None] >= 0

Free variables

\[Alpha]

n

x

Constraints

(x >= - 1 , \[Alpha] >= 1/4)

Symbol info

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

SymPy

Translation: Sum(((Symbol('{C}_{j}'))**(Symbol('alpha'))(x))/(binomial(2*Symbol('alpha')+ j - 1,j)), (j, 0, n)) >= 0

Information

Sub Equations

Sum(((Symbol('{C}_{j}'))**(Symbol('alpha'))(x))/(binomial(2*Symbol('alpha')+ j - 1,j)), (j, 0, n)) >= 0

Free variables

Symbol('alpha')

n

x

Constraints

(x >= - 1 , Symbol('alpha') >= 1/4)

Symbol info

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

Maple

Translation: sum(((C[j])^(alpha)(x))/(binomial(2*alpha + j - 1,j)), j = 0..n) >= 0

Information

Sub Equations

sum(((C[j])^(alpha)(x))/(binomial(2*alpha + j - 1,j)), j = 0..n) >= 0

Free variables

alpha

n

x

Constraints

(x >= - 1 , alpha >= 1/4)

Symbol info

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

$\sum _{j=0}^{n}{\frac {C_{j}^{\alpha }(x)}{2\alpha +j-1 \choose j}}\geq 0\qquad (x\geq -1,\,\alpha \geq 1/4)$

$\alpha$

$\mathbf {R} ^{n}$

$n$

Is part of

$\sum _{j=0}^{n}{\frac {C_{j}^{\alpha }(x)}{2\alpha +j-1 \choose j}}\geq 0\qquad (x\geq -1,\,\alpha \geq 1/4)$

Complete translation information:

{
  "id" : "FORMULA_bd3d9875c6263638d9c05af78041b61b",
  "formula" : "\\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)",
  "semanticFormula" : "\\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)",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "Sum[Divide[(Subscript[C, j])^\\[Alpha][x],Binomial[2*\\[Alpha]+ j - 1,j]], {j, 0, n}, GenerateConditions->None] >= 0",
      "translationInformation" : {
        "subEquations" : [ "Sum[Divide[(Subscript[C, j])^\\[Alpha][x],Binomial[2*\\[Alpha]+ j - 1,j]], {j, 0, n}, GenerateConditions->None] >= 0" ],
        "freeVariables" : [ "\\[Alpha]", "n", "x" ],
        "constraints" : [ "(x >= - 1 , \\[Alpha] >= 1/4)" ],
        "tokenTranslations" : {
          "\\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",
          "C" : "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" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "SymPy" : {
      "translation" : "Sum(((Symbol('{C}_{j}'))**(Symbol('alpha'))(x))/(binomial(2*Symbol('alpha')+ j - 1,j)), (j, 0, n)) >= 0",
      "translationInformation" : {
        "subEquations" : [ "Sum(((Symbol('{C}_{j}'))**(Symbol('alpha'))(x))/(binomial(2*Symbol('alpha')+ j - 1,j)), (j, 0, n)) >= 0" ],
        "freeVariables" : [ "Symbol('alpha')", "n", "x" ],
        "constraints" : [ "(x >= - 1 , Symbol('alpha') >= 1/4)" ],
        "tokenTranslations" : {
          "\\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",
          "C" : "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" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "Maple" : {
      "translation" : "sum(((C[j])^(alpha)(x))/(binomial(2*alpha + j - 1,j)), j = 0..n) >= 0",
      "translationInformation" : {
        "subEquations" : [ "sum(((C[j])^(alpha)(x))/(binomial(2*alpha + j - 1,j)), j = 0..n) >= 0" ],
        "freeVariables" : [ "alpha", "n", "x" ],
        "constraints" : [ "(x >= - 1 , alpha >= 1/4)" ],
        "tokenTranslations" : {
          "\\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",
          "C" : "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" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    }
  },
  "positions" : [ ],
  "includes" : [ "\\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)", "\\alpha", "\\mathbf{R}^{n}", "n" ],
  "isPartOf" : [ "\\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)" ],
  "definiens" : [ ]
}

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]]
	Purge cached results.

LaTeX to CAS translator

Mathematica

Information

Tests

Symbolic

Numeric

SymPy

Information

Tests

Symbolic

Numeric

Maple

Information

Tests

Symbolic

Numeric

Dependency Graph Information

Specify your own input

Navigation menu

Search