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 \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} }

${\displaystyle {\begin{aligned}C_{0}^{\alpha }(x)&=1\\C_{1}^{\alpha }(x)&=2\alpha x\\C_{n}^{\alpha }(x)&={\frac {1}{n}}[2x(n+\alpha -1)C_{n-1}^{\alpha }(x)-(n+2\alpha -2)C_{n-2}^{\alpha }(x)].\end{aligned}}}$

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

Semantic latex: \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}

Confidence: 0

Mathematica

Translation: (Subscript[C, 0])^\[Alpha][x] == 1 (Subscript[C, 1])^\[Alpha][x] == 2*\[Alpha]*x (Subscript[C, n])^\[Alpha][x] == Divide[1,n]*(2*x*(n + \[Alpha]- 1)*(Subscript[C, n - 1])^\[Alpha][x]-(n + 2*\[Alpha]- 2)*(Subscript[C, n - 2])^\[Alpha][x])

Information

Sub Equations

(Subscript[C, 0])^\[Alpha][x] = 1

(Subscript[C, 1])^\[Alpha][x] = 2*\[Alpha]*x

(Subscript[C, n])^\[Alpha][x] = Divide[1,n]*(2*x*(n + \[Alpha]- 1)*(Subscript[C, n - 1])^\[Alpha][x]-(n + 2*\[Alpha]- 2)*(Subscript[C, n - 2])^\[Alpha][x])

Free variables

\[Alpha]

n

x

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: (Symbol('{C}_{0}'))**(Symbol('alpha'))(x) == 1 (Symbol('{C}_{1}'))**(Symbol('alpha'))(x) == 2*Symbol('alpha')*x (Symbol('{C}_{n}'))**(Symbol('alpha'))(x) == (1)/(n)*(2*x*(n + Symbol('alpha')- 1)*(Symbol('{C}_{n - 1}'))**(Symbol('alpha'))(x)-(n + 2*Symbol('alpha')- 2)*(Symbol('{C}_{n - 2}'))**(Symbol('alpha'))(x))

Information

Sub Equations

(Symbol('{C}_{0}'))**(Symbol('alpha'))(x) = 1

(Symbol('{C}_{1}'))**(Symbol('alpha'))(x) = 2*Symbol('alpha')*x

(Symbol('{C}_{n}'))**(Symbol('alpha'))(x) = (1)/(n)*(2*x*(n + Symbol('alpha')- 1)*(Symbol('{C}_{n - 1}'))**(Symbol('alpha'))(x)-(n + 2*Symbol('alpha')- 2)*(Symbol('{C}_{n - 2}'))**(Symbol('alpha'))(x))

Free variables

Symbol('alpha')

n

x

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: (C[0])^(alpha)(x) = 1; (C[1])^(alpha)(x) = 2*alpha*x; (C[n])^(alpha)(x) = (1)/(n)*(2*x*(n + alpha - 1)*(C[n - 1])^(alpha)(x)-(n + 2*alpha - 2)*(C[n - 2])^(alpha)(x))

Information

Sub Equations

(C[0])^(alpha)(x) = 1

(C[1])^(alpha)(x) = 2*alpha*x

(C[n])^(alpha)(x) = (1)/(n)*(2*x*(n + alpha - 1)*(C[n - 1])^(alpha)(x)-(n + 2*alpha - 2)*(C[n - 2])^(alpha)(x))

Free variables

alpha

n

x

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

$\alpha$

$n$

${\begin{aligned}C_{0}^{\alpha }(x)&=1\\C_{1}^{\alpha }(x)&=2\alpha x\\C_{n}^{\alpha }(x)&={\frac {1}{n}}[2x(n+\alpha -1)C_{n-1}^{\alpha }(x)-(n+2\alpha -2)C_{n-2}^{\alpha }(x)].\end{aligned}}$

$=2$

$=1$

Is part of

${\begin{aligned}C_{0}^{\alpha }(x)&=1\\C_{1}^{\alpha }(x)&=2\alpha x\\C_{n}^{\alpha }(x)&={\frac {1}{n}}[2x(n+\alpha -1)C_{n-1}^{\alpha }(x)-(n+2\alpha -2)C_{n-2}^{\alpha }(x)].\end{aligned}}$

Complete translation information:

{
  "id" : "FORMULA_0b7f6b1e6d5c1f0d12a9de0b565bddd3",
  "formula" : "\\begin{align}\nC_0^\\alpha(x) & = 1 \\\\\nC_1^\\alpha(x) & = 2 \\alpha x \\\\\nC_n^\\alpha(x) & = \\frac{1}{n}[2x(n+\\alpha-1)C_{n-1}^\\alpha(x) - (n+2\\alpha-2)C_{n-2}^\\alpha(x)].\n\\end{align}",
  "semanticFormula" : "\\begin{align}\nC_0^\\alpha(x) & = 1 \\\\\nC_1^\\alpha(x) & = 2 \\alpha x \\\\\nC_n^\\alpha(x) & = \\frac{1}{n}[2x(n+\\alpha-1)C_{n-1}^\\alpha(x) - (n+2\\alpha-2)C_{n-2}^\\alpha(x)].\n\\end{align}",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "(Subscript[C, 0])^\\[Alpha][x] == 1\n(Subscript[C, 1])^\\[Alpha][x] == 2*\\[Alpha]*x\n(Subscript[C, n])^\\[Alpha][x] == Divide[1,n]*(2*x*(n + \\[Alpha]- 1)*(Subscript[C, n - 1])^\\[Alpha][x]-(n + 2*\\[Alpha]- 2)*(Subscript[C, n - 2])^\\[Alpha][x])",
      "translationInformation" : {
        "subEquations" : [ "(Subscript[C, 0])^\\[Alpha][x] = 1", "(Subscript[C, 1])^\\[Alpha][x] = 2*\\[Alpha]*x", "(Subscript[C, n])^\\[Alpha][x] = Divide[1,n]*(2*x*(n + \\[Alpha]- 1)*(Subscript[C, n - 1])^\\[Alpha][x]-(n + 2*\\[Alpha]- 2)*(Subscript[C, n - 2])^\\[Alpha][x])" ],
        "freeVariables" : [ "\\[Alpha]", "n", "x" ],
        "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" : "(Symbol('{C}_{0}'))**(Symbol('alpha'))(x) == 1\n(Symbol('{C}_{1}'))**(Symbol('alpha'))(x) == 2*Symbol('alpha')*x\n(Symbol('{C}_{n}'))**(Symbol('alpha'))(x) == (1)/(n)*(2*x*(n + Symbol('alpha')- 1)*(Symbol('{C}_{n - 1}'))**(Symbol('alpha'))(x)-(n + 2*Symbol('alpha')- 2)*(Symbol('{C}_{n - 2}'))**(Symbol('alpha'))(x))",
      "translationInformation" : {
        "subEquations" : [ "(Symbol('{C}_{0}'))**(Symbol('alpha'))(x) = 1", "(Symbol('{C}_{1}'))**(Symbol('alpha'))(x) = 2*Symbol('alpha')*x", "(Symbol('{C}_{n}'))**(Symbol('alpha'))(x) = (1)/(n)*(2*x*(n + Symbol('alpha')- 1)*(Symbol('{C}_{n - 1}'))**(Symbol('alpha'))(x)-(n + 2*Symbol('alpha')- 2)*(Symbol('{C}_{n - 2}'))**(Symbol('alpha'))(x))" ],
        "freeVariables" : [ "Symbol('alpha')", "n", "x" ],
        "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" : "(C[0])^(alpha)(x) = 1; (C[1])^(alpha)(x) = 2*alpha*x; (C[n])^(alpha)(x) = (1)/(n)*(2*x*(n + alpha - 1)*(C[n - 1])^(alpha)(x)-(n + 2*alpha - 2)*(C[n - 2])^(alpha)(x))",
      "translationInformation" : {
        "subEquations" : [ "(C[0])^(alpha)(x) = 1", "(C[1])^(alpha)(x) = 2*alpha*x", "(C[n])^(alpha)(x) = (1)/(n)*(2*x*(n + alpha - 1)*(C[n - 1])^(alpha)(x)-(n + 2*alpha - 2)*(C[n - 2])^(alpha)(x))" ],
        "freeVariables" : [ "alpha", "n", "x" ],
        "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" : [ "\\alpha", "n", "\\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}", "=2", "=1" ],
  "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" : [ ]
}

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

Information

Tests

Symbolic

Numeric

SymPy

Information

Tests

Symbolic

Numeric

Maple

Information

Tests

Symbolic

Numeric

Dependency Graph Information

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