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 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).}

${\displaystyle P_{n}^{(\alpha ,\beta )}=\lim _{t\to \infty }t^{-n}p_{n}\left({\tfrac {1}{2}}xt;{\tfrac {1}{2}}(\alpha +1-it),{\tfrac {1}{2}}(\beta +1+it),{\tfrac {1}{2}}(\alpha +1+it),{\tfrac {1}{2}}(\beta +1-it)\right).}$

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

Semantic latex: P_n^{(\alpha,\beta)} = \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: 0.90419950349709

Mathematica

Translation: (Subscript[P, n])^(\[Alpha], \[Beta]) == 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]

Information

Sub Equations

(Subscript[P, n])^(\[Alpha], \[Beta]) = 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]

Free variables

Subscript[P, n]

\[Alpha]

\[Beta]

n

x

Symbol info

Continuous Hahn polynomial; Example: \contHahnpolyp{n}@{x}{a}{b}{\conj{a}}{\conj{b}}

Will be translated to: Alternative translations: [I^($0)*Divide[Pochhammer[$2 + $4, $0]*Pochhammer[$2 + $5, $0], ($0)!] * HypergeometricPFQ[{-($0), $0 + 2*Re[$2 + $3] - 1, $2 + I*($1)}, {$2 + $4, $2 + $5}, 1]]Relevant links to definitions: DLMF: http://dlmf.nist.gov/18.19#P2.p1 Mathematica:

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.

Imaginary unit was translated to: I

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

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

Tests

Symbolic

Numeric

Maple

Translation:

Information

Symbol info

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

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

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

$F_{n}$

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

Description

case of the continuous Hahn polynomial

Jacobi polynomial

x

Complete translation information:

{
  "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" : "P_n^{(\\alpha,\\beta)} = \\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" : "(Subscript[P, n])^(\\[Alpha], \\[Beta]) == 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]",
      "translationInformation" : {
        "subEquations" : [ "(Subscript[P, n])^(\\[Alpha], \\[Beta]) = 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]" ],
        "freeVariables" : [ "Subscript[P, n]", "\\[Alpha]", "\\[Beta]", "n", "x" ],
        "tokenTranslations" : {
          "\\contHahnpolyp" : "Continuous Hahn polynomial; Example: \\contHahnpolyp{n}@{x}{a}{b}{\\conj{a}}{\\conj{b}}\nWill be translated to: \nAlternative translations: [I^($0)*Divide[Pochhammer[$2 + $4, $0]*Pochhammer[$2 + $5, $0], ($0)!] * HypergeometricPFQ[{-($0), $0 + 2*Re[$2 + $3] - 1, $2 + I*($1)}, {$2 + $4, $2 + $5}, 1]]Relevant links to definitions:\nDLMF:         http://dlmf.nist.gov/18.19#P2.p1\nMathematica:  ",
          "\\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",
          "\\iunit" : "Imaginary unit was translated to: I"
        }
      },
      "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" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \\contHahnpolyp [\\contHahnpolyp]"
        }
      },
      "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" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> Maple) No translation possible for given token: Cannot extract information from feature set: \\contHahnpolyp [\\contHahnpolyp]"
        }
      },
      "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" : [ {
    "section" : 5,
    "sentence" : 2,
    "word" : 20
  } ],
  "includes" : [ "p_{n}(x;a,b,c,d)", "F_{n}", "P_{n}^{(\\alpha,\\beta)}" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "case of the continuous Hahn polynomial",
    "score" : 0.7125985104912714
  }, {
    "definition" : "Jacobi polynomial",
    "score" : 0.6460746792928004
  }, {
    "definition" : "x",
    "score" : 0.5988174995334326
  } ]
}

Specify your own input

Formula input
Wikitext context	In mathematics, the '''continuous Hahn polynomials''' are a family of [[orthogonal polynomials]] in the [[Askey scheme]] of hypergeometric orthogonal polynomials. They are defined in terms of [[generalized hypergeometric function]]s by :<math>p_n(x;a,b,c,d)= i^n\frac{(a+c)_n(a+d)_n}{n!}{}_3F_2\left( \begin{array}{c} -n, n+a+b+c+d-1, a+ix \\ a+c, a+d \end{array} ; 1\right)</math> {{harvs\|txt \| last1=Koekoek \| first1=Roelof \| last2=Lesky \| first2=Peter A. \| last3=Swarttouw \| first3=René F. \| title=Hypergeometric orthogonal polynomials and their q-analogues \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Springer Monographs in Mathematics \| isbn=978-3-642-05013-8 \| doi=10.1007/978-3-642-05014-5 \| mr=2656096 \| year=2010\|loc=14}} give a detailed list of their properties. Closely related polynomials include the [[dual Hahn polynomials]] ''R''<sub>''n''</sub>(''x'';γ,δ,''N''), the [[Hahn polynomials]] ''Q''<sub>''n''</sub>(''x'';''a'',''b'',''c''), and the [[continuous dual Hahn polynomials]] ''S''<sub>''n''</sub>(''x'';''a'',''b'',''c''). These polynomials all have ''q''-analogs with an extra parameter ''q'', such as the [[q-Hahn polynomials]] ''Q''<sub>''n''</sub>(''x'';α,β, ''N'';''q''), and so on. ==Orthogonality== The continuous Hahn polynomials ''p''<sub>''n''</sub>(''x'';''a'',''b'',''c'',''d'') are orthogonal with respect to the weight function :<math>w(x)=\Gamma(a+ix)\,\Gamma(b+ix)\,\Gamma(c-ix)\,\Gamma(d-ix).</math> In particular, they satisfy the orthogonality relation<ref>Koekoek, Lesky, & Swarttouw (2010), p. 200.</ref><ref>Askey, R. (1985), "Continuous Hahn polynomials", ''J. Phys. A: Math. Gen.'' '''18''': pp. L1017-L1019.</ref><ref>Andrews, Askey, & Roy (1999), p. 333.</ref> :<math>\begin{align}&\frac{1}{2\pi}\int_{-\infty}^{\infty}\Gamma(a+ix)\,\Gamma(b+ix)\,\Gamma(c-ix)\,\Gamma(d-ix)\,p_m(x;a,b,c,d)\,p_n(x;a,b,c,d)\,dx\\ &\qquad\qquad=\frac{\Gamma(n+a+c)\,\Gamma(n+a+d)\,\Gamma(n+b+c)\,\Gamma(n+b+d)}{n!(2n+a+b+c+d-1)\,\Gamma(n+a+b+c+d-1)}\,\delta_{n m}\end{align}</math> for <math>\Re(a)>0</math>, <math>\Re(b)>0</math>, <math>\Re(c)>0</math>, <math>\Re(d)>0</math>, <math>c = \overline{a}</math>, <math>d = \overline{b}</math>. ==Recurrence and difference relations== The sequence of continuous Hahn polynomials satisfies the recurrence relation<ref>Koekoek, Lesky, & Swarttouw (2010), p. 201.</ref> :<math>xp_n(x)=p_{n+1}(x)+i(A_n+C_n)p_{n}(x)-A_{n-1}C_n p_{n-1}(x),</math> :<math>\begin{align} \text{where}\quad&p_n(x)=\frac{n!(n+a+b+c+d-1)!}{(2n+a+b+c+d-1)!}p_n(x;a,b,c,d),\\ &A_n=-\frac{(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)},\\ \text{and}\quad&C_n=\frac{n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}. \end{align}</math> ==Rodrigues formula== The continuous Hahn polynomials are given by the Rodrigues-like formula<ref>Koekoek, Lesky, & Swarttouw (2010), p. 202.</ref> :<math>\begin{align}&\Gamma(a+ix)\,\Gamma(b+ix)\,\Gamma(c-ix)\,\Gamma(d-ix)\,p_n(x;a,b,c,d)\\ &\qquad=\frac{(-1)^n}{n!}\frac{d^n}{dx^n}\left(\Gamma\left(a+\frac{n}{2}+ix\right)\,\Gamma\left(b+\frac{n}{2}+ix\right)\,\Gamma\left(c+\frac{n}{2}-ix\right)\,\Gamma\left(d+\frac{n}{2}-ix\right)\right).\end{align}</math> ==Generating functions== The continuous Hahn polynomials have the following generating function:<ref>Koekoek, Lesky, & Swarttouw (2010), p. 202.</ref> :<math>\begin{align}&\sum_{n=0}^{\infty}\frac{\Gamma(n+a+b+c+d)\,\Gamma(a+c+1)\,\Gamma(a+d+1)}{\Gamma(a+b+c+d)\,\Gamma(n+a+c+1)\,\Gamma(n+a+d+1)}(-it)^n p_n(x;a,b,c,d)\\ &\qquad=(1-t)^{1-a-b-c-d}{}_3F_2\left( \begin{array}{c} \frac12(a+b+c+d-1), \frac12(a+b+c+d), a+ix\\ a+c, a+d\end{array} ; -\frac{4t}{(1-t)^2} \right).\end{align}</math> A second, distinct generating function is given by :<math>\sum_{n=0}^{\infty}\frac{\Gamma(a+c+1)\,\Gamma(b+d+1)}{\Gamma(n+a+c+1)\,\Gamma(n+b+d+1)}t^n p_n(x;a,b,c,d)=\,_1F_1\left( \begin{array}{c} a + ix \\ a + c\end{array} ; -it\right)\,_1F_1\left( \begin{array}{c} d - ix \\ b + d\end{array} ; it\right).</math> ==Relation to other polynomials== * The [[Wilson polynomials]] are a generalization of the continuous Hahn polynomials. * The [[Bateman polynomials]] ''F''<sub>''n''</sub>(x) are related to the special case ''a''=''b''=''c''=''d''=1/2 of the continuous Hahn polynomials by :<math>p_n\left(x;\tfrac12,\tfrac12,\tfrac12,\tfrac12\right) = i^n n!F_n\left(2ix\right).</math> * The [[Jacobi polynomials]] ''P''<sub>''n''</sub><sup>(α,β)</sup>(x) can be obtained as a limiting case of the continuous Hahn polynomials:<ref>Koekoek, Lesky, & Swarttouw (2010), p. 203.</ref> :<math>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).</math> ==References== {{Reflist}} {{Citation \| last1=Hahn \| first1=Wolfgang \| title=Über Orthogonalpolynome, die q-Differenzengleichungen genügen \| doi=10.1002/mana.19490020103 \| mr=0030647 \| year=1949 \| journal=Mathematische Nachrichten \| issn=0025-584X \| volume=2 \| pages=4–34}} {{Citation \| last1=Koekoek \| first1=Roelof \| last2=Lesky \| first2=Peter A. \| last3=Swarttouw \| first3=René F. \| title=Hypergeometric orthogonal polynomials and their q-analogues \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Springer Monographs in Mathematics \| isbn=978-3-642-05013-8 \| doi=10.1007/978-3-642-05014-5 \| mr=2656096 \| year=2010}} {{dlmf\|id=18.19\|title=Hahn Class: Definitions\|first=Tom H. \|last=Koornwinder\|first2=Roderick S. C.\|last2= Wong\|first3=Roelof \|last3=Koekoek\|\|first4=René F. \|last4=Swarttouw}} {{Citation \| last1=Andrews \| first1=George E. \| last2=Askey \| first2 = Richard \| last3=Roy \| first3=Ranjan \| title=Special functions \| publisher=[[Cambridge University Press]] \| location=Cambridge \| series=Encyclopedia of Mathematics and its Applications 71 \| isbn=978-0-521-62321-6 \| year=1999}} [[Category:Special hypergeometric functions]] [[Category:Orthogonal polynomials]]
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LaTeX to CAS translator

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