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 \int_{-\infty}^{\infty}P_n^{(\lambda)}(x;\phi)P_m^{(\lambda)}(x;\phi)w(x; \lambda, \phi)dx=\frac{2\pi\Gamma(n+2\lambda)}{(2\sin\phi)^{2\lambda}n!}\delta_{mn},\quad \lambda>0,\quad 0<\phi<\pi.}

${\displaystyle \int _{-\infty }^{\infty }P_{n}^{(\lambda )}(x;\phi )P_{m}^{(\lambda )}(x;\phi )w(x;\lambda ,\phi )dx={\frac {2\pi \Gamma (n+2\lambda )}{(2\sin \phi )^{2\lambda }n!}}\delta _{mn},\quad \lambda >0,\quad 0<\phi <\pi .}$

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

Semantic latex: \int_{-\infty}^{\infty} \MeixnerPollaczekpolyP{\lambda}{n}@{x}{\phi} \MeixnerPollaczekpolyP{\lambda}{m}@{x}{\phi} w(x ; \lambda , \phi) \diff{x} = \frac{2 \cpi \Gamma(n + 2 \lambda)}{(2\sin\phi)^{2\lambda}n!} \delta_{mn} , \quad \lambda > 0 , \quad 0 < \phi < \cpi

Confidence: 0.8820248930976

Mathematica

Translation:

Information

Symbol info

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

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

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

Tests

Symbolic

Numeric

Maple

Translation:

Information

Symbol info

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

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

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

$w(x;\lambda ,\varphi )$

Description

orthogonality relation

Meixner -- Pollaczek polynomial

real line with respect

weight function

Complete translation information:

{
  "id" : "FORMULA_8eabb8bac3299d20ac4433f08df82dbf",
  "formula" : "\\int_{-\\infty}^{\\infty}P_n^{(\\lambda)}(x;\\phi)P_m^{(\\lambda)}(x;\\phi)w(x; \\lambda, \\phi)dx=\\frac{2\\pi\\Gamma(n+2\\lambda)}{(2\\sin\\phi)^{2\\lambda}n!}\\delta_{mn},\\quad \\lambda>0,\\quad 0<\\phi<\\pi",
  "semanticFormula" : "\\int_{-\\infty}^{\\infty} \\MeixnerPollaczekpolyP{\\lambda}{n}@{x}{\\phi} \\MeixnerPollaczekpolyP{\\lambda}{m}@{x}{\\phi} w(x ; \\lambda , \\phi) \\diff{x} = \\frac{2 \\cpi \\Gamma(n + 2 \\lambda)}{(2\\sin\\phi)^{2\\lambda}n!} \\delta_{mn} , \\quad \\lambda > 0 , \\quad 0 < \\phi < \\cpi",
  "confidence" : 0.8820248930976001,
  "translations" : {
    "Mathematica" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> Mathematica) No translation possible for given token: Cannot extract information from feature set: \\MeixnerPollaczekpolyP [\\MeixnerPollaczekpolyP]"
        }
      },
      "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: \\MeixnerPollaczekpolyP [\\MeixnerPollaczekpolyP]"
        }
      },
      "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: \\MeixnerPollaczekpolyP [\\MeixnerPollaczekpolyP]"
        }
      },
      "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" : 2,
    "sentence" : 0,
    "word" : 26
  } ],
  "includes" : [ "P_{m}^{(\\lambda)}(x;\\varphi)", "w(x;\\lambda,\\varphi)" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "orthogonality relation",
    "score" : 0.6859086196238077
  }, {
    "definition" : "Meixner -- Pollaczek polynomial",
    "score" : 0.6460746792928004
  }, {
    "definition" : "real line with respect",
    "score" : 0.6460746792928004
  }, {
    "definition" : "weight function",
    "score" : 0.6460746792928004
  } ]
}

Specify your own input

Formula input
Wikitext context	{{distinguish\|Meixner polynomials}} {{More citations needed\|date=March 2016}} In mathematics, the '''Meixner–Pollaczek polynomials''' are a family of [[orthogonal polynomials]] ''P''{{su\|b=''n''\|p=(λ)}}(''x'',φ) introduced by {{harvs\|txt\|authorlink=Josef Meixner\|last=Meixner\|year=1934}}, which up to elementary changes of variables are the same as the '''Pollaczek polynomials''' ''P''{{su\|b=''n''\|p=λ}}(''x'',''a'',''b'') rediscovered by {{harvs\|txt\|authorlink=Felix Pollaczek\|last=Pollaczek\|year=1949}} in the case λ=1/2, and later generalized by him. They are defined by :<math>P_n^{(\lambda)}(x;\phi) = \frac{(2\lambda)_n}{n!}e^{in\phi}{}_2F_1\left(\begin{array}{c} -n,~\lambda+ix\\ 2\lambda \end{array}; 1-e^{-2i\phi}\right)</math> :<math>P_n^{\lambda}(\cos \phi;a,b) = \frac{(2\lambda)_n}{n!}e^{in\phi}{}_2F_1\left(\begin{array}{c}-n,~\lambda+i(a\cos \phi+b)/\sin \phi\\ 2\lambda \end{array};1-e^{-2i\phi}\right)</math> ==Examples== The first few Meixner–Pollaczek polynomials are :<math>P_0^{(\lambda)}(x;\phi)=1</math> :<math>P_1^{(\lambda)}(x;\phi)=2(\lambda\cos\phi + x\sin\phi)</math> :<math>P_2^{(\lambda)}(x;\phi)=x^2+\lambda^2+(\lambda^2+\lambda-x^2)\cos(2\phi)+(1+2\lambda)x\sin(2\phi).</math> ==Properties== ===Orthogonality=== The Meixner–Pollaczek polynomials ''P''<sub>m</sub><sup>(λ)</sup>(''x'';φ) are orthogonal on the real line with respect to the weight function :<math> w(x; \lambda, \phi)= \|\Gamma(\lambda+ix)\|^2 e^{(2\phi-\pi)x}</math> and the orthogonality relation is given by<ref>Koekoek, Lesky, & Swarttouw (2010), p. 213.</ref> :<math>\int_{-\infty}^{\infty}P_n^{(\lambda)}(x;\phi)P_m^{(\lambda)}(x;\phi)w(x; \lambda, \phi)dx=\frac{2\pi\Gamma(n+2\lambda)}{(2\sin\phi)^{2\lambda}n!}\delta_{mn},\quad \lambda>0,\quad 0<\phi<\pi.</math> ===Recurrence relation=== The sequence of Meixner–Pollaczek polynomials satisfies the recurrence relation<ref>Koekoek, Lesky, & Swarttouw (2010), p. 213.</ref> :<math>(n+1)P_{n+1}^{(\lambda)}(x;\phi)=2\bigl(x\sin\phi + (n+\lambda)\cos\phi\bigr)P_n^{(\lambda)}(x;\phi)-(n+2\lambda-1)P_{n-1}(x;\phi).</math> ===Rodrigues formula=== The Meixner–Pollaczek polynomials are given by the Rodrigues-like formula<ref>Koekoek, Lesky, & Swarttouw (2010), p. 214.</ref> :<math>P_n^{(\lambda)}(x;\phi)=\frac{(-1)^n}{n!\,w(x;\lambda,\phi)}\frac{d^n}{dx^n}w\left(x;\lambda+\tfrac12n,\phi\right),</math> where ''w''(''x'';λ,φ) is the weight function given above. ===Generating function=== The Meixner–Pollaczek polynomials have the generating function<ref>Koekoek, Lesky, & Swarttouw (2010), p. 215.</ref> :<math>\sum_{n=0}^{\infty}t^n P_n^{(\lambda)}(x;\phi) = (1-e^{i\phi}t)^{-\lambda+ix}(1-e^{-i\phi}t)^{-\lambda-ix}.</math> ==See also== [[Sieved Pollaczek polynomials]] ==References== {{Reflist}} {{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=18.35\|title=Pollaczek Polynomials\|first=Tom H. \|last=Koornwinder\|first2=Roderick S. C.\|last2= Wong\|first3=Roelof \|last3=Koekoek\|\|first4=René F. \|last4=Swarttouw}} {{Citation \| last1=Meixner \| first1=J. \| title=Orthogonale Polynomsysteme Mit Einer Besonderen Gestalt Der Erzeugenden Funktion \| doi=10.1112/jlms/s1-9.1.6 \| year=1934 \| journal=J. London Math. Soc. \| volume=s1-9 \| pages=6–13}} *{{Citation \| last1=Pollaczek \| first1=Félix \| title=Sur une généralisation des polynomes de Legendre \| url=http://gallica.bnf.fr/ark:/12148/bpt6k31801/f1363 \| mr=0030037 \| year=1949 \| journal=[[Les Comptes rendus de l'Académie des sciences]] \| volume=228 \| pages=1363–1365}} {{DEFAULTSORT:Meixner-Pollaczek polynomials}} [[Category:Orthogonal polynomials]]
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LaTeX to CAS translator

Mathematica

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Symbolic

Numeric

SymPy

Information

Tests

Symbolic

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Maple

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Tests

Symbolic

Numeric

Dependency Graph Information

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