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} \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}}

${\displaystyle {\begin{aligned}{\text{where}}\quad &p_{n}(x)={\frac {n!(n+a+b+c+d-1)!}{(2n+a+b+c+d-1)!}}p_{n}(x;a,b,c,d),\\&A_{n}=-{\frac {(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}},\\{\text{and}}\quad &C_{n}={\frac {n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}}.\end{aligned}}}$

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

Semantic latex: \begin{align}where}\quad &p_n(x) = \frac{n!(n+a+b+c+d-1)!}{(2n+a+b+c+d-1)!} \contHahnpolyp{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)}, \\ 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}

Confidence: 0.71685960350077

Mathematica

Translation:

Information

Symbol info

(LaTeX -> Mathematica) CRITICAL! POM-Tagger Error Reached: Encountered " "}" "} "" at line 1, column 18.

Was expecting one of:

   <LEFTDELIM> ...
   <RIGHTDELIM> ...
   <LARGEDELIM> ...
   "\\begin" ...
   "\\end" ...
   "array" ...
   "\\array" ...
   "subarray" ...
   "tabular" ...
   <MATRIX> ...
   "\\matrix" ...
   "\\pmatrix" ...
   "bordermatrix" ...
   "\\bordermatrix" ...
   "equation" ...
   "multline" ...
   "align" ...
   "aligned" ...
   "gather" ...
   "gathered" ...
   "split" ...
   "CD" ...
   "picture" ...
   "pspicture" ...
   <AMPERSAND> ...
   "cases" ...
   "\\cases" ...
   <XYMATRIX> ...
   "\\eqalign" ...
   "\\eqalignno" ...
   "\\oalign" ...
   "\\ooalign" ...
   <LPAREN> ...
   <RPAREN> ...
   "{" ...
   <LBRACKET> ...
   <RBRACKET> ...
   <ESCAPED_LBRACE> ...
   <ESCAPED_RBRACE> ...
   "\\backslash" ...
   <ALPHANUM> ...
   <NUMERIC> ...
   <LETTER> ...
   <DIGIT> ...
   "`" ...
   "!" ...
   "@" ...
   "*" ...
   "^" ...
   "_" ...
   "-" ...
   "=" ...
   "+" ...
   "|" ...
   ";" ...
   <COLON> ...
   "\'" ...
   "\"" ...
   "," ...
   <LESS_THEN> ...
   "." ...
   <GREATER_THAN> ...
   "/" ...
   "?" ...
   "\\And" ...
   "\\#" ...
   "\\%" ...
   "\\$" ...
   "\\_" ...
   "\\&" ...
   <ESCAPED_SYMBOL_ALLOWED> ...
   <ESCAPED_SYMBOL_DISALLOWED> ...
   <FRAC> ...
   <BINOM> ...
   "\\stackrel" ...
   "\\overset" ...
   "\\underset" ...
   "\\sqrt" ...
   "\\MultiIntegral" ...
   <PHANTOM> ...
   "\\pmod" ...
   <MOD> ...
   "\\pod" ...
   "\\footnote" ...
   "\\not" ...
   <ELLIPSIS> ...
   "\\dots" ...
   <ACCENT> ...
   <MATH_AND_TEXT_ACCENT> ...
   <WIDEACCENT> ...
   "\\underline" ...
   <FONT_STYLE> ...
   <FONT_WEIGHT> ...
   <FONT_TYPEFACE> ...
   <TEXT_FONT_TYPEFACE> ...
   <TEXT_FONT_SIZE> ...
   <CONTROLLED_SPACE> ...
   "\\newline" ...
   "\\\\*" ...
   "\\linebreak" ...
   "\\nolinebreak" ...
   "\\pagebreak" ...
   "\\nopagebreak" ...
   "\\text" ...
   "\\txt" ...
   "\\mbox" ...
   "\\fbox" ...
   "\\hbox" ...
   "\\vbox" ...
   "\\par" ...
   "\\makebox" ...
   "\\framebox" ...
   <OTHER_LATEX> ...

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

(LaTeX -> SymPy) CRITICAL! POM-Tagger Error Reached: Encountered " "}" "} "" at line 1, column 18.

Was expecting one of:

   <LEFTDELIM> ...
   <RIGHTDELIM> ...
   <LARGEDELIM> ...
   "\\begin" ...
   "\\end" ...
   "array" ...
   "\\array" ...
   "subarray" ...
   "tabular" ...
   <MATRIX> ...
   "\\matrix" ...
   "\\pmatrix" ...
   "bordermatrix" ...
   "\\bordermatrix" ...
   "equation" ...
   "multline" ...
   "align" ...
   "aligned" ...
   "gather" ...
   "gathered" ...
   "split" ...
   "CD" ...
   "picture" ...
   "pspicture" ...
   <AMPERSAND> ...
   "cases" ...
   "\\cases" ...
   <XYMATRIX> ...
   "\\eqalign" ...
   "\\eqalignno" ...
   "\\oalign" ...
   "\\ooalign" ...
   <LPAREN> ...
   <RPAREN> ...
   "{" ...
   <LBRACKET> ...
   <RBRACKET> ...
   <ESCAPED_LBRACE> ...
   <ESCAPED_RBRACE> ...
   "\\backslash" ...
   <ALPHANUM> ...
   <NUMERIC> ...
   <LETTER> ...
   <DIGIT> ...
   "`" ...
   "!" ...
   "@" ...
   "*" ...
   "^" ...
   "_" ...
   "-" ...
   "=" ...
   "+" ...
   "|" ...
   ";" ...
   <COLON> ...
   "\'" ...
   "\"" ...
   "," ...
   <LESS_THEN> ...
   "." ...
   <GREATER_THAN> ...
   "/" ...
   "?" ...
   "\\And" ...
   "\\#" ...
   "\\%" ...
   "\\$" ...
   "\\_" ...
   "\\&" ...
   <ESCAPED_SYMBOL_ALLOWED> ...
   <ESCAPED_SYMBOL_DISALLOWED> ...
   <FRAC> ...
   <BINOM> ...
   "\\stackrel" ...
   "\\overset" ...
   "\\underset" ...
   "\\sqrt" ...
   "\\MultiIntegral" ...
   <PHANTOM> ...
   "\\pmod" ...
   <MOD> ...
   "\\pod" ...
   "\\footnote" ...
   "\\not" ...
   <ELLIPSIS> ...
   "\\dots" ...
   <ACCENT> ...
   <MATH_AND_TEXT_ACCENT> ...
   <WIDEACCENT> ...
   "\\underline" ...
   <FONT_STYLE> ...
   <FONT_WEIGHT> ...
   <FONT_TYPEFACE> ...
   <TEXT_FONT_TYPEFACE> ...
   <TEXT_FONT_SIZE> ...
   <CONTROLLED_SPACE> ...
   "\\newline" ...
   "\\\\*" ...
   "\\linebreak" ...
   "\\nolinebreak" ...
   "\\pagebreak" ...
   "\\nopagebreak" ...
   "\\text" ...
   "\\txt" ...
   "\\mbox" ...
   "\\fbox" ...
   "\\hbox" ...
   "\\vbox" ...
   "\\par" ...
   "\\makebox" ...
   "\\framebox" ...
   <OTHER_LATEX> ...

Tests

Symbolic

Numeric

Maple

Translation:

Information

Symbol info

(LaTeX -> Maple) CRITICAL! POM-Tagger Error Reached: Encountered " "}" "} "" at line 1, column 18.

Was expecting one of:

   <LEFTDELIM> ...
   <RIGHTDELIM> ...
   <LARGEDELIM> ...
   "\\begin" ...
   "\\end" ...
   "array" ...
   "\\array" ...
   "subarray" ...
   "tabular" ...
   <MATRIX> ...
   "\\matrix" ...
   "\\pmatrix" ...
   "bordermatrix" ...
   "\\bordermatrix" ...
   "equation" ...
   "multline" ...
   "align" ...
   "aligned" ...
   "gather" ...
   "gathered" ...
   "split" ...
   "CD" ...
   "picture" ...
   "pspicture" ...
   <AMPERSAND> ...
   "cases" ...
   "\\cases" ...
   <XYMATRIX> ...
   "\\eqalign" ...
   "\\eqalignno" ...
   "\\oalign" ...
   "\\ooalign" ...
   <LPAREN> ...
   <RPAREN> ...
   "{" ...
   <LBRACKET> ...
   <RBRACKET> ...
   <ESCAPED_LBRACE> ...
   <ESCAPED_RBRACE> ...
   "\\backslash" ...
   <ALPHANUM> ...
   <NUMERIC> ...
   <LETTER> ...
   <DIGIT> ...
   "`" ...
   "!" ...
   "@" ...
   "*" ...
   "^" ...
   "_" ...
   "-" ...
   "=" ...
   "+" ...
   "|" ...
   ";" ...
   <COLON> ...
   "\'" ...
   "\"" ...
   "," ...
   <LESS_THEN> ...
   "." ...
   <GREATER_THAN> ...
   "/" ...
   "?" ...
   "\\And" ...
   "\\#" ...
   "\\%" ...
   "\\$" ...
   "\\_" ...
   "\\&" ...
   <ESCAPED_SYMBOL_ALLOWED> ...
   <ESCAPED_SYMBOL_DISALLOWED> ...
   <FRAC> ...
   <BINOM> ...
   "\\stackrel" ...
   "\\overset" ...
   "\\underset" ...
   "\\sqrt" ...
   "\\MultiIntegral" ...
   <PHANTOM> ...
   "\\pmod" ...
   <MOD> ...
   "\\pod" ...
   "\\footnote" ...
   "\\not" ...
   <ELLIPSIS> ...
   "\\dots" ...
   <ACCENT> ...
   <MATH_AND_TEXT_ACCENT> ...
   <WIDEACCENT> ...
   "\\underline" ...
   <FONT_STYLE> ...
   <FONT_WEIGHT> ...
   <FONT_TYPEFACE> ...
   <TEXT_FONT_TYPEFACE> ...
   <TEXT_FONT_SIZE> ...
   <CONTROLLED_SPACE> ...
   "\\newline" ...
   "\\\\*" ...
   "\\linebreak" ...
   "\\nolinebreak" ...
   "\\pagebreak" ...
   "\\nopagebreak" ...
   "\\text" ...
   "\\txt" ...
   "\\mbox" ...
   "\\fbox" ...
   "\\hbox" ...
   "\\vbox" ...
   "\\par" ...
   "\\makebox" ...
   "\\framebox" ...
   <OTHER_LATEX> ...

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

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

${\begin{aligned}{\text{where}}\quad &p_{n}(x)={\frac {n!(n+a+b+c+d-1)!}{(2n+a+b+c+d-1)!}}p_{n}(x;a,b,c,d),\\&A_{n}=-{\frac {(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}},\\{\text{and}}\quad &C_{n}={\frac {n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}}.\end{aligned}}$

$F_{n}$

Is part of

${\begin{aligned}{\text{where}}\quad &p_{n}(x)={\frac {n!(n+a+b+c+d-1)!}{(2n+a+b+c+d-1)!}}p_{n}(x;a,b,c,d),\\&A_{n}=-{\frac {(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}},\\{\text{and}}\quad &C_{n}={\frac {n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}}.\end{aligned}}$

Complete translation information:

{
  "id" : "FORMULA_c4ad46205cf6ad97b12a6dfceaddf650",
  "formula" : "\\begin{align}\n\\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),\\\\\n&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)},\\\\\n\\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)}.\n\\end{align}",
  "semanticFormula" : "\\begin{align}where}\\quad &p_n(x) = \\frac{n!(n+a+b+c+d-1)!}{(2n+a+b+c+d-1)!} \\contHahnpolyp{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)}, \\\\ 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}",
  "confidence" : 0.7168596035007688,
  "translations" : {
    "Mathematica" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> Mathematica) CRITICAL! POM-Tagger Error Reached: Encountered \" \"}\" \"} \"\" at line 1, column 18.\nWas expecting one of:\n    <LEFTDELIM> ...\n    <RIGHTDELIM> ...\n    <LARGEDELIM> ...\n    \"\\\\begin\" ...\n    \"\\\\end\" ...\n    \"array\" ...\n    \"\\\\array\" ...\n    \"subarray\" ...\n    \"tabular\" ...\n    <MATRIX> ...\n    \"\\\\matrix\" ...\n    \"\\\\pmatrix\" ...\n    \"bordermatrix\" ...\n    \"\\\\bordermatrix\" ...\n    \"equation\" ...\n    \"multline\" ...\n    \"align\" ...\n    \"aligned\" ...\n    \"gather\" ...\n    \"gathered\" ...\n    \"split\" ...\n    \"CD\" ...\n    \"picture\" ...\n    \"pspicture\" ...\n    <AMPERSAND> ...\n    \"cases\" ...\n    \"\\\\cases\" ...\n    <XYMATRIX> ...\n    \"\\\\eqalign\" ...\n    \"\\\\eqalignno\" ...\n    \"\\\\oalign\" ...\n    \"\\\\ooalign\" ...\n    <LPAREN> ...\n    <RPAREN> ...\n    \"{\" ...\n    <LBRACKET> ...\n    <RBRACKET> ...\n    <ESCAPED_LBRACE> ...\n    <ESCAPED_RBRACE> ...\n    \"\\\\backslash\" ...\n    <ALPHANUM> ...\n    <NUMERIC> ...\n    <LETTER> ...\n    <DIGIT> ...\n    \"`\" ...\n    \"!\" ...\n    \"@\" ...\n    \"*\" ...\n    \"^\" ...\n    \"_\" ...\n    \"-\" ...\n    \"=\" ...\n    \"+\" ...\n    \"|\" ...\n    \";\" ...\n    <COLON> ...\n    \"\\'\" ...\n    \"\\\"\" ...\n    \",\" ...\n    <LESS_THEN> ...\n    \".\" ...\n    <GREATER_THAN> ...\n    \"/\" ...\n    \"?\" ...\n    \"\\\\And\" ...\n    \"\\\\#\" ...\n    \"\\\\%\" ...\n    \"\\\\$\" ...\n    \"\\\\_\" ...\n    \"\\\\&\" ...\n    <ESCAPED_SYMBOL_ALLOWED> ...\n    <ESCAPED_SYMBOL_DISALLOWED> ...\n    <FRAC> ...\n    <BINOM> ...\n    \"\\\\stackrel\" ...\n    \"\\\\overset\" ...\n    \"\\\\underset\" ...\n    \"\\\\sqrt\" ...\n    \"\\\\MultiIntegral\" ...\n    <PHANTOM> ...\n    \"\\\\pmod\" ...\n    <MOD> ...\n    \"\\\\pod\" ...\n    \"\\\\footnote\" ...\n    \"\\\\not\" ...\n    <ELLIPSIS> ...\n    \"\\\\dots\" ...\n    <ACCENT> ...\n    <MATH_AND_TEXT_ACCENT> ...\n    <WIDEACCENT> ...\n    \"\\\\underline\" ...\n    <FONT_STYLE> ...\n    <FONT_WEIGHT> ...\n    <FONT_TYPEFACE> ...\n    <TEXT_FONT_TYPEFACE> ...\n    <TEXT_FONT_SIZE> ...\n    <CONTROLLED_SPACE> ...\n    \"\\\\newline\" ...\n    \"\\\\\\\\*\" ...\n    \"\\\\linebreak\" ...\n    \"\\\\nolinebreak\" ...\n    \"\\\\pagebreak\" ...\n    \"\\\\nopagebreak\" ...\n    \"\\\\text\" ...\n    \"\\\\txt\" ...\n    \"\\\\mbox\" ...\n    \"\\\\fbox\" ...\n    \"\\\\hbox\" ...\n    \"\\\\vbox\" ...\n    \"\\\\par\" ...\n    \"\\\\makebox\" ...\n    \"\\\\framebox\" ...\n    <OTHER_LATEX> ...\n    "
        }
      },
      "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) CRITICAL! POM-Tagger Error Reached: Encountered \" \"}\" \"} \"\" at line 1, column 18.\nWas expecting one of:\n    <LEFTDELIM> ...\n    <RIGHTDELIM> ...\n    <LARGEDELIM> ...\n    \"\\\\begin\" ...\n    \"\\\\end\" ...\n    \"array\" ...\n    \"\\\\array\" ...\n    \"subarray\" ...\n    \"tabular\" ...\n    <MATRIX> ...\n    \"\\\\matrix\" ...\n    \"\\\\pmatrix\" ...\n    \"bordermatrix\" ...\n    \"\\\\bordermatrix\" ...\n    \"equation\" ...\n    \"multline\" ...\n    \"align\" ...\n    \"aligned\" ...\n    \"gather\" ...\n    \"gathered\" ...\n    \"split\" ...\n    \"CD\" ...\n    \"picture\" ...\n    \"pspicture\" ...\n    <AMPERSAND> ...\n    \"cases\" ...\n    \"\\\\cases\" ...\n    <XYMATRIX> ...\n    \"\\\\eqalign\" ...\n    \"\\\\eqalignno\" ...\n    \"\\\\oalign\" ...\n    \"\\\\ooalign\" ...\n    <LPAREN> ...\n    <RPAREN> ...\n    \"{\" ...\n    <LBRACKET> ...\n    <RBRACKET> ...\n    <ESCAPED_LBRACE> ...\n    <ESCAPED_RBRACE> ...\n    \"\\\\backslash\" ...\n    <ALPHANUM> ...\n    <NUMERIC> ...\n    <LETTER> ...\n    <DIGIT> ...\n    \"`\" ...\n    \"!\" ...\n    \"@\" ...\n    \"*\" ...\n    \"^\" ...\n    \"_\" ...\n    \"-\" ...\n    \"=\" ...\n    \"+\" ...\n    \"|\" ...\n    \";\" ...\n    <COLON> ...\n    \"\\'\" ...\n    \"\\\"\" ...\n    \",\" ...\n    <LESS_THEN> ...\n    \".\" ...\n    <GREATER_THAN> ...\n    \"/\" ...\n    \"?\" ...\n    \"\\\\And\" ...\n    \"\\\\#\" ...\n    \"\\\\%\" ...\n    \"\\\\$\" ...\n    \"\\\\_\" ...\n    \"\\\\&\" ...\n    <ESCAPED_SYMBOL_ALLOWED> ...\n    <ESCAPED_SYMBOL_DISALLOWED> ...\n    <FRAC> ...\n    <BINOM> ...\n    \"\\\\stackrel\" ...\n    \"\\\\overset\" ...\n    \"\\\\underset\" ...\n    \"\\\\sqrt\" ...\n    \"\\\\MultiIntegral\" ...\n    <PHANTOM> ...\n    \"\\\\pmod\" ...\n    <MOD> ...\n    \"\\\\pod\" ...\n    \"\\\\footnote\" ...\n    \"\\\\not\" ...\n    <ELLIPSIS> ...\n    \"\\\\dots\" ...\n    <ACCENT> ...\n    <MATH_AND_TEXT_ACCENT> ...\n    <WIDEACCENT> ...\n    \"\\\\underline\" ...\n    <FONT_STYLE> ...\n    <FONT_WEIGHT> ...\n    <FONT_TYPEFACE> ...\n    <TEXT_FONT_TYPEFACE> ...\n    <TEXT_FONT_SIZE> ...\n    <CONTROLLED_SPACE> ...\n    \"\\\\newline\" ...\n    \"\\\\\\\\*\" ...\n    \"\\\\linebreak\" ...\n    \"\\\\nolinebreak\" ...\n    \"\\\\pagebreak\" ...\n    \"\\\\nopagebreak\" ...\n    \"\\\\text\" ...\n    \"\\\\txt\" ...\n    \"\\\\mbox\" ...\n    \"\\\\fbox\" ...\n    \"\\\\hbox\" ...\n    \"\\\\vbox\" ...\n    \"\\\\par\" ...\n    \"\\\\makebox\" ...\n    \"\\\\framebox\" ...\n    <OTHER_LATEX> ...\n    "
        }
      },
      "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" : "",
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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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