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 \rho(s)=\frac{\Gamma(a+s+1)\Gamma(c+s+1)}{\Gamma(s-a+1)\Gamma(b-s)\Gamma(b+s+1)\Gamma(s-c+1)}}

${\displaystyle \rho (s)={\frac {\Gamma (a+s+1)\Gamma (c+s+1)}{\Gamma (s-a+1)\Gamma (b-s)\Gamma (b+s+1)\Gamma (s-c+1)}}}$

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

Semantic latex: \rho(s)=\frac{\Gamma(a+s+1)\Gamma(c+s+1)}{\Gamma(s-a+1)\Gamma(b-s)\Gamma(b+s+1)\Gamma(s-c+1)}

Confidence: 0

Mathematica

Translation: \[Rho][s] == Divide[\[CapitalGamma]*(a + s + 1)*\[CapitalGamma]*(c + s + 1),\[CapitalGamma]*(s - a + 1)*\[CapitalGamma]*(b - s)*\[CapitalGamma]*(b + s + 1)*\[CapitalGamma]*(s - c + 1)]

Information

Sub Equations

\[Rho][s] = Divide[\[CapitalGamma]*(a + s + 1)*\[CapitalGamma]*(c + s + 1),\[CapitalGamma]*(s - a + 1)*\[CapitalGamma]*(b - s)*\[CapitalGamma]*(b + s + 1)*\[CapitalGamma]*(s - c + 1)]

Free variables

\[CapitalGamma]

\[Rho]

a

b

c

s

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Test expression: (\[Rho]*(s))-(Divide[\[CapitalGamma]*(a + s + 1)*\[CapitalGamma]*(c + s + 1),\[CapitalGamma]*(s - a + 1)*\[CapitalGamma]*(b - s)*\[CapitalGamma]*(b + s + 1)*\[CapitalGamma]*(s - c + 1)])

ERROR:

{
    "result": "ERROR",
    "testTitle": "Simple",
    "testExpression": null,
    "resultExpression": null,
    "wasAborted": false,
    "conditionallySuccessful": false
}

Numeric

SymPy

Translation: Symbol('rho')(s) == (Symbol('Gamma')*(a + s + 1)*Symbol('Gamma')*(c + s + 1))/(Symbol('Gamma')*(s - a + 1)*Symbol('Gamma')*(b - s)*Symbol('Gamma')*(b + s + 1)*Symbol('Gamma')*(s - c + 1))

Information

Sub Equations

Symbol('rho')(s) = (Symbol('Gamma')*(a + s + 1)*Symbol('Gamma')*(c + s + 1))/(Symbol('Gamma')*(s - a + 1)*Symbol('Gamma')*(b - s)*Symbol('Gamma')*(b + s + 1)*Symbol('Gamma')*(s - c + 1))

Free variables

Symbol('Gamma')

Symbol('rho')

a

b

c

s

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Numeric

Maple

Translation: rho(s) = (Gamma*(a + s + 1)*Gamma*(c + s + 1))/(Gamma*(s - a + 1)*Gamma*(b - s)*Gamma*(b + s + 1)*Gamma*(s - c + 1))

Information

Sub Equations

rho(s) = (Gamma*(a + s + 1)*Gamma*(c + s + 1))/(Gamma*(s - a + 1)*Gamma*(b - s)*Gamma*(b + s + 1)*Gamma*(s - c + 1))

Free variables

Gamma

a

b

c

rho

s

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Numeric

Dependency Graph Information

Complete translation information:

{
  "id" : "FORMULA_dfe146818f10303553b70545069b789d",
  "formula" : "\\rho(s)=\\frac{\\Gamma(a+s+1)\\Gamma(c+s+1)}{\\Gamma(s-a+1)\\Gamma(b-s)\\Gamma(b+s+1)\\Gamma(s-c+1)}",
  "semanticFormula" : "\\rho(s)=\\frac{\\Gamma(a+s+1)\\Gamma(c+s+1)}{\\Gamma(s-a+1)\\Gamma(b-s)\\Gamma(b+s+1)\\Gamma(s-c+1)}",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "\\[Rho][s] == Divide[\\[CapitalGamma]*(a + s + 1)*\\[CapitalGamma]*(c + s + 1),\\[CapitalGamma]*(s - a + 1)*\\[CapitalGamma]*(b - s)*\\[CapitalGamma]*(b + s + 1)*\\[CapitalGamma]*(s - c + 1)]",
      "translationInformation" : {
        "subEquations" : [ "\\[Rho][s] = Divide[\\[CapitalGamma]*(a + s + 1)*\\[CapitalGamma]*(c + s + 1),\\[CapitalGamma]*(s - a + 1)*\\[CapitalGamma]*(b - s)*\\[CapitalGamma]*(b + s + 1)*\\[CapitalGamma]*(s - c + 1)]" ],
        "freeVariables" : [ "\\[CapitalGamma]", "\\[Rho]", "a", "b", "c", "s" ],
        "tokenTranslations" : {
          "\\rho" : "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" : "ERROR",
        "numberOfTests" : 1,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 1,
        "crashed" : false,
        "testCalculationsGroup" : [ {
          "lhs" : "\\[Rho]*(s)",
          "rhs" : "Divide[\\[CapitalGamma]*(a + s + 1)*\\[CapitalGamma]*(c + s + 1),\\[CapitalGamma]*(s - a + 1)*\\[CapitalGamma]*(b - s)*\\[CapitalGamma]*(b + s + 1)*\\[CapitalGamma]*(s - c + 1)]",
          "testExpression" : "(\\[Rho]*(s))-(Divide[\\[CapitalGamma]*(a + s + 1)*\\[CapitalGamma]*(c + s + 1),\\[CapitalGamma]*(s - a + 1)*\\[CapitalGamma]*(b - s)*\\[CapitalGamma]*(b + s + 1)*\\[CapitalGamma]*(s - c + 1)])",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        } ]
      }
    },
    "SymPy" : {
      "translation" : "Symbol('rho')(s) == (Symbol('Gamma')*(a + s + 1)*Symbol('Gamma')*(c + s + 1))/(Symbol('Gamma')*(s - a + 1)*Symbol('Gamma')*(b - s)*Symbol('Gamma')*(b + s + 1)*Symbol('Gamma')*(s - c + 1))",
      "translationInformation" : {
        "subEquations" : [ "Symbol('rho')(s) = (Symbol('Gamma')*(a + s + 1)*Symbol('Gamma')*(c + s + 1))/(Symbol('Gamma')*(s - a + 1)*Symbol('Gamma')*(b - s)*Symbol('Gamma')*(b + s + 1)*Symbol('Gamma')*(s - c + 1))" ],
        "freeVariables" : [ "Symbol('Gamma')", "Symbol('rho')", "a", "b", "c", "s" ],
        "tokenTranslations" : {
          "\\rho" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      }
    },
    "Maple" : {
      "translation" : "rho(s) = (Gamma*(a + s + 1)*Gamma*(c + s + 1))/(Gamma*(s - a + 1)*Gamma*(b - s)*Gamma*(b + s + 1)*Gamma*(s - c + 1))",
      "translationInformation" : {
        "subEquations" : [ "rho(s) = (Gamma*(a + s + 1)*Gamma*(c + s + 1))/(Gamma*(s - a + 1)*Gamma*(b - s)*Gamma*(b + s + 1)*Gamma*(s - c + 1))" ],
        "freeVariables" : [ "Gamma", "a", "b", "c", "rho", "s" ],
        "tokenTranslations" : {
          "\\rho" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      }
    }
  },
  "positions" : [ {
    "section" : 1,
    "sentence" : 1,
    "word" : 3
  } ],
  "includes" : [ ],
  "isPartOf" : [ ],
  "definiens" : [ ]
}

Specify your own input

Formula input
Wikitext context	In mathematics, the '''dual Hahn polynomials''' are a family of [[orthogonal polynomials]] in the [[Askey scheme]] of hypergeometric orthogonal polynomials. They are defined on a non-uniform lattice <math>x(s)=s(s+1)</math> and are defined as :<math>w_n^{(c)} (s,a,b)=\frac{(a-b+1)_n(a+c+1)_n}{n!} {}_3F_2(-n,a-s,a+s+1;a-b+a,a+c+1;1)</math> for <math>n=0,1,...,N-1</math> and the parameters <math>a,b,c</math> are restricted to <math>-\frac{1}{2}<a<b, \|c\|<1+a, b=a+N</math>. Note that <math>(u)_k</math> is the [[falling and rising factorials]], otherwise known as the Pochhammer symbol, and <math>{}_3F_2(\cdot)</math> is the [[generalized hypergeometric function]]s {{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. ==Orthogonality== The Dual Hahn polynomials have the orthogonality condition :<math>\sum^{b-1}_{s=a}w_n^{(c)}(s,a,b)w_m^{(c)}(s,a,b)\rho(s)[\Delta x(s-\frac{1}{2}) ]=\delta_{nm}d_n^2</math> for <math>n,m=0,1,...,N-1</math>. Where <math>\Delta x(s)=x(s+1)-x(s)</math>, :<math>\rho(s)=\frac{\Gamma(a+s+1)\Gamma(c+s+1)}{\Gamma(s-a+1)\Gamma(b-s)\Gamma(b+s+1)\Gamma(s-c+1)}</math> and :<math>d_n^2=\frac{\Gamma(a+c+n+a)}{n!(b-a-n-1)!\Gamma(b-c-n)}</math> ==Numerical Instability== As the value of <math>n</math> increases, the values that the discrete polynomials obtain also increases. As a result, to obtain [[numerical stability]] in calculating the polynomials you would use the renormalised Dual Hahn Polynomial as defined as :<math>\hat w_n^{(c)}(s,a,b)=w_n^{(c)}(s,a,b)\sqrt{\frac{\rho(s)}{d_n^2}[\Delta x(s-\frac{1}{2})]}</math> for <math>n=0,1,...,N-1</math>. Then the orthogonality condition becomes :<math>\sum^{b-1}_{s=a}\hat w_n^{(c)}(s,a,b)\hat w_m^{(c)}(s,a,b)=\delta_{m,n}</math> for <math>n,m=0,1,...,N-1</math> ==Recurrence and difference relations== {{Empty section\|date=September 2011}} ==Rodrigues formula== {{Empty section\|date=September 2011}} ==Generating function== {{Empty section\|date=September 2011}} ==Relation to other polynomials== The Hahn polynomials, <math>h_n(x,N;\alpha,\beta)</math>, is defined on the uniform lattice <math>x(s)=s</math>, and the parameters <math>a,b,c</math> are defined as <math>a=(\alpha+\beta)/2,b=a+N,c=(\beta-\alpha)/2</math>. Then setting <math>\alpha=\beta=0</math> the [[Hahn polynomials]] become the [[Tchebichef polynomials]]. Note that the Dual Hahn polynomials have a ''q''-analog with an extra parameter ''q'' known as the [[Dual Hahn Q-polynomials]] [[Racah polynomials]] are a generalization of dual Hahn polynomials ==References== {{Citation\| last1= Zhu\| first1 = Hongqing\| title=Image analysis by discrete orthogonal dual Hahn moments\| journal = Pattern Recognition Letters\| volume = 28\| issue = 13\| pages = 1688–1704\|doi = 10.1016/j.patrec.2007.04.013\|year=2007\| url = https://www.hal.inserm.fr/inserm-00189813/file/Image_analysis_Hahn.pdf}} {{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 \| issue=1–2 \| 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}} [[Category:Special hypergeometric functions]] [[Category:Orthogonal polynomials]]
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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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