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 h_n(x,N;\alpha,\beta)}

${\displaystyle h_{n}(x,N;\alpha ,\beta )}$

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

Semantic latex: h_n(x,N;\alpha,\beta)

Confidence: 0

Mathematica

Translation: Subscript[h, n][x , N ; \[Alpha], \[Beta]]

Information

Sub Equations

Subscript[h, n][x , N ; \[Alpha], \[Beta]]

Free variables

N

\[Alpha]

\[Beta]

n

x

Symbol info

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

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.

Tests

Symbolic

Numeric

SymPy

Translation: Symbol('{h}_{n}')(x , N ; Symbol('alpha'), Symbol('beta'))

Information

Sub Equations

Symbol('{h}_{n}')(x , N ; Symbol('alpha'), Symbol('beta'))

Free variables

N

Symbol('alpha')

Symbol('beta')

n

x

Symbol info

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

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.

Tests

Symbolic

Numeric

Maple

Translation: h[n](x , N ; alpha , beta)

Information

Sub Equations

h[n](x , N ; alpha , beta)

Free variables

N

alpha

beta

n

x

Symbol info

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

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.

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$n$

Description

Hahn polynomial

uniform lattice

parameter

Complete translation information:

{
  "id" : "FORMULA_6543938cb0edbe57fec0672c056b0d1b",
  "formula" : "h_n(x,N;\\alpha,\\beta)",
  "semanticFormula" : "h_n(x,N;\\alpha,\\beta)",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "Subscript[h, n][x , N ; \\[Alpha], \\[Beta]]",
      "translationInformation" : {
        "subEquations" : [ "Subscript[h, n][x , N ; \\[Alpha], \\[Beta]]" ],
        "freeVariables" : [ "N", "\\[Alpha]", "\\[Beta]", "n", "x" ],
        "tokenTranslations" : {
          "h" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\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"
        }
      },
      "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('{h}_{n}')(x , N ; Symbol('alpha'), Symbol('beta'))",
      "translationInformation" : {
        "subEquations" : [ "Symbol('{h}_{n}')(x , N ; Symbol('alpha'), Symbol('beta'))" ],
        "freeVariables" : [ "N", "Symbol('alpha')", "Symbol('beta')", "n", "x" ],
        "tokenTranslations" : {
          "h" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\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"
        }
      },
      "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" : "h[n](x , N ; alpha , beta)",
      "translationInformation" : {
        "subEquations" : [ "h[n](x , N ; alpha , beta)" ],
        "freeVariables" : [ "N", "alpha", "beta", "n", "x" ],
        "tokenTranslations" : {
          "h" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\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"
        }
      },
      "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" : 3,
    "sentence" : 0,
    "word" : 4
  } ],
  "includes" : [ "n" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "Hahn polynomial",
    "score" : 0.7125985104912714
  }, {
    "definition" : "uniform lattice",
    "score" : 0.6460746792928004
  }, {
    "definition" : "parameter",
    "score" : 0.5988174995334326
  } ]
}

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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