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 \mathcal{K}_k(x; n,q) = \mathcal{K}_k(x) = \sum_{j=0}^{k}(-1)^j (q-1)^{k-j} \binom {x}{j} \binom{n-x}{k-j}, \quad k=0,1, \ldots, n.}

${\displaystyle {\mathcal {K}}_{k}(x;n,q)={\mathcal {K}}_{k}(x)=\sum _{j=0}^{k}(-1)^{j}(q-1)^{k-j}{\binom {x}{j}}{\binom {n-x}{k-j}},\quad k=0,1,\ldots ,n.}$

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

Semantic latex: \mathcal{K}_k(x; n,q) = \mathcal{K}_k(x) = \sum_{j=0}^{k}(-1)^j (q-1)^{k-j} \binom {x}{j} \binom{n-x}{k-j}, \quad k=0,1, \ldots, n

Confidence: 0

Mathematica

Translation: Subscript[K, k][x ; n , q] == Subscript[K, k][x] == Sum[(- 1)^(j)*(q - 1)^(k - j)*Binomial[x,j]*Binomial[n - x,k - j], {j, 0, k}, GenerateConditions->None]

Information

Sub Equations

Subscript[K, k][x ; n , q] = Subscript[K, k][x]

Subscript[K, k][x] = Sum[(- 1)^(j)*(q - 1)^(k - j)*Binomial[x,j]*Binomial[n - x,k - j], {j, 0, k}, GenerateConditions->None]

Free variables

k

n

q

x

Constraints

k == 0 , 1 , \[Ellipsis], n

Symbol info

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

Tests

Symbolic

Numeric

SymPy

Translation: Symbol('{K}_{k}')(x ; n , q) == Symbol('{K}_{k}')(x) == Sum((- 1)**(j)*(q - 1)**(k - j)*binomial(x,j)*binomial(n - x,k - j), (j, 0, k))

Information

Sub Equations

Symbol('{K}_{k}')(x ; n , q) = Symbol('{K}_{k}')(x)

Symbol('{K}_{k}')(x) = Sum((- 1)**(j)*(q - 1)**(k - j)*binomial(x,j)*binomial(n - x,k - j), (j, 0, k))

Free variables

k

n

q

x

Constraints

k == 0 , 1 , null , n

Symbol info

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

Tests

Symbolic

Numeric

Maple

Translation: K[k](x ; n , q) = K[k](x) = sum((- 1)^(j)*(q - 1)^(k - j)*binomial(x,j)*binomial(n - x,k - j), j = 0..k)

Information

Sub Equations

K[k](x ; n , q) = K[k](x)

K[k](x) = sum((- 1)^(j)*(q - 1)^(k - j)*binomial(x,j)*binomial(n - x,k - j), j = 0..k)

Free variables

k

n

q

x

Constraints

k = 0 , 1 , .. , n

Symbol info

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

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$q$

$n$

Description

Kravchuk polynomial

prime power

positive integer

Complete translation information:

{
  "id" : "FORMULA_835dfbc367f2ff51a494b24a3cb81809",
  "formula" : "\\mathcal{K}_k(x; n,q) = \\mathcal{K}_k(x) = \\sum_{j=0}^{k}(-1)^j (q-1)^{k-j} \\binom {x}{j} \\binom{n-x}{k-j}, \\quad k=0,1, \\ldots, n",
  "semanticFormula" : "\\mathcal{K}_k(x; n,q) = \\mathcal{K}_k(x) = \\sum_{j=0}^{k}(-1)^j (q-1)^{k-j} \\binom {x}{j} \\binom{n-x}{k-j}, \\quad k=0,1, \\ldots, n",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "Subscript[K, k][x ; n , q] == Subscript[K, k][x] == Sum[(- 1)^(j)*(q - 1)^(k - j)*Binomial[x,j]*Binomial[n - x,k - j], {j, 0, k}, GenerateConditions->None]",
      "translationInformation" : {
        "subEquations" : [ "Subscript[K, k][x ; n , q] = Subscript[K, k][x]", "Subscript[K, k][x] = Sum[(- 1)^(j)*(q - 1)^(k - j)*Binomial[x,j]*Binomial[n - x,k - j], {j, 0, k}, GenerateConditions->None]" ],
        "freeVariables" : [ "k", "n", "q", "x" ],
        "constraints" : [ "k == 0 , 1 , \\[Ellipsis], n" ],
        "tokenTranslations" : {
          "K" : "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" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "SymPy" : {
      "translation" : "Symbol('{K}_{k}')(x ; n , q) == Symbol('{K}_{k}')(x) == Sum((- 1)**(j)*(q - 1)**(k - j)*binomial(x,j)*binomial(n - x,k - j), (j, 0, k))",
      "translationInformation" : {
        "subEquations" : [ "Symbol('{K}_{k}')(x ; n , q) = Symbol('{K}_{k}')(x)", "Symbol('{K}_{k}')(x) = Sum((- 1)**(j)*(q - 1)**(k - j)*binomial(x,j)*binomial(n - x,k - j), (j, 0, k))" ],
        "freeVariables" : [ "k", "n", "q", "x" ],
        "constraints" : [ "k == 0 , 1 , null , n" ],
        "tokenTranslations" : {
          "K" : "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" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "Maple" : {
      "translation" : "K[k](x ; n , q) = K[k](x) = sum((- 1)^(j)*(q - 1)^(k - j)*binomial(x,j)*binomial(n - x,k - j), j = 0..k)",
      "translationInformation" : {
        "subEquations" : [ "K[k](x ; n , q) = K[k](x)", "K[k](x) = sum((- 1)^(j)*(q - 1)^(k - j)*binomial(x,j)*binomial(n - x,k - j), j = 0..k)" ],
        "freeVariables" : [ "k", "n", "q", "x" ],
        "constraints" : [ "k = 0 , 1 , .. , n" ],
        "tokenTranslations" : {
          "K" : "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" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    }
  },
  "positions" : [ {
    "section" : 1,
    "sentence" : 0,
    "word" : 14
  } ],
  "includes" : [ "q", "n" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "Kravchuk polynomial",
    "score" : 0.722
  }, {
    "definition" : "prime power",
    "score" : 0.6859086196238077
  }, {
    "definition" : "positive integer",
    "score" : 0.6460746792928004
  } ]
}

Specify your own input

Formula input
Wikitext context	'''Kravchuk polynomials''' or '''Krawtchouk polynomials''' (also written using several other transliterations of the Ukrainian surname "Кравчу́к") are [[discrete orthogonal polynomials\|discrete]] [[orthogonal polynomials]] associated with the [[binomial distribution]], introduced by {{harvs\|txt\|authorlink=Mikhail Kravchuk\|first=Mykhailo\|last=Kravchuk\|year= 1929}}. The first few polynomials are (for ''q''=2): * <math>\mathcal{K}_0(x; n) = 1</math> * <math>\mathcal{K}_1(x; n) = -2x + n</math> * <math>\mathcal{K}_2(x; n) = 2x^2 - 2nx + {n\choose 2}</math> * <math>\mathcal{K}_3(x; n) = -\frac{4}{3}x^3 + 2nx^2 - (n^2 - n + \frac{2}{3})x + {n \choose 3}.</math> The Kravchuk polynomials are a special case of the [[Meixner polynomials]] of the first kind. ==Definition== For any [[prime power]] ''q'' and positive integer ''n'', define the Kravchuk polynomial :<math>\mathcal{K}_k(x; n,q) = \mathcal{K}_k(x) = \sum_{j=0}^{k}(-1)^j (q-1)^{k-j} \binom {x}{j} \binom{n-x}{k-j}, \quad k=0,1, \ldots, n.</math> ==Properties== The Kravchuk polynomial has the following alternative expressions: :<math>\mathcal{K}_k(x; n,q) = \sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \binom {n-j}{k-j} \binom{x}{j}. </math> :<math>\mathcal{K}_k(x; n,q) = \sum_{j=0}^{k}(-1)^j q^{k-j} \binom {n-k+j}{j} \binom{n-x}{k-j}. </math> === Symmetry relations === For integers <math>i,k \ge 0</math>, we have that :<math>\begin{align} (q-1)^{i} {n \choose i} \mathcal{K}_k(i;n,q) = (q-1)^{k}{n \choose k} \mathcal{K}_i(k;n,q). \end{align}</math> ===Orthogonality relations=== For non-negative integers ''r'', ''s'', :<math>\sum_{i=0}^n\binom{n}{i}(q-1)^i\mathcal{K}_r(i; n,q)\mathcal{K}_s(i; n,q) = q^n(q-1)^r\binom{n}{r}\delta_{r,s}. </math> ===Generating function=== The [[generating series]] of Kravchuk polynomials is given as below. Here <math>z</math> is a formal variable. :<math>\begin{align} (1+(q-1)z)^{n-x}(1-z)^x &= \sum_{k=0}^\infty \mathcal{K}_k(x;n,q) {z^k}. \end{align}</math> ==See also== * [[Hermite polynomials]] ==References== <references/> {{Citation \| last1=Kravchuk \| first1=M. \| authorlink = Mikhail Kravchuk \| title=Sur une généralisation des polynomes d'Hermite. \| url=http://gallica.bnf.fr/ark:/12148/bpt6k3142j.pleinepage.f620.langEN \| language=French \| jfm=55.0799.01 \| year=1929 \| journal=Comptes Rendus Mathématique \| volume=189 \| pages=620–622}} {{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 = Nikiforov \| first1 = A. F. \| last2 = Suslov \| first2 = S. K. \| last3 = Uvarov \| first3 = V. B. \| isbn = 3-540-51123-7 \| location = Berlin \| mr = 1149380 \| publisher = Springer-Verlag \| series = Springer Series in Computational Physics \| title = Classical Orthogonal Polynomials of a Discrete Variable \| year = 1991}}. {{citation \| last = Levenshtein \| first = Vladimir I. \| author-link = Vladimir Levenshtein \| doi = 10.1109/18.412678 \| issue = 5 \| journal = IEEE Transactions on Information Theory \| mr = 1366326 \| pages = 1303–1321 \| title = Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces \| volume = 41 \| year = 1995}}. {{Citation \| first1=F. J. \| last1=MacWilliams \| first2=N. J. A. \| last2=Sloane \| title=The Theory of Error-Correcting Codes \| publisher=North-Holland \| year=1977 \| isbn=0-444-85193-3 \| url-access=registration \| url=https://archive.org/details/theoryoferrorcor0000macw }} ==External links== {{commons category}} [https://web.archive.org/web/20070205055023/http://orthpol.narod.ru/ Krawtchouk Polynomials Home Page] *[http://mathworld.wolfram.com/KrawtchoukPolynomial.html "Krawtchouk polynomial"] at [[MathWorld]] [[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

Specify your own input

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