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 F_2(x,y)=C_1F_2(a,b_1,b_2,c_1,c_2;x,y)+C_2x^{1-c_1}F_2(a-c_1+1,b_1-c_1+1,b_2,2-c_1,c_2;x,y)+C_3y^{1-c_2}F_2(a-c_2+1,b_1,b_2-c_2+1,c_1,2-c_2;x,y)+C_4x^{1-c_1}y^{1-c_2}F_2(a-c_1-c_2+2,b_1-c_1+1,b_2-c_2+1,2-c_1,2-c_2;x,y)}

${\displaystyle F_{2}(x,y)=C_{1}F_{2}(a,b_{1},b_{2},c_{1},c_{2};x,y)+C_{2}x^{1-c_{1}}F_{2}(a-c_{1}+1,b_{1}-c_{1}+1,b_{2},2-c_{1},c_{2};x,y)+C_{3}y^{1-c_{2}}F_{2}(a-c_{2}+1,b_{1},b_{2}-c_{2}+1,c_{1},2-c_{2};x,y)+C_{4}x^{1-c_{1}}y^{1-c_{2}}F_{2}(a-c_{1}-c_{2}+2,b_{1}-c_{1}+1,b_{2}-c_{2}+1,2-c_{1},2-c_{2};x,y)}$

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

Semantic latex: F_2(x,y)=C_1F_2(a,b_1,b_2,c_1,c_2;x,y)+C_2x^{1-c_1}F_2(a-c_1+1,b_1-c_1+1,b_2,2-c_1,c_2;x,y)+C_3y^{1-c_2}F_2(a-c_2+1,b_1,b_2-c_2+1,c_1,2-c_2;x,y)+C_4x^{1-c_1}y^{1-c_2}F_2(a-c_1-c_2+2,b_1-c_1+1,b_2-c_2+1,2-c_1,2-c_2;x,y)

Confidence: 0

Mathematica

Translation: Subscript[F, 2][x , y] == Subscript[C, 1]*Subscript[F, 2][a , Subscript[b, 1], Subscript[b, 2], Subscript[c, 1], Subscript[c, 2]; x , y]+ Subscript[C, 2]*(x)^(1 - Subscript[c, 1])* Subscript[F, 2][a - Subscript[c, 1]+ 1 , Subscript[b, 1]- Subscript[c, 1]+ 1 , Subscript[b, 2], 2 - Subscript[c, 1], Subscript[c, 2]; x , y]+ Subscript[C, 3]*(y)^(1 - Subscript[c, 2])* Subscript[F, 2][a - Subscript[c, 2]+ 1 , Subscript[b, 1], Subscript[b, 2]- Subscript[c, 2]+ 1 , Subscript[c, 1], 2 - Subscript[c, 2]; x , y]+ Subscript[C, 4]*(x)^(1 - Subscript[c, 1])* (y)^(1 - Subscript[c, 2])* Subscript[F, 2][a - Subscript[c, 1]- Subscript[c, 2]+ 2 , Subscript[b, 1]- Subscript[c, 1]+ 1 , Subscript[b, 2]- Subscript[c, 2]+ 1 , 2 - Subscript[c, 1], 2 - Subscript[c, 2]; x , y]

Information

Sub Equations

Subscript[F, 2][x , y] = Subscript[C, 1]*Subscript[F, 2][a , Subscript[b, 1], Subscript[b, 2], Subscript[c, 1], Subscript[c, 2]; x , y]+ Subscript[C, 2]*(x)^(1 - Subscript[c, 1])* Subscript[F, 2][a - Subscript[c, 1]+ 1 , Subscript[b, 1]- Subscript[c, 1]+ 1 , Subscript[b, 2], 2 - Subscript[c, 1], Subscript[c, 2]; x , y]+ Subscript[C, 3]*(y)^(1 - Subscript[c, 2])* Subscript[F, 2][a - Subscript[c, 2]+ 1 , Subscript[b, 1], Subscript[b, 2]- Subscript[c, 2]+ 1 , Subscript[c, 1], 2 - Subscript[c, 2]; x , y]+ Subscript[C, 4]*(x)^(1 - Subscript[c, 1])* (y)^(1 - Subscript[c, 2])* Subscript[F, 2][a - Subscript[c, 1]- Subscript[c, 2]+ 2 , Subscript[b, 1]- Subscript[c, 1]+ 1 , Subscript[b, 2]- Subscript[c, 2]+ 1 , 2 - Subscript[c, 1], 2 - Subscript[c, 2]; x , y]

Free variables

Subscript[C, 1]

Subscript[C, 2]

Subscript[C, 3]

Subscript[C, 4]

Subscript[b, 1]

Subscript[b, 2]

Subscript[c, 1]

Subscript[c, 2]

a

x

y

Symbol info

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

Tests

Symbolic

Numeric

SymPy

Translation: Symbol('{F}_{2}')(x , y) == Symbol('{C}_{1}')*Symbol('{F}_{2}')(a , Symbol('{b}_{1}'), Symbol('{b}_{2}'), Symbol('{c}_{1}'), Symbol('{c}_{2}'); x , y)+ Symbol('{C}_{2}')*(x)**(1 - Symbol('{c}_{1}'))* Symbol('{F}_{2}')(a - Symbol('{c}_{1}')+ 1 , Symbol('{b}_{1}')- Symbol('{c}_{1}')+ 1 , Symbol('{b}_{2}'), 2 - Symbol('{c}_{1}'), Symbol('{c}_{2}'); x , y)+ Symbol('{C}_{3}')*(y)**(1 - Symbol('{c}_{2}'))* Symbol('{F}_{2}')(a - Symbol('{c}_{2}')+ 1 , Symbol('{b}_{1}'), Symbol('{b}_{2}')- Symbol('{c}_{2}')+ 1 , Symbol('{c}_{1}'), 2 - Symbol('{c}_{2}'); x , y)+ Symbol('{C}_{4}')*(x)**(1 - Symbol('{c}_{1}'))* (y)**(1 - Symbol('{c}_{2}'))* Symbol('{F}_{2}')(a - Symbol('{c}_{1}')- Symbol('{c}_{2}')+ 2 , Symbol('{b}_{1}')- Symbol('{c}_{1}')+ 1 , Symbol('{b}_{2}')- Symbol('{c}_{2}')+ 1 , 2 - Symbol('{c}_{1}'), 2 - Symbol('{c}_{2}'); x , y)

Information

Sub Equations

Symbol('{F}_{2}')(x , y) = Symbol('{C}_{1}')*Symbol('{F}_{2}')(a , Symbol('{b}_{1}'), Symbol('{b}_{2}'), Symbol('{c}_{1}'), Symbol('{c}_{2}'); x , y)+ Symbol('{C}_{2}')*(x)**(1 - Symbol('{c}_{1}'))* Symbol('{F}_{2}')(a - Symbol('{c}_{1}')+ 1 , Symbol('{b}_{1}')- Symbol('{c}_{1}')+ 1 , Symbol('{b}_{2}'), 2 - Symbol('{c}_{1}'), Symbol('{c}_{2}'); x , y)+ Symbol('{C}_{3}')*(y)**(1 - Symbol('{c}_{2}'))* Symbol('{F}_{2}')(a - Symbol('{c}_{2}')+ 1 , Symbol('{b}_{1}'), Symbol('{b}_{2}')- Symbol('{c}_{2}')+ 1 , Symbol('{c}_{1}'), 2 - Symbol('{c}_{2}'); x , y)+ Symbol('{C}_{4}')*(x)**(1 - Symbol('{c}_{1}'))* (y)**(1 - Symbol('{c}_{2}'))* Symbol('{F}_{2}')(a - Symbol('{c}_{1}')- Symbol('{c}_{2}')+ 2 , Symbol('{b}_{1}')- Symbol('{c}_{1}')+ 1 , Symbol('{b}_{2}')- Symbol('{c}_{2}')+ 1 , 2 - Symbol('{c}_{1}'), 2 - Symbol('{c}_{2}'); x , y)

Free variables

Symbol('{C}_{1}')

Symbol('{C}_{2}')

Symbol('{C}_{3}')

Symbol('{C}_{4}')

Symbol('{b}_{1}')

Symbol('{b}_{2}')

Symbol('{c}_{1}')

Symbol('{c}_{2}')

a

x

y

Symbol info

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

Tests

Symbolic

Numeric

Maple

Translation: F[2](x , y) = C[1]*F[2](a , b[1], b[2], c[1], c[2]; x , y)+ C[2]*(x)^(1 - c[1])* F[2](a - c[1]+ 1 , b[1]- c[1]+ 1 , b[2], 2 - c[1], c[2]; x , y)+ C[3]*(y)^(1 - c[2])* F[2](a - c[2]+ 1 , b[1], b[2]- c[2]+ 1 , c[1], 2 - c[2]; x , y)+ C[4]*(x)^(1 - c[1])* (y)^(1 - c[2])* F[2](a - c[1]- c[2]+ 2 , b[1]- c[1]+ 1 , b[2]- c[2]+ 1 , 2 - c[1], 2 - c[2]; x , y)

Information

Sub Equations

F[2](x , y) = C[1]*F[2](a , b[1], b[2], c[1], c[2]; x , y)+ C[2]*(x)^(1 - c[1])* F[2](a - c[1]+ 1 , b[1]- c[1]+ 1 , b[2], 2 - c[1], c[2]; x , y)+ C[3]*(y)^(1 - c[2])* F[2](a - c[2]+ 1 , b[1], b[2]- c[2]+ 1 , c[1], 2 - c[2]; x , y)+ C[4]*(x)^(1 - c[1])* (y)^(1 - c[2])* F[2](a - c[1]- c[2]+ 2 , b[1]- c[1]+ 1 , b[2]- c[2]+ 1 , 2 - c[1], 2 - c[2]; x , y)

Free variables

C[1]

C[2]

C[3]

C[4]

a

b[1]

b[2]

c[1]

c[2]

x

y

Symbol info

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

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$F_{2}$

$y$

$x$

$F$

Description

solution

A system

Appell

definition

partial differential equation

system

derivative

derivative result from the definition

double series

system of differential equation

system of second-order differential equation

Complete translation information:

{
  "id" : "FORMULA_26fb9e69275fe26d56459576839eeb84",
  "formula" : "F_2(x,y)=C_1F_2(a,b_1,b_2,c_1,c_2;x,y)+C_2x^{1-c_1}F_2(a-c_1+1,b_1-c_1+1,b_2,2-c_1,c_2;x,y)+C_3y^{1-c_2}F_2(a-c_2+1,b_1,b_2-c_2+1,c_1,2-c_2;x,y)+C_4x^{1-c_1}y^{1-c_2}F_2(a-c_1-c_2+2,b_1-c_1+1,b_2-c_2+1,2-c_1,2-c_2;x,y)",
  "semanticFormula" : "F_2(x,y)=C_1F_2(a,b_1,b_2,c_1,c_2;x,y)+C_2x^{1-c_1}F_2(a-c_1+1,b_1-c_1+1,b_2,2-c_1,c_2;x,y)+C_3y^{1-c_2}F_2(a-c_2+1,b_1,b_2-c_2+1,c_1,2-c_2;x,y)+C_4x^{1-c_1}y^{1-c_2}F_2(a-c_1-c_2+2,b_1-c_1+1,b_2-c_2+1,2-c_1,2-c_2;x,y)",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "Subscript[F, 2][x , y] == Subscript[C, 1]*Subscript[F, 2][a , Subscript[b, 1], Subscript[b, 2], Subscript[c, 1], Subscript[c, 2]; x , y]+ Subscript[C, 2]*(x)^(1 - Subscript[c, 1])* Subscript[F, 2][a - Subscript[c, 1]+ 1 , Subscript[b, 1]- Subscript[c, 1]+ 1 , Subscript[b, 2], 2 - Subscript[c, 1], Subscript[c, 2]; x , y]+ Subscript[C, 3]*(y)^(1 - Subscript[c, 2])* Subscript[F, 2][a - Subscript[c, 2]+ 1 , Subscript[b, 1], Subscript[b, 2]- Subscript[c, 2]+ 1 , Subscript[c, 1], 2 - Subscript[c, 2]; x , y]+ Subscript[C, 4]*(x)^(1 - Subscript[c, 1])* (y)^(1 - Subscript[c, 2])* Subscript[F, 2][a - Subscript[c, 1]- Subscript[c, 2]+ 2 , Subscript[b, 1]- Subscript[c, 1]+ 1 , Subscript[b, 2]- Subscript[c, 2]+ 1 , 2 - Subscript[c, 1], 2 - Subscript[c, 2]; x , y]",
      "translationInformation" : {
        "subEquations" : [ "Subscript[F, 2][x , y] = Subscript[C, 1]*Subscript[F, 2][a , Subscript[b, 1], Subscript[b, 2], Subscript[c, 1], Subscript[c, 2]; x , y]+ Subscript[C, 2]*(x)^(1 - Subscript[c, 1])* Subscript[F, 2][a - Subscript[c, 1]+ 1 , Subscript[b, 1]- Subscript[c, 1]+ 1 , Subscript[b, 2], 2 - Subscript[c, 1], Subscript[c, 2]; x , y]+ Subscript[C, 3]*(y)^(1 - Subscript[c, 2])* Subscript[F, 2][a - Subscript[c, 2]+ 1 , Subscript[b, 1], Subscript[b, 2]- Subscript[c, 2]+ 1 , Subscript[c, 1], 2 - Subscript[c, 2]; x , y]+ Subscript[C, 4]*(x)^(1 - Subscript[c, 1])* (y)^(1 - Subscript[c, 2])* Subscript[F, 2][a - Subscript[c, 1]- Subscript[c, 2]+ 2 , Subscript[b, 1]- Subscript[c, 1]+ 1 , Subscript[b, 2]- Subscript[c, 2]+ 1 , 2 - Subscript[c, 1], 2 - Subscript[c, 2]; x , y]" ],
        "freeVariables" : [ "Subscript[C, 1]", "Subscript[C, 2]", "Subscript[C, 3]", "Subscript[C, 4]", "Subscript[b, 1]", "Subscript[b, 2]", "Subscript[c, 1]", "Subscript[c, 2]", "a", "x", "y" ],
        "tokenTranslations" : {
          "F" : "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('{F}_{2}')(x , y) == Symbol('{C}_{1}')*Symbol('{F}_{2}')(a , Symbol('{b}_{1}'), Symbol('{b}_{2}'), Symbol('{c}_{1}'), Symbol('{c}_{2}'); x , y)+ Symbol('{C}_{2}')*(x)**(1 - Symbol('{c}_{1}'))* Symbol('{F}_{2}')(a - Symbol('{c}_{1}')+ 1 , Symbol('{b}_{1}')- Symbol('{c}_{1}')+ 1 , Symbol('{b}_{2}'), 2 - Symbol('{c}_{1}'), Symbol('{c}_{2}'); x , y)+ Symbol('{C}_{3}')*(y)**(1 - Symbol('{c}_{2}'))* Symbol('{F}_{2}')(a - Symbol('{c}_{2}')+ 1 , Symbol('{b}_{1}'), Symbol('{b}_{2}')- Symbol('{c}_{2}')+ 1 , Symbol('{c}_{1}'), 2 - Symbol('{c}_{2}'); x , y)+ Symbol('{C}_{4}')*(x)**(1 - Symbol('{c}_{1}'))* (y)**(1 - Symbol('{c}_{2}'))* Symbol('{F}_{2}')(a - Symbol('{c}_{1}')- Symbol('{c}_{2}')+ 2 , Symbol('{b}_{1}')- Symbol('{c}_{1}')+ 1 , Symbol('{b}_{2}')- Symbol('{c}_{2}')+ 1 , 2 - Symbol('{c}_{1}'), 2 - Symbol('{c}_{2}'); x , y)",
      "translationInformation" : {
        "subEquations" : [ "Symbol('{F}_{2}')(x , y) = Symbol('{C}_{1}')*Symbol('{F}_{2}')(a , Symbol('{b}_{1}'), Symbol('{b}_{2}'), Symbol('{c}_{1}'), Symbol('{c}_{2}'); x , y)+ Symbol('{C}_{2}')*(x)**(1 - Symbol('{c}_{1}'))* Symbol('{F}_{2}')(a - Symbol('{c}_{1}')+ 1 , Symbol('{b}_{1}')- Symbol('{c}_{1}')+ 1 , Symbol('{b}_{2}'), 2 - Symbol('{c}_{1}'), Symbol('{c}_{2}'); x , y)+ Symbol('{C}_{3}')*(y)**(1 - Symbol('{c}_{2}'))* Symbol('{F}_{2}')(a - Symbol('{c}_{2}')+ 1 , Symbol('{b}_{1}'), Symbol('{b}_{2}')- Symbol('{c}_{2}')+ 1 , Symbol('{c}_{1}'), 2 - Symbol('{c}_{2}'); x , y)+ Symbol('{C}_{4}')*(x)**(1 - Symbol('{c}_{1}'))* (y)**(1 - Symbol('{c}_{2}'))* Symbol('{F}_{2}')(a - Symbol('{c}_{1}')- Symbol('{c}_{2}')+ 2 , Symbol('{b}_{1}')- Symbol('{c}_{1}')+ 1 , Symbol('{b}_{2}')- Symbol('{c}_{2}')+ 1 , 2 - Symbol('{c}_{1}'), 2 - Symbol('{c}_{2}'); x , y)" ],
        "freeVariables" : [ "Symbol('{C}_{1}')", "Symbol('{C}_{2}')", "Symbol('{C}_{3}')", "Symbol('{C}_{4}')", "Symbol('{b}_{1}')", "Symbol('{b}_{2}')", "Symbol('{c}_{1}')", "Symbol('{c}_{2}')", "a", "x", "y" ],
        "tokenTranslations" : {
          "F" : "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" : "F[2](x , y) = C[1]*F[2](a , b[1], b[2], c[1], c[2]; x , y)+ C[2]*(x)^(1 - c[1])* F[2](a - c[1]+ 1 , b[1]- c[1]+ 1 , b[2], 2 - c[1], c[2]; x , y)+ C[3]*(y)^(1 - c[2])* F[2](a - c[2]+ 1 , b[1], b[2]- c[2]+ 1 , c[1], 2 - c[2]; x , y)+ C[4]*(x)^(1 - c[1])* (y)^(1 - c[2])* F[2](a - c[1]- c[2]+ 2 , b[1]- c[1]+ 1 , b[2]- c[2]+ 1 , 2 - c[1], 2 - c[2]; x , y)",
      "translationInformation" : {
        "subEquations" : [ "F[2](x , y) = C[1]*F[2](a , b[1], b[2], c[1], c[2]; x , y)+ C[2]*(x)^(1 - c[1])* F[2](a - c[1]+ 1 , b[1]- c[1]+ 1 , b[2], 2 - c[1], c[2]; x , y)+ C[3]*(y)^(1 - c[2])* F[2](a - c[2]+ 1 , b[1], b[2]- c[2]+ 1 , c[1], 2 - c[2]; x , y)+ C[4]*(x)^(1 - c[1])* (y)^(1 - c[2])* F[2](a - c[1]- c[2]+ 2 , b[1]- c[1]+ 1 , b[2]- c[2]+ 1 , 2 - c[1], 2 - c[2]; x , y)" ],
        "freeVariables" : [ "C[1]", "C[2]", "C[3]", "C[4]", "a", "b[1]", "b[2]", "c[1]", "c[2]", "x", "y" ],
        "tokenTranslations" : {
          "F" : "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" : 3,
    "sentence" : 0,
    "word" : 55
  } ],
  "includes" : [ "F_{2}", "y", "x", "F" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "solution",
    "score" : 0.8869384888466118
  }, {
    "definition" : "A system",
    "score" : 0.8426021531523621
  }, {
    "definition" : "Appell",
    "score" : 0.8426021531523621
  }, {
    "definition" : "definition",
    "score" : 0.8426021531523621
  }, {
    "definition" : "partial differential equation",
    "score" : 0.8426021531523621
  }, {
    "definition" : "system",
    "score" : 0.8426021531523621
  }, {
    "definition" : "derivative",
    "score" : 0.6687181434333315
  }, {
    "definition" : "derivative result from the definition",
    "score" : 0.6687181434333315
  }, {
    "definition" : "double series",
    "score" : 0.6687181434333315
  }, {
    "definition" : "system of differential equation",
    "score" : 0.6687181434333315
  }, {
    "definition" : "system of second-order differential equation",
    "score" : 0.6687181434333315
  } ]
}

Specify your own input

Formula input
Wikitext context	{{dablink\|For generalizations of Lambert series see [[Appell–Lerch series]].}} In mathematics, '''Appell [[series (mathematics)\|series]]''' are a set of four [[hypergeometric series]] ''F''<sub>1</sub>, ''F''<sub>2</sub>, ''F''<sub>3</sub>, ''F''<sub>4</sub> of two [[variable (mathematics)\|variable]]s that were introduced by {{harvs \| txt \| authorlink= Paul Émile Appell \| first= Paul \| last= Appell \| year= 1880}} and that generalize [[hypergeometric function\|Gauss's hypergeometric series]] <sub>2</sub>''F''<sub>1</sub> of one variable. Appell established the set of [[partial differential equation]]s of which these [[function (mathematics)\|function]]s are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variable. ==Definitions== The Appell series ''F''<sub>1</sub> is defined for \|''x''\| < 1, \|''y''\| < 1 by the double series :<math> F_1(a,b_1,b_2;c;x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n} (b_1)_m (b_2)_n} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~, </math> where <math>(q)_n</math> is the [[Hypergeometric function#The_hypergeometric_series\|Pochhammer symbol]]. For other values of ''x'' and ''y'' the function ''F''<sub>1</sub> can be defined by [[analytic continuation]]. It can be shown<ref>See Burchnall & Chaundy (1940), formula (30).</ref> that :<math>F_1(a,b_1,b_2;c;x,y) = \sum_{r=0}^\infty \frac{(a)_r (b_1)_r (b_2)_r (c-a)_r} {(c+r-1)_r (c)_{2r} r!} \,x^r y^r {}_2F_{1}\left(a+r,b_1+r;c+2r;x\right){}_2F_{1}\left(a+r,b_2+r;c+2r;y\right)~.</math> Similarly, the function ''F''<sub>2</sub> is defined for \|''x''\| + \|''y''\| < 1 by the series :<math> F_2(a,b_1,b_2;c_1,c_2;x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n} (b_1)_m (b_2)_n} {(c_1)_m (c_2)_n \,m! \,n!} \,x^m y^n </math> and it can be shown<ref>See Burchnall & Chaundy (1940), formula (26) or Erdélyi (1953), formula 5.12(9).</ref> that :<math> F_2(a,b_1,b_2;c_1,c_2;x,y) = \sum_{r=0}^\infty \frac{(a)_r (b_1)_r (b_2)_r} {(c_1)_r (c_2)_r r!} \,x^r y^r {}_2F_{1}\left(a+r,b_1+r;c_1+r;x\right){}_2F_{1}\left(a+r,b_2+r;c_2+r;y\right)~.</math> Also the function ''F''<sub>3</sub> for \|''x''\| < 1, \|''y''\| < 1 can be defined by the series :<math> F_3(a_1,a_2,b_1,b_2;c;x,y) = \sum_{m,n=0}^\infty \frac{(a_1)_m (a_2)_n (b_1)_m (b_2)_n} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~, </math> and the function ''F''<sub>4</sub> for \|''x''\|<sup>½</sup> + \|''y''\|<sup>½</sup> < 1 by the series :<math> F_4(a,b;c_1,c_2;x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n} (b)_{m+n}} {(c_1)_m (c_2)_n \,m! \,n!} \,x^m y^n ~. </math> ==Recurrence relations== Like the Gauss hypergeometric series <sub>2</sub>''F''<sub>1</sub>, the Appell double series entail [[recurrence relation]]s among contiguous functions. For example, a basic set of such relations for Appell's ''F''<sub>1</sub> is given by: :<math> (a-b_1-b_2) F_1(a,b_1,b_2,c; x,y) - a \,F_1(a+1,b_1,b_2,c; x,y) + b_1 F_1(a,b_1+1,b_2,c; x,y) + b_2 F_1(a,b_1,b_2+1,c; x,y) = 0 ~, </math> :<math> c \,F_1(a,b_1,b_2,c; x,y) - (c-a) F_1(a,b_1,b_2,c+1; x,y) - a \,F_1(a+1,b_1,b_2,c+1; x,y) = 0 ~, </math> :<math> c \,F_1(a,b_1,b_2,c; x,y) + c(x-1) F_1(a,b_1+1,b_2,c; x,y) - (c-a)x \,F_1(a,b_1+1,b_2,c+1; x,y) = 0 ~, </math> :<math> c \,F_1(a,b_1,b_2,c; x,y) + c(y-1) F_1(a,b_1,b_2+1,c; x,y) - (c-a)y \,F_1(a,b_1,b_2+1,c+1; x,y) = 0 ~. </math> Any other relation<ref>For example, <math>(y-x) F_1(a, b_1+1, b_2+1,c,x,y) = y \, F_1(a,b_1,b_2+1,c,x,y) - x \, F_1(a,b_1+1,b_2,c,x,y)</math></ref> valid for ''F''<sub>1</sub> can be derived from these four. Similarly, all recurrence relations for Appell's ''F''<sub>3</sub> follow from this set of five: :<math> c \,F_3(a_1,a_2,b_1,b_2,c; x,y) + (a_1+a_2-c) F_3(a_1,a_2,b_1,b_2,c+1; x,y) - a_1 F_3(a_1+1,a_2,b_1,b_2,c+1; x,y) - a_2 F_3(a_1,a_2+1,b_1,b_2,c+1; x,y) = 0 ~, </math> :<math> c \,F_3(a_1,a_2,b_1,b_2,c; x,y) - c \,F_3(a_1+1,a_2,b_1,b_2,c; x,y) + b_1 x \,F_3(a_1+1,a_2,b_1+1,b_2,c+1; x,y) = 0 ~, </math> :<math> c \,F_3(a_1,a_2,b_1,b_2,c; x,y) - c \,F_3(a_1,a_2+1,b_1,b_2,c; x,y) + b_2 y \,F_3(a_1,a_2+1,b_1,b_2+1,c+1; x,y) = 0 ~, </math> :<math> c \,F_3(a_1,a_2,b_1,b_2,c; x,y) - c \,F_3(a_1,a_2,b_1+1,b_2,c; x,y) + a_1 x \,F_3(a_1+1,a_2,b_1+1,b_2,c+1; x,y) = 0 ~, </math> :<math> c \,F_3(a_1,a_2,b_1,b_2,c; x,y) - c \,F_3(a_1,a_2,b_1,b_2+1,c; x,y) + a_2 y \,F_3(a_1,a_2+1,b_1,b_2+1,c+1; x,y) = 0 ~. </math> ==Derivatives and differential equations== For Appell's ''F''<sub>1</sub>, the following [[derivative]]s result from the definition by a double series: :<math> \frac {\partial^n} {\partial x^n} F_1(a,b_1,b_2,c; x,y) = \frac {\left(a\right)_n \left(b_1\right)_n} {\left(c\right)_n} F_1(a+n,b_1+n,b_2,c+n; x,y) </math> :<math> \frac {\partial^n} {\partial y^n} F_1(a,b_1,b_2,c; x,y) = \frac {\left(a\right)_n \left(b_2\right)_n} {\left(c\right)_n} F_1(a+n,b_1,b_2+n,c+n; x,y) </math> From its definition, Appell's ''F''<sub>1</sub> is further found to satisfy the following system of second-order [[partial differential equation\|differential equations]]: :<math> x(1-x) \frac {\partial^2F_1(x,y)} {\partial x^2} + y(1-x) \frac {\partial^2F_1(x,y)} {\partial x \partial y} + [c - (a+b_1+1) x] \frac {\partial F_1(x,y)} {\partial x} - b_1 y \frac {\partial F_1(x,y)} {\partial y} - a b_1 F_1(x,y) = 0 </math> :<math> y(1-y) \frac {\partial^2F_1(x,y)} {\partial y^2} + x(1-y) \frac {\partial^2F_1(x,y)} {\partial x \partial y} + [c - (a+b_2+1) y] \frac {\partial F_1(x,y)} {\partial y} - b_2 x \frac {\partial F_1(x,y)} {\partial x} - a b_2 F_1(x,y)= 0 </math> A system partial differential equations for ''F''<sub>2</sub> is :<math> x(1-x) \frac {\partial^2F_2(x,y)} {\partial x^2} - xy \frac {\partial^2F_2(x,y)} {\partial x \partial y} + [c_1 - (a+b_1+1) x] \frac {\partial F_2(x,y)} {\partial x} -b_1 y \frac {\partial F_2(x,y)} {\partial y}- a b_1 F_2(x,y) = 0 </math> :<math> y(1-y) \frac {\partial^2F_2(x,y)} {\partial y^2} - xy \frac {\partial^2F_2(x,y)} {\partial x \partial y} + [c_2 - (a+b_2+1) x] \frac {\partial F_2(x,y)} {\partial y} -b_2 x \frac {\partial F_2(x,y)} {\partial x}- a b_2 F_2(x,y) = 0 </math> The system have solution :<math>F_2(x,y)=C_1F_2(a,b_1,b_2,c_1,c_2;x,y)+C_2x^{1-c_1}F_2(a-c_1+1,b_1-c_1+1,b_2,2-c_1,c_2;x,y)+C_3y^{1-c_2}F_2(a-c_2+1,b_1,b_2-c_2+1,c_1,2-c_2;x,y)+C_4x^{1-c_1}y^{1-c_2}F_2(a-c_1-c_2+2,b_1-c_1+1,b_2-c_2+1,2-c_1,2-c_2;x,y)</math> Similarly, for ''F''<sub>3</sub> the following derivatives result from the definition: :<math> \frac {\partial} {\partial x} F_3(a_1,a_2,b_1,b_2,c; x,y) = \frac {a_1 b_1} {c} F_3(a_1+1,a_2,b_1+1,b_2,c+1; x,y) </math> :<math> \frac {\partial} {\partial y} F_3(a_1,a_2,b_1,b_2,c; x,y) = \frac {a_2 b_2} {c} F_3(a_1,a_2+1,b_1,b_2+1,c+1; x,y) </math> And for ''F''<sub>3</sub> the following system of differential equations is obtained: :<math> x(1-x) \frac {\partial^2F_3(x,y)} {\partial x^2} + y \frac {\partial^2F_3(x,y)} {\partial x \partial y} + [c - (a_1+b_1+1) x] \frac {\partial F_3(x,y)} {\partial x} - a_1 b_1 F_3(x,y) = 0 </math> :<math> y(1-y) \frac {\partial^2F_3(x,y)} {\partial y^2} + x \frac {\partial^2F_3(x,y)} {\partial x \partial y} + [c - (a_2+b_2+1) y] \frac {\partial F_3(x,y)} {\partial y} - a_2 b_2 F_3(x,y) = 0 </math> A system partial differential equations for ''F''<sub>4</sub> is :<math> x(1-x) \frac {\partial^2F_4(x,y)} {\partial x^2} - y^2 \frac {\partial^2F_4(x,y)} {\partial y^2} -2xy\frac {\partial^2F_4(x,y)} {\partial x \partial y}+[c_1 - (a+b+1) x] \frac {\partial F_4(x,y)} {\partial x} - (a+b+1) y \frac {\partial F_4(x,y)} {\partial y}-a b F_4(x,y)= 0 </math> :<math> y(1-y) \frac {\partial^2F_4(x,y)} {\partial y^2} - x^2 \frac {\partial^2F_4(x,y)} {\partial x^2} -2xy\frac {\partial^2F_4(x,y)} {\partial x \partial y}+[c_2 - (a+b+1) y] \frac {\partial F_4(x,y)} {\partial y} - (a+b+1) x \frac {\partial F_4(x,y)} {\partial x}-a b F_4(x,y)= 0 </math> The system have solution :<math>F_4(x,y)=C_1F_4(a,b,c_1,c_2;x,y)+C_2x^{1-c_1}F_4(a-c_1+1,b-c_1+1,2-c_1,c_2;x,y)+C_3y^{1-c_2}F_4(a-c_2+1,b-c_2+1,c_1,2-c_2;x,y)+C_4x^{1-c_1}y^{1-c_2}F_4(2+a-c_1-c_2,2+b-c_1-c_2,2-c_1,2-c_2;x,y)</math> ==Integral representations== The four functions defined by Appell's double series can be represented in terms of [[multiple integral\|double integral]]s involving [[elementary functions]] only {{harv\|Gradshteyn\|Ryzhik\|2015\|loc=§9.184}}. However, {{harvs \| txt \| authorlink= Charles Émile Picard \| first= Émile \| last= Picard \| year= 1881}} discovered that Appell's ''F''<sub>1</sub> can also be written as a one-dimensional [[Leonhard Euler\|Euler]]-type [[integral]]: :<math> F_1(a,b_1,b_2,c; x,y) = \frac{\Gamma(c)} {\Gamma(a)\Gamma(c-a)} \int_0^1 t^{a-1} (1-t)^{c-a-1} (1-xt)^{-b_1} (1-yt)^{-b_2} \,\mathrm{d}t, \quad \real \,c > \real \,a > 0 ~. </math> This representation can be verified by means of [[Taylor series\|Taylor expansion]] of the integrand, followed by termwise integration. ==Special cases== Picard's integral representation implies that the [[elliptic integral\|incomplete elliptic integral]]s ''F'' and ''E'' as well as the [[elliptic integral\|complete elliptic integral]] Π are special cases of Appell's ''F''<sub>1</sub>: :<math> F(\phi,k) = \int_0^\phi \frac{\mathrm{d} \theta} {\sqrt{1 - k^2 \sin^2 \theta}} = \sin (\phi) \,F_1(\tfrac 1 2, \tfrac 1 2, \tfrac 1 2, \tfrac 3 2; \sin^2 \phi, k^2 \sin^2 \phi), \quad \|\real \,\phi\| < \frac \pi 2 ~, </math> :<math> E(\phi, k) = \int_0^\phi \sqrt{1 - k^2 \sin^2 \theta} \,\mathrm{d} \theta = \sin (\phi) \,F_1(\tfrac 1 2, \tfrac 1 2, -\tfrac 1 2, \tfrac 3 2; \sin^2 \phi, k^2 \sin^2 \phi), \quad \|\real \,\phi\| < \frac \pi 2 ~, </math> :<math> \Pi(n,k) = \int_0^{\pi/2} \frac{\mathrm{d} \theta} {(1 - n \sin^2 \theta) \sqrt{1 - k^2 \sin^2 \theta}} = \frac {\pi} {2} \,F_1(\tfrac 1 2, 1, \tfrac 1 2, 1; n,k^2) ~. </math> ==Related series== * {{main\|Humbert series}} :There are seven related series of two variables, Φ<sub>1</sub>, Φ<sub>2</sub>, Φ<sub>3</sub>, Ψ<sub>1</sub>, Ψ<sub>2</sub>, Ξ<sub>1</sub>, and Ξ<sub>2</sub>, which generalize [[confluent hypergeometric function\|Kummer's confluent hypergeometric function]] <sub>1</sub>''F''<sub>1</sub> of one variable and the [[confluent hypergeometric limit function]] <sub>0</sub>''F''<sub>1</sub> of one variable in a similar manner. The first of these was introduced by [[Pierre Humbert (mathematician)\|Pierre Humbert]] in [[#{{harvid\|Humbert\|1920}}\|1920]]. * {{main\|Lauricella hypergeometric series}} :{{harvs \| txt \| authorlink= Giuseppe Lauricella \| first= Giuseppe \| last= Lauricella \| year= 1893}} defined four functions similar to the Appell series, but depending on many variables rather than just the two variables ''x'' and ''y''. These series were also studied by Appell. They satisfy certain partial differential equations, and can also be given in terms of Euler-type integrals and [[line integral\|contour integral]]s. ==References== {{Reflist}} * {{cite journal \| last= Appell \| first= Paul \| authorlink= Paul Émile Appell \| title= Sur les séries hypergéométriques de deux variables et sur des équations différentielles linéaires aux dérivées partielles \| language= French \| journal= Comptes rendus hebdomadaires des séances de l'Académie des sciences \| year= 1880 \| volume= 90 \| pages= 296–298 and 731–735 \| jfm= 12.0296.01 \| ref= harv}} (see also "Sur la série F<sub>3</sub>(α,α',β,β',γ; x,y)" in ''C. R. Acad. Sci.'' '''90''', pp. 977–980) * {{cite journal \| last= Appell \| first= Paul \| title= Sur les fonctions hypergéométriques de deux variables \| language= French \| url= http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1882_3_8_A8_0 \| journal= [[Journal de Mathématiques Pures et Appliquées]] \| series= (3ème série) \| year= 1882 \| volume= 8 \| pages= 173–216 \| ref= harv }}{{Dead link\|date=May 2019 \|bot=InternetArchiveBot \|fix-attempted=yes }} * {{cite book \| last1= Appell \| first1= Paul \| last2= Kampé de Fériet \| first2= Joseph \| author2-link= Joseph Kampé de Fériet \| title= Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite \| language= French \| location= Paris \| publisher= Gauthier–Villars \| year= 1926 \| jfm= 52.0361.13 \| ref= harv}} (see p. 14) * {{dlmf \| id= 16.13 \| first= R. A. \| last= Askey \| first2= A. B. \| last2= Olde Daalhuis}} * {{cite journal \| last1= Burchnall \| first1= J. L. \| last2= Chaundy \| first2= T. W. \| title= Expansions of Appell's double hypergeometric functions \| journal= Quart. J. Math., Oxford Ser. \| year= 1940 \| volume= 11 \| pages= 249–270 \| doi= 10.1093/qmath/os-11.1.249 \| ref= harv}} * {{cite book \| first= A. \| last= Erdélyi \| author-link= Arthur Erdélyi \| title= Higher Transcendental Functions, Vol. I \| url= http://apps.nrbook.com/bateman/Vol1.pdf \| location= New York \| publisher= McGraw–Hill \| year= 1953 \| ref = harv}} (see p. 224) * {{cite book \|author-first1=Izrail Solomonovich \|author-last1=Gradshteyn \|author-link1=Izrail Solomonovich Gradshteyn \|author-first2=Iosif Moiseevich \|author-last2=Ryzhik \|author-link2=Iosif Moiseevich Ryzhik \|author-first3=Yuri Veniaminovich \|author-last3=Geronimus \|author-link3=Yuri Veniaminovich Geronimus \|author-first4=Michail Yulyevich \|author-last4=Tseytlin \|author-link4=Michail Yulyevich Tseytlin \|author-first5=Alan \|author-last5=Jeffrey \|editor-first1=Daniel \|editor-last1=Zwillinger \|editor-first2=Victor Hugo \|editor-last2=Moll \|translator=Scripta Technica, Inc. \|title=Table of Integrals, Series, and Products \|publisher=[[Academic Press, Inc.]] \|date=2015 \|orig-year=October 2014 \|edition=8 \|language=English \|isbn=978-0-12-384933-5 \|lccn=2014010276 <!-- \|url=https://books.google.com/books?id=NjnLAwAAQBAJ \|access-date=2016-02-21-->\|title-link=Gradshteyn and Ryzhik \|chapter=9.18. \|ref=harv}} * {{cite journal \| last= Humbert \| first= Pierre \| authorlink= Pierre Humbert (mathematician) \| title= Sur les fonctions hypercylindriques \| language= French \| journal= Comptes rendus hebdomadaires des séances de l'Académie des sciences \| year= 1920 \| volume= 171 \| pages= 490–492 \| jfm= 47.0348.01 \| ref= harv}} * {{cite journal \| last= Lauricella \| first= Giuseppe \| authorlink= Giuseppe Lauricella \| title= Sulle funzioni ipergeometriche a più variabili \| language= Italian \| journal= [[Rendiconti del Circolo Matematico di Palermo]] \| year= 1893 \| volume= 7 \| pages= 111–158 \| doi= 10.1007/BF03012437 \| jfm= 25.0756.01 \| s2cid= 122316343 \| ref= harv}} * {{cite journal \| last= Picard \| first= Émile \| authorlink= Charles Émile Picard \| title= Sur une extension aux fonctions de deux variables du problème de Riemann relativ aux fonctions hypergéométriques \| language= French \| url= http://www.numdam.org/item?id=ASENS_1881_2_10__305_0 \| journal= Annales Scientifiques de l'École Normale Supérieure \| series=Série 2 \| year= 1881 \| volume= 10 \| pages= 305–322 \| doi= 10.24033/asens.203 \| jfm= 13.0389.01 \| ref= harv\| doi-access= free }} (see also ''C. R. Acad. Sci.'' '''90''' (1880), pp. 1119–1121 and 1267–1269) * {{cite book \| last= Slater \| first= Lucy Joan \| authorlink= Lucy Joan Slater \| title= Generalized hypergeometric functions \| url= https://archive.org/details/generalizedhyper0000unse_g0b6 \| url-access= registration \| location= Cambridge, UK \| publisher= Cambridge University Press \| year= 1966 \| isbn= 0-521-06483-X \| mr= 0201688 \| ref= harv}} (there is a 2008 paperback with {{ISBN\|978-0-521-09061-2}}) ==External links== * {{mathworld \| urlname= LauricellaFunctions \| title= Lauricella Functions \| author= Aarts, Ronald M.}} * {{mathworld \| urlname= AppellHypergeometricFunction \| title= Appell Hypergeometric Function}} {{DEFAULTSORT:Appell Series}} [[Category:Hypergeometric functions]] [[Category:Mathematical series]]
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