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 AppellF1

 http://functions.wolfram.com/07.36.06.0025.01

 Input Form

 AppellF1[a, Subscript[b, 1], Subscript[b, 2], c, Subscript[z, 1], Subscript[z, 2]] == Subscript[F, Infinity][Subscript[z, 2], a, Subscript[b, 1], Subscript[b, 2], c, Subscript[z, 1]] /; Subscript[F, n][Subscript[z, 2], a, Subscript[b, 1], Subscript[b, 2], c, Subscript[z, 1]] == Sum[((Pochhammer[a, k] Pochhammer[Subscript[b, 2], k] Subscript[z, 2]^k)/(Pochhammer[c, k] k!)) Hypergeometric2F1[a + k, Subscript[b, 1], c + k, Subscript[z, 1]], {k, 0, n}] == AppellF1[a, Subscript[b, 1], Subscript[b, 2], c, Subscript[z, 1], Subscript[z, 2]] - ((Pochhammer[a, n + 1] Pochhammer[Subscript[b, 2], n + 1] Subscript[z, 2]^(n + 1))/(Pochhammer[c, n + 1] (n + 1)!)) HypergeometricPFQ[{{a + n + 1}, {Subscript[b, 1]}, {1, Subscript[b, 2] + n + 1}}, {{c + n + 1}, {}, {n + 2}}, Subscript[z, 1], Subscript[z, 2]] && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["AppellF1", "[", RowBox[List["a", ",", SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", "c", ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List[SubscriptBox["z", "2"], ",", "a", ",", SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", "c", ",", SubscriptBox["z", "1"]]], "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", RowBox[List[SubscriptBox["z", "2"], ",", "a", ",", SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", "c", ",", SubscriptBox["z", "1"]]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "2"], ",", "k"]], "]"]], " ", SubsuperscriptBox["z", "2", "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["c", ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["a", "+", "k"]], ",", SubscriptBox["b", "1"], ",", RowBox[List["c", "+", "k"]], ",", SubscriptBox["z", "1"]]], "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List["AppellF1", "[", RowBox[List["a", ",", SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", "c", ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", RowBox[List["n", "+", "1"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "2"], ",", RowBox[List["n", "+", "1"]]]], "]"]], " ", SubsuperscriptBox["z", "2", RowBox[List["n", "+", "1"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["c", ",", RowBox[List["n", "+", "1"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["a", "+", "n", "+", "1"]], "}"]], ",", RowBox[List["{", SubscriptBox["b", "1"], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List[SubscriptBox["b", "2"], "+", "n", "+", "1"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["c", "+", "n", "+", "1"]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["n", "+", "2"]], "}"]]]], "}"]], ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]]]]]]]], StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]], ")"]]]]]]

 MathML Form

 F 1 AppellF1 ( a ; b 1 , b 2 ; c ; z 1 , z 2 ) F ( z 2 , a , b 1 , b 2 , c , z 1 ) /; ( ( F n ( z 2 , a , b 1 , b 2 , c , z 1 ) k = 0 n ( a ) k TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "k"], Pochhammer] ( b 2 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "2"], ")"]], "k"], Pochhammer] z 2 k ( c ) k TagBox[SubscriptBox[RowBox[List["(", "c", ")"]], "k"], Pochhammer] k ! 2 F 1 ( a + k , b 1 ; c + k ; z 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["a", "+", "k"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "1"], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["c", "+", "k"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SubscriptBox["z", "1"], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] F 1 AppellF1 ( a ; b 1 , b 2 ; c ; z 1 , z 2 ) - ( a ) n + 1 TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] ( b 2 ) n + 1 TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "2"], ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] z 2 n + 1 ( c ) n + 1 TagBox[SubscriptBox[RowBox[List["(", "c", ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] ( n + 1 ) ! F 1 2 0 1 2 1 ( a + n + 1 ; 1 , b 1 ; b 2 + n + 1 ; c + n + 1 ; ; n + 2 ; z 1 , z 2 ) ) n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] ) FormBox RowBox RowBox RowBox InterpretationBox SubscriptBox F 1 AppellF1 Rule Editable Rule Selectable ( RowBox RowBox a ; SubscriptBox b 1 , RowBox SubscriptBox b 2 ; c ; SubscriptBox z 1 , SubscriptBox z 2 ) RowBox SubscriptBox F ( RowBox SubscriptBox z 2 , a , SubscriptBox b 1 , SubscriptBox b 2 , c , SubscriptBox z 1 ) /; RowBox ( RowBox RowBox ( RowBox RowBox SubscriptBox F n ( RowBox SubscriptBox z 2 , a , SubscriptBox b 1 , SubscriptBox b 2 , c , SubscriptBox z 1 ) RowBox UnderoverscriptBox RowBox k = 0 n RowBox FractionBox RowBox TagBox SubscriptBox RowBox ( a ) k Pochhammer TagBox SubscriptBox RowBox ( SubscriptBox b 2 ) k Pochhammer SubsuperscriptBox z 2 k RowBox TagBox SubscriptBox RowBox ( c ) k Pochhammer RowBox k ! TagBox TagBox RowBox RowBox SubscriptBox FormBox 2 TraditionalForm SubscriptBox F FormBox 1 TraditionalForm RowBox ( RowBox TagBox TagBox RowBox TagBox RowBox a + k Hypergeometric2F1 Rule Editable , TagBox SubscriptBox b 1 Hypergeometric2F1 Rule Editable InterpretTemplate Function SlotSequence 1 Hypergeometric2F1 Rule Editable ; TagBox TagBox TagBox RowBox c + k Hypergeometric2F1 Rule Editable InterpretTemplate Function SlotSequence 1 Hypergeometric2F1 Rule Editable ; TagBox SubscriptBox z 1 Hypergeometric2F1 Rule Editable ) InterpretTemplate Function HypergeometricPFQ Slot 1 Slot 2 Slot 3 Rule Editable Hypergeometric2F1 RowBox RowBox InterpretationBox SubscriptBox F 1 AppellF1 Rule Editable Rule Selectable ( RowBox RowBox a ; SubscriptBox b 1 , RowBox SubscriptBox b 2 ; c ; SubscriptBox z 1 , SubscriptBox z 2 ) - RowBox FractionBox RowBox TagBox SubscriptBox RowBox ( a ) RowBox n + 1 Pochhammer TagBox SubscriptBox RowBox ( SubscriptBox b 2 ) RowBox n + 1 Pochhammer SubsuperscriptBox z 2 RowBox n + 1 RowBox TagBox SubscriptBox RowBox ( c ) RowBox n + 1 Pochhammer RowBox RowBox ( RowBox n + 1 ) ! RowBox SubsuperscriptBox F RowBox 1 2 0 RowBox 1 2 1 [ RowBox GridBox ErrorBox RowBox RowBox RowBox a + n + 1 ; 1 , RowBox SubscriptBox b 1 ; RowBox SubscriptBox b 2 + n + 1 ; RowBox RowBox RowBox c + n + 1 ; ; RowBox n + 2 ; SubscriptBox z 1 , SubscriptBox z 2 ] ) RowBox n TagBox Function ) TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AppellF1", "[", RowBox[List["a_", ",", SubscriptBox["b_", "1"], ",", SubscriptBox["b_", "2"], ",", "c_", ",", SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List[SubscriptBox["zz", "2"], ",", "a", ",", SubscriptBox["bb", "1"], ",", SubscriptBox["bb", "2"], ",", "c", ",", SubscriptBox["zz", "1"]]], "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", RowBox[List[SubscriptBox["zz", "2"], ",", "a", ",", SubscriptBox["bb", "1"], ",", SubscriptBox["bb", "2"], ",", "c", ",", SubscriptBox["zz", "1"]]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["bb", "2"], ",", "k"]], "]"]], " ", SubsuperscriptBox["zz", "2", "k"]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["a", "+", "k"]], ",", SubscriptBox["bb", "1"], ",", RowBox[List["c", "+", "k"]], ",", SubscriptBox["zz", "1"]]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["c", ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]], "\[Equal]", RowBox[List[RowBox[List["AppellF1", "[", RowBox[List["a", ",", SubscriptBox["bb", "1"], ",", SubscriptBox["bb", "2"], ",", "c", ",", SubscriptBox["zz", "1"], ",", SubscriptBox["zz", "2"]]], "]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", RowBox[List["n", "+", "1"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["bb", "2"], ",", RowBox[List["n", "+", "1"]]]], "]"]], " ", SubsuperscriptBox["zz", "2", RowBox[List["n", "+", "1"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["a", "+", "n", "+", "1"]], "}"]], ",", RowBox[List["{", SubscriptBox["bb", "1"], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List[SubscriptBox["bb", "2"], "+", "n", "+", "1"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["c", "+", "n", "+", "1"]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["n", "+", "2"]], "}"]]]], "}"]], ",", SubscriptBox["zz", "1"], ",", SubscriptBox["zz", "2"]]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["c", ",", RowBox[List["n", "+", "1"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02

© 1998-2013 Wolfram Research, Inc.