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 AppellF1

 http://functions.wolfram.com/07.36.06.0015.02

 Input Form

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

 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[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "c", "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "1"], "-", "a"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["c", "-", "a"]], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["b", "1"], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", SubscriptBox["z", "1"]]], ")"]], RowBox[List["-", "a"]]], RowBox[List["AppellF1", "[", RowBox[List["a", ",", RowBox[List["1", "+", "a", "-", "c"]], ",", SubscriptBox["b", "2"], ",", RowBox[List["1", "+", "a", "-", SubscriptBox["b", "1"]]], ",", FractionBox["1", SubscriptBox["z", "1"]], ",", FractionBox[SubscriptBox["z", "2"], SubscriptBox["z", "1"]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["Gamma", "[", "c", "]"]], RowBox[List["Gamma", "[", "a", "]"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", SubscriptBox["z", "1"]]], ")"]], RowBox[List["-", SubscriptBox["b", "1"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "-", SubscriptBox["b", "1"], "+", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "2"], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "-", SubscriptBox["b", "1"], "+", "k"]], "]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[SubscriptBox["b", "1"], ",", RowBox[List["1", "-", "c", "+", SubscriptBox["b", "1"], "-", "k"]], ",", RowBox[List["1", "-", "a", "+", SubscriptBox["b", "1"], "-", "k"]], ",", FractionBox["1", SubscriptBox["z", "1"]]]], "]"]], " ", SubsuperscriptBox["z", "2", "k"]]]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["a", "-", SubscriptBox["b", "1"]]], ",", "Integers"]], "]"]], "]"]]]]]]

 MathML Form

 F 1 AppellF1 ( a ; b 1 , b 2 ; c ; z 1 , z 2 ) Γ ( c ) Γ ( b 1 - a ) Γ ( c - a ) Γ ( b 1 ) ( - z 1 ) - a F 1 AppellF1 ( a ; a - c + 1 , b 2 ; a - b 1 + 1 ; 1 z 1 , z 2 z 1 ) + Γ ( c ) Γ ( a ) ( - z 1 ) - b 1 k = 0 Γ ( a + k - b 1 ) ( b 2 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "2"], ")"]], "k"], Pochhammer] k ! Γ ( c + k - b 1 + k ) 2 F 1 ( b 1 , b 1 - c - k + 1 ; b 1 - a - k + 1 ; 1 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[SubscriptBox["b", "1"], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "1"], "-", "c", "-", "k", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[SubscriptBox["b", "1"], "-", "a", "-", "k", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["1", SubscriptBox["z", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] z 2 k /; a - b 1 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition AppellF1 a Subscript b 1 Subscript b 2 c Subscript z 1 Subscript z 2 Gamma c Gamma Subscript b 1 -1 a Gamma c -1 a Gamma Subscript b 1 -1 -1 Subscript z 1 -1 a AppellF1 a a -1 c 1 Subscript b 2 a -1 Subscript b 1 1 1 Subscript z 1 -1 Subscript z 2 Subscript z 1 -1 Gamma c Gamma a -1 -1 Subscript z 1 -1 Subscript b 1 k 0 Gamma a k -1 Subscript b 1 Pochhammer Subscript b 2 k k Gamma c k -1 Subscript b 1 k -1 Hypergeometric2F1 Subscript b 1 Subscript b 1 -1 c -1 k 1 Subscript b 1 -1 a -1 k 1 1 Subscript z 1 -1 Subscript z 2 k a -1 Subscript b 1 [/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[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "c", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["bb", "1"], "-", "a"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SubscriptBox["zz", "1"]]], ")"]], RowBox[List["-", "a"]]], " ", RowBox[List["AppellF1", "[", RowBox[List["a", ",", RowBox[List["1", "+", "a", "-", "c"]], ",", SubscriptBox["bb", "2"], ",", RowBox[List["1", "+", "a", "-", SubscriptBox["bb", "1"]]], ",", FractionBox["1", SubscriptBox["zz", "1"]], ",", FractionBox[SubscriptBox["zz", "2"], SubscriptBox["zz", "1"]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["c", "-", "a"]], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["bb", "1"], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["Gamma", "[", "c", "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SubscriptBox["zz", "1"]]], ")"]], RowBox[List["-", SubscriptBox["bb", "1"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "-", SubscriptBox["bb", "1"], "+", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["bb", "2"], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[SubscriptBox["bb", "1"], ",", RowBox[List["1", "-", "c", "+", SubscriptBox["bb", "1"], "-", "k"]], ",", RowBox[List["1", "-", "a", "+", SubscriptBox["bb", "1"], "-", "k"]], ",", FractionBox["1", SubscriptBox["zz", "1"]]]], "]"]], " ", SubsuperscriptBox["zz", "2", "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "-", SubscriptBox["bb", "1"], "+", "k"]], "]"]]]]]]]]], RowBox[List["Gamma", "[", "a", "]"]]]]], "/;", RowBox[List["!", RowBox[List[RowBox[List["a", "-", SubscriptBox["bb", "1"]]], "\[Element]", "Integers"]]]]]]]]]]

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

 2001-10-29

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