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 Hypergeometric2F1

 http://functions.wolfram.com/07.23.17.0150.01

 Input Form

 Hypergeometric2F1[a, b, c, z] == (Gamma[c]/((-z)^\[Mu] (Gamma[a] Gamma[b]))) (Sum[(Gamma[\[Mu] + k] (n - k - 1)!)/(z^k (k! (c - \[Mu] - k - 1)!)), {k, 0, n - 1}] + (-1)^n Sum[(Gamma[\[Mu] + k]/((-z)^k (k! (k - n)! (c - \[Mu] - k - 1)!))) (PolyGamma[k + 1] + PolyGamma[k - n + 1] - PolyGamma[\[Mu] + k] - PolyGamma[c - \[Mu] - k] + Log[-z]), {k, n, c - \[Mu] - 1}] + (-1)^n (-z)^(-c + \[Mu]) Gamma[c] HypergeometricPFQRegularized[{1, 1, c}, {1 + c - \[Mu], 1 + c - \[Mu] - n}, 1/z]) /; Abs[z] > 1 && Element[b - a, Integers] && n == Abs[b - a] && Element[c - a, Integers] && \[Mu] == a + Min[0, b - a]

 Standard Form

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 MathML Form

 2 F 1 ( a , b ; c ; z ) 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["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ( - z ) - μ Γ ( c ) Γ ( a ) Γ ( b ) ( ( - 1 ) n Γ ( c ) ( - z ) μ - c 3 F ~ 2 ( 1 , 1 , c ; c - μ + 1 , c - n - μ + 1 ; 1 z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["c", HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["c", "-", "\[Mu]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["c", "-", "n", "-", "\[Mu]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[FractionBox["1", "z"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] + k = 0 n - 1 z - k Γ ( k + μ ) ( n - k - 1 ) ! k ! ( c - k - μ - 1 ) ! + ( - 1 ) n k = n c - μ - 1 ( - z ) - k Γ ( k + μ ) k ! ( k - n ) ! ( c - k - μ - 1 ) ! ( log ( - z ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( c - k - μ ) - ψ TagBox["\[Psi]", PolyGamma] ( k + μ ) ) ) /; "\[LeftBracketingBar]" z "\[RightBracketingBar]" > 1 b - a TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] n "\[LeftBracketingBar]" b - a "\[RightBracketingBar]" c - a TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] μ a + min ( 0 , b - a ) Condition Hypergeometric2F1 a b c z -1 z -1 μ Gamma c Gamma a Gamma b -1 -1 n Gamma c -1 z μ -1 c HypergeometricPFQRegularized 1 1 c c -1 μ 1 c -1 n -1 μ 1 1 z -1 k 0 n -1 z -1 k Gamma k μ n -1 k -1 k c -1 k -1 μ -1 -1 -1 n k n c -1 μ -1 -1 z -1 k Gamma k μ k k -1 n c -1 k -1 μ -1 -1 -1 z PolyGamma k 1 PolyGamma k -1 n 1 -1 PolyGamma c -1 k -1 μ -1 PolyGamma k μ z 1 b -1 a n b -1 a c -1 a μ a 0 b -1 a [/itex]

 Rule Form

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

 2007-05-02