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 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.06.0025.01

 Input Form

 Hypergeometric2F1Regularized[a, a + n, c, z] == ((-z)^(-a - n)/(Gamma[a + n] Gamma[c - a])) Sum[(((Pochhammer[a, k + n] Pochhammer[1 + a - c, k + n])/(k! (k + n)!)) (PolyGamma[k + n + 1] + PolyGamma[k + 1] - PolyGamma[a + n + k] - PolyGamma[c - a - n - k]))/z^k, {k, 0, Infinity}] + (Sin[(c - a) Pi]/(Pi n! Gamma[a])) Gamma[1 + a - c + n] Log[-z] (-z)^(-a - n) Hypergeometric2F1[a + n, 1 + a - c + n, 1 + n, 1/z] + ((1/Gamma[a + n]) Sum[(Pochhammer[a, k] Gamma[n - k])/ (k! Gamma[c - a - k])/z^k, {k, 0, n - 1}])/(-z)^a /; Abs[z] > 1 && Element[n, Integers] && n >= 0 && !Element[c - a, Integers]

 Standard Form

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 MathML Form

 2 F ~ 1 ( a , a + n ; c ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "n"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] ( - z ) - a Γ ( a + n ) k = 0 n - 1 ( a ) k TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "k"], Pochhammer] Γ ( n - k ) z - k k ! Γ ( c - a - k ) + sin ( ( c - a ) π ) Γ ( a - c + n + 1 ) π n ! Γ ( a ) log ( - z ) ( - z ) - a - n 2 F 1 ( a + n , a - c + n + 1 ; n + 1 ; 1 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[RowBox[List["a", "+", "n"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "-", "c", "+", "n", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["n", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["1", "z"], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + ( - z ) - a - n Γ ( a + n ) Γ ( c - a ) k = 0 ( a ) k + n TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], RowBox[List["k", "+", "n"]]], Pochhammer] ( a - c + 1 ) k + n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", "c", "+", "1"]], ")"]], RowBox[List["k", "+", "n"]]], Pochhammer] k ! ( k + n ) ! ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + n + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( c - a - k - n ) - ψ TagBox["\[Psi]", PolyGamma] ( a + k + n ) ) z - k /; "\[LeftBracketingBar]" z "\[RightBracketingBar]" > 1 n c - a TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Hypergeometric2F1Regularized a a n c z -1 z -1 a Gamma a n -1 k 0 n -1 Pochhammer a k Gamma n -1 k z -1 k k Gamma c -1 a -1 k -1 c -1 a Gamma a -1 c n 1 n Gamma a -1 log -1 z -1 z -1 a -1 n Hypergeometric2F1 a n a -1 c n 1 n 1 1 z -1 -1 z -1 a -1 n Gamma a n Gamma c -1 a -1 k 0 Pochhammer a k n Pochhammer a -1 c 1 k n k k n -1 PolyGamma k 1 PolyGamma k n 1 -1 PolyGamma c -1 a -1 k -1 n -1 PolyGamma a k n z -1 k z 1 n c -1 a [/itex]

 Rule Form

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

 2001-10-29