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 Hypergeometric2F1

 http://functions.wolfram.com/07.23.17.0151.01

 Input Form

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

 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] ( - 1 ) m - n - 1 Γ ( c ) Γ ( a + m - n ) Γ ( b + m - n ) n ! log ( 1 - z ) ( z - 1 ) m 2 F 1 ( a + m , b + m ; 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", "+", "m"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", "m"]], 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[RowBox[List["1", "-", "z"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] + ( - 1 ) m - n Γ ( c ) Γ ( a + m - n ) Γ ( b + m - n ) ( z - 1 ) m k = 0 ( a + m ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "m"]], ")"]], "k"], Pochhammer] ( b + m ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "+", "m"]], ")"]], "k"], Pochhammer] ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + n + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( a + k + m ) - ψ TagBox["\[Psi]", PolyGamma] ( b + k + m ) ) ( 1 - z ) k k ! ( k + n ) ! + Γ ( c ) ( 1 - z ) m - n Γ ( a + m ) Γ ( b + m ) k = 0 n - 1 ( n - 1 ) ! ( a + m - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "m", "-", "n"]], ")"]], "k"], Pochhammer] ( b + m - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "+", "m", "-", "n"]], ")"]], "k"], Pochhammer] ( 1 - z ) k k ! ( 1 - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "n"]], ")"]], "k"], Pochhammer] /; c - a - b TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] "\[LeftBracketingBar]" 1 - z "\[RightBracketingBar]" < 1 n = "\[LeftBracketingBar]" c - a - b "\[RightBracketingBar]" m = max ( c - a - b , 0 ) Set Condition Hypergeometric2F1 a b c z -1 m -1 n -1 Gamma c Gamma a m -1 n Gamma b m -1 n n -1 log 1 -1 z z -1 m Hypergeometric2F1 a m b m n 1 1 -1 z -1 m -1 n Gamma c Gamma a m -1 n Gamma b m -1 n -1 z -1 m k 0 Pochhammer a m k Pochhammer b m k PolyGamma k 1 PolyGamma k n 1 -1 PolyGamma a k m -1 PolyGamma b k m 1 -1 z k k k n -1 Gamma c 1 -1 z m -1 n Gamma a m Gamma b m -1 k 0 n -1 n -1 Pochhammer a m -1 n k Pochhammer b m -1 n k 1 -1 z k k Pochhammer 1 -1 n k -1 c -1 a -1 b 1 -1 z 1 n Set c -1 a -1 b m c -1 a -1 b 0 [/itex]

 Rule Form

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

 2007-05-02