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 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.06.0061.01

 Input Form

 Hypergeometric2F1Regularized[a, b, a + b + n, z] == Subscript[F, Infinity][z, a, b, n] /; (Subscript[F, m][z, a, b, n] == (1/(Gamma[a + n] Gamma[b + n])) Sum[(1/k!) Gamma[n - k] Pochhammer[a, k] Pochhammer[b, k] (z - 1)^k, {k, 0, n - 1}] + ((z - 1)^n/(Gamma[a] Gamma[b])) Sum[((Pochhammer[a + n, k] Pochhammer[b + n, k])/(k! (n + k)!)) (PolyGamma[k + 1] + PolyGamma[k + n + 1] - PolyGamma[a + k + n] - PolyGamma[b + k + n] - Log[1 - z]) (1 - z)^k, {k, 0, m}] == Hypergeometric2F1Regularized[a, b, a + b + n, z] - ((-1)^n/(Gamma[a] Gamma[b] Gamma[a + n] Gamma[b + n])) MeijerG[{{1 + m + n, 1 + m + n, 1 - a, 1 - b}, {}}, {{n + m + 1, n + m + 1}, {0, n}}, 1 - z] && Element[m, Integers] && m >= 0) && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 2 F ~ 1 ( a , b ; a + b + n ; 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["b", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "+", "b", "+", "n"]], 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] F ( z , a , b , n ) /; ( ( F m ( z , a , b , n ) 1 Γ ( a + n ) Γ ( b + n ) k = 0 n - 1 Γ ( n - k ) ( a ) k TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "k"], Pochhammer] ( b ) k TagBox[SubscriptBox[RowBox[List["(", "b", ")"]], "k"], Pochhammer] k ! ( z - 1 ) k + ( z - 1 ) n Γ ( a ) Γ ( b ) k = 0 m ( a + n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "n"]], ")"]], "k"], Pochhammer] ( b + n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "+", "n"]], ")"]], "k"], Pochhammer] k ! ( k + n ) ! ( - log ( 1 - z ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + n + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( a + k + n ) - ψ TagBox["\[Psi]", PolyGamma] ( b + k + n ) ) ( 1 - z ) k 2 F ~ 1 ( a , b ; a + b + n ; 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["b", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "+", "b", "+", "n"]], 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] - ( - 1 ) n Γ ( a ) Γ ( b ) Γ ( a + n ) Γ ( b + n ) G 4 , 4 2 , 4 ( 1 - z m + n + 1 , m + n + 1 , 1 - a , 1 - b m + n + 1 , m + n + 1 , 0 , n ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "4"]], RowBox[List["2", ",", "4"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["1", "-", "z"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["m", "+", "n", "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "n", "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "a"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "b"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List["m", "+", "n", "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "n", "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox["n", MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) m ) n Condition Hypergeometric2F1Regularized a b a b n z Subscript F z a b n Subscript F m z a b n 1 Gamma a n Gamma b n -1 k 0 n -1 Gamma n -1 k Pochhammer a k Pochhammer b k k -1 z -1 k z -1 n Gamma a Gamma b -1 k 0 m Pochhammer a n k Pochhammer b n k k k n -1 -1 1 -1 z PolyGamma k 1 PolyGamma k n 1 -1 PolyGamma a k n -1 PolyGamma b k n 1 -1 z k Hypergeometric2F1Regularized a b a b n z -1 -1 n Gamma a Gamma b Gamma a n Gamma b n -1 MeijerG m n 1 m n 1 1 -1 a 1 -1 b m n 1 m n 1 0 n 1 -1 z m n [/itex]

 Rule Form

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

 2007-05-02