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 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.06.0059.01

 Input Form

 Hypergeometric2F1Regularized[a, b, c, z] == Subscript[F, Infinity][z, a, b, c] /; (Subscript[F, n][z, a, b, c] == (Gamma[c - a - b]/(Gamma[c - a] Gamma[c - b])) Sum[((Pochhammer[a, k] Pochhammer[b, k])/(Pochhammer[a + b - c + 1, k] k!)) (1 - z)^k, {k, 0, n}] + (Gamma[a + b - c]/(Gamma[a] Gamma[b])) (1 - z)^(c - a - b) Sum[((Pochhammer[c - a, k] Pochhammer[c - b, k])/ (Pochhammer[1 - a - b + c, k] k!)) (1 - z)^k, {k, 0, n}] == Hypergeometric2F1Regularized[a, b, c, z] - (1 - z)^(1 + n) ((Gamma[-a - b + c] Pochhammer[a, 1 + n] Pochhammer[b, 1 + n])/ ((1 + n)! Gamma[-a + c] Gamma[-b + c] Pochhammer[1 + a + b - c, 1 + n])) HypergeometricPFQ[{1, 1 + a + n, 1 + b + n}, {2 + n, 2 + a + b - c + n}, 1 - z] - (1 - z)^(1 - a - b + c + n) ((Gamma[a + b - c] Pochhammer[-a + c, 1 + n] Pochhammer[-b + c, 1 + n])/ ((1 + n)! Gamma[a] Gamma[b] Pochhammer[1 - a - b + c, 1 + n])) HypergeometricPFQ[{1, 1 - a + c + n, 1 - b + c + n}, {2 + n, 2 - a - b + c + n}, 1 - z] && Element[n, Integers] && n >= 0) && !Element[c - a - b, Integers]

 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[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["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] F ( z , a , b , c ) /; ( ( F n ( z , a , b , c ) Γ ( a + b - c ) Γ ( a ) Γ ( b ) ( 1 - z ) c - a - b k = 0 n ( c - a ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["c", "-", "a"]], ")"]], "k"], Pochhammer] ( c - b ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["c", "-", "b"]], ")"]], "k"], Pochhammer] ( c - a - b + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["c", "-", "a", "-", "b", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( 1 - z ) k + Γ ( c - a - b ) Γ ( c - a ) Γ ( c - b ) k = 0 n ( a ) k TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "k"], Pochhammer] ( b ) k TagBox[SubscriptBox[RowBox[List["(", "b", ")"]], "k"], Pochhammer] ( a + b - c + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "-", "c", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( 1 - z ) k 2 F ~ 1 ( a , b ; 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["b", 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] - Γ ( c - a - b ) ( a ) n + 1 TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] ( b ) n + 1 TagBox[SubscriptBox[RowBox[List["(", "b", ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] ( n + 1 ) ! Γ ( c - a ) Γ ( c - b ) ( a + b - c + 1 ) n + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "b", "-", "c", "+", "1"]], ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] ( 1 - z ) n + 1 3 F 2 ( 1 , a + n + 1 , b + n + 1 ; n + 2 , a + b - c + n + 2 ; 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "n", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", "n", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "b", "-", "c", "+", "n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["1", "-", "z"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - Γ ( a + b - c ) ( c - a ) n + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["c", "-", "a"]], ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] ( c - b ) n + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["c", "-", "b"]], ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] ( n + 1 ) ! Γ ( a ) Γ ( b ) ( c - a - b + 1 ) n + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["c", "-", "a", "-", "b", "+", "1"]], ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] ( 1 - z ) c - a - b + n + 1 3 F 2 ( 1 , - a + c + n + 1 , - b + c + n + 1 ; n + 2 , c - a - b + n + 2 ; 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "a"]], "+", "c", "+", "n", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "b"]], "+", "c", "+", "n", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["c", "-", "a", "-", "b", "+", "n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["1", "-", "z"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] ) c - a - b TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Hypergeometric2F1Regularized a b c z Subscript F z a b c Subscript F n z a b c Gamma a b -1 c Gamma a Gamma b -1 1 -1 z c -1 a -1 b k 0 n Pochhammer c -1 a k Pochhammer c -1 b k Pochhammer c -1 a -1 b 1 k k -1 1 -1 z k Gamma c -1 a -1 b Gamma c -1 a Gamma c -1 b -1 k 0 n Pochhammer a k Pochhammer b k Pochhammer a b -1 c 1 k k -1 1 -1 z k Hypergeometric2F1Regularized a b c z -1 Gamma c -1 a -1 b Pochhammer a n 1 Pochhammer b n 1 n 1 Gamma c -1 a Gamma c -1 b Pochhammer a b -1 c 1 n 1 -1 1 -1 z n 1 HypergeometricPFQ 1 a n 1 b n 1 n 2 a b -1 c n 2 1 -1 z -1 Gamma a b -1 c Pochhammer c -1 a n 1 Pochhammer c -1 b n 1 n 1 Gamma a Gamma b Pochhammer c -1 a -1 b 1 n 1 -1 1 -1 z c -1 a -1 b n 1 HypergeometricPFQ 1 -1 a c n 1 -1 b c n 1 n 2 c -1 a -1 b n 2 1 -1 z n c -1 a -1 b [/itex]

 Rule Form

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

 2007-05-02