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 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.06.0056.01

 Input Form

 Hypergeometric2F1Regularized[a, b, -n, z] == Subscript[F, Infinity][z, a, b, n] /; (Subscript[F, m][z, a, b, n] == ((Pochhammer[a, n + 1] Pochhammer[b, n + 1])/(n + 1)!) z^(1 + n) Sum[(Pochhammer[1 + a + n, k] Pochhammer[1 + b + n, k] z^k)/ (Pochhammer[2 + n, k] k!), {k, 0, m}] == z^(1 + n) Pochhammer[a, n + 1] Pochhammer[b, n + 1] Hypergeometric2F1Regularized[1 + a + n, 1 + b + n, 2 + n, z] - z^(2 + m + n) Pochhammer[a, m + n + 2] Pochhammer[b, m + n + 2] HypergeometricPFQRegularized[{1, 2 + a + m + n, 2 + b + m + n}, {2 + m, 3 + m + n}, z] && Element[m, Integers] && m >= 0) && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 2 F ~ 1 ( a , b ; - n ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[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["-", "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 ) ( 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] z n + 1 ( n + 1 ) ! k = 0 m ( a + n + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "n", "+", "1"]], ")"]], "k"], Pochhammer] ( b + n + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "+", "n", "+", "1"]], ")"]], "k"], Pochhammer] z k ( n + 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", "2"]], ")"]], "k"], Pochhammer] k ! z n + 1 ( 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] 2 F ~ 1 ( a + n + 1 , b + n + 1 ; n + 2 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["a", "+", "n", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", "n", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["n", "+", "2"]], 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 m + n + 2 ( a ) m + n + 2 TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], RowBox[List["m", "+", "n", "+", "2"]]], Pochhammer] ( b ) m + n + 2 TagBox[SubscriptBox[RowBox[List["(", "b", ")"]], RowBox[List["m", "+", "n", "+", "2"]]], Pochhammer] 3 F ~ 2 ( 1 , a + m + n + 2 , b + m + n + 2 ; m + 2 , m + n + 3 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "m", "+", "n", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", "m", "+", "n", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["m", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "n", "+", "3"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) m ) n Condition Hypergeometric2F1Regularized a b -1 n z Subscript F z a b n Subscript F m z a b n Pochhammer a n 1 Pochhammer b n 1 z n 1 n 1 -1 k 0 m Pochhammer a n 1 k Pochhammer b n 1 k z k Pochhammer n 2 k k -1 z n 1 Pochhammer a n 1 Pochhammer b n 1 Hypergeometric2F1Regularized a n 1 b n 1 n 2 z -1 z m n 2 Pochhammer a m n 2 Pochhammer b m n 2 HypergeometricPFQRegularized 1 a m n 2 b m n 2 m 2 m n 3 z m n [/itex]

 Rule Form

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

 2007-05-02