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 Hypergeometric2F1

 http://functions.wolfram.com/07.23.03.0146.01

 Input Form

 Hypergeometric2F1[n, b, m, z] == (((m - 1)! Gamma[b - m + n])/((n - 1)! Gamma[b])) (1 - z)^(m - n - b) z^(n - m) Sum[((Pochhammer[1 - n, k] Pochhammer[m - n, k])/ (Pochhammer[1 - b + m - n, k] k!)) ((z - 1)/z)^k, {k, 0, n - 1}] + ((((m - 1)! Gamma[-b + m - n])/((m - n - 1)! Gamma[-b + m])) Sum[((Pochhammer[1 - m + n, k] Pochhammer[n, k])/ (Pochhammer[1 + b - m + n, k] k!)) ((z - 1)/z)^k, {k, 0, m - n - 1}])/ z^n /; Element[m, Integers] && m > 0 && Element[n, Integers] && n >= 0 && m > n

 Standard Form

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 MathML Form

 2 F 1 ( n , b ; m ; 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["n", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["m", 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] ( m - 1 ) ! Γ ( b - m + n ) ( n - 1 ) ! Γ ( b ) ( 1 - z ) m - n - b z n - m k = 0 n - 1 ( 1 - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "n"]], ")"]], "k"], Pochhammer] ( m - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]], "k"], Pochhammer] ( m - b - n + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "-", "b", "-", "n", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( z - 1 z ) k + z - n ( m - 1 ) ! Γ ( m - n - b ) ( m - n - 1 ) ! Γ ( m - b ) k = 0 m - n - 1 ( n - m + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "-", "m", "+", "1"]], ")"]], "k"], Pochhammer] ( n ) k TagBox[SubscriptBox[RowBox[List["(", "n", ")"]], "k"], Pochhammer] ( b - m + n + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "-", "m", "+", "n", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( z - 1 z ) k /; m + n m > n Condition Hypergeometric2F1 n b m z m -1 Gamma b -1 m n n -1 Gamma b -1 1 -1 z m -1 n -1 b z n -1 m k 0 n -1 Pochhammer 1 -1 n k Pochhammer m -1 n k Pochhammer m -1 b -1 n 1 k k -1 z -1 z -1 k z -1 n m -1 Gamma m -1 n -1 b m -1 n -1 Gamma m -1 b -1 k 0 m -1 n -1 Pochhammer n -1 m 1 k Pochhammer n k Pochhammer b -1 m n 1 k k -1 z -1 z -1 k m SuperPlus n m n [/itex]

 Rule Form

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

 2001-10-29