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 Hypergeometric2F1

 http://functions.wolfram.com/07.23.06.0009.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", "b", ",", RowBox[List["a", "+", "b", "-", "n"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]], RowBox[List["Gamma", "[", RowBox[List["a", "+", "b", "-", "n"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], RowBox[List["Gamma", "[", "b", "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["-", "n"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "-", "n"]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["b", "-", "n"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "n"]], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List["Gamma", "[", RowBox[List["a", "+", "b", "-", "n"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "-", "n"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["b", "-", "n"]], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List["b", ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "k"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "n", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["b", "+", "k"]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["1", "-", "z"]], "]"]], "<", "1"]]]]]]]]

 MathML Form

 2 F 1 ( a , b ; a + b - n ; 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[RowBox[List["a", "+", "b", "-", "n"]], 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] ( n - 1 ) ! Γ ( a + b - n ) Γ ( a ) Γ ( b ) ( 1 - z ) - n k = 0 n - 1 ( 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] ( 1 - z ) k k ! ( 1 - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "n"]], ")"]], "k"], Pochhammer] + ( - 1 ) n Γ ( a + b - n ) Γ ( a - n ) Γ ( b - n ) k = 0 ( a ) k TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "k"], Pochhammer] ( b ) k TagBox[SubscriptBox[RowBox[List["(", "b", ")"]], "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 ) - ψ TagBox["\[Psi]", PolyGamma] ( b + k ) ) ( 1 - z ) k /; n + "\[LeftBracketingBar]" 1 - z "\[RightBracketingBar]" < 1 Condition Hypergeometric2F1 a b a b -1 n z n -1 Gamma a b -1 n Gamma a Gamma b -1 1 -1 z -1 n k 0 n -1 Pochhammer a -1 n k Pochhammer b -1 n k 1 -1 z k k Pochhammer 1 -1 n k -1 -1 n Gamma a b -1 n Gamma a -1 n Gamma b -1 n -1 k 0 Pochhammer a k Pochhammer b k k k n -1 -1 log 1 -1 z PolyGamma k 1 PolyGamma k n 1 -1 PolyGamma a k -1 PolyGamma b k 1 -1 z k n SuperPlus 1 -1 z 1 [/itex]

 Rule Form

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

 2001-10-29