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 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.06.0027.01

 Input Form

 Hypergeometric2F1Regularized[a, a + n, a + m, z] == ((-1)^n/(Gamma[a] (m - n - 1)!)) (-z)^(-a - n) Sum[(((Pochhammer[a + n, k] Pochhammer[1 - m + n, k])/(k! (k + n)!)) (Log[-z] + PolyGamma[1 + k] - PolyGamma[m - n - k] + PolyGamma[1 + n + k] - PolyGamma[a + n + k]))/z^k, {k, 0, m - n - 1}] + ((Gamma[n]/(Gamma[m] Gamma[a + n])) Sum[(Pochhammer[a, k] Pochhammer[1 - m, k])/(k! Pochhammer[1 - n, k])/ z^k, {k, 0, n - 1}])/(-z)^a + (((-1)^n Gamma[a + m])/(Gamma[a] Gamma[a + n])) (-z)^(-a - m) HypergeometricPFQRegularized[{1, 1, a + m}, {1 + m, 1 + m - n}, 1/z] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m >= 0 && m >= n && !IntervalMemberQ[Interval[{0, 1}], z]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", RowBox[List["a", "+", "n"]], ",", RowBox[List["a", "+", "m"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "n", "-", "1"]], ")"]], "!"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "a"]], "-", "n"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "-", "n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "m", "+", "n"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["m", "-", "n", "-", "k"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "n", "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "n", "+", "k"]], "]"]]]], ")"]], SuperscriptBox["z", RowBox[List["-", "k"]]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List["Gamma", "[", "n", "]"]], RowBox[List[RowBox[List["Gamma", "[", "m", "]"]], RowBox[List["Gamma", "[", RowBox[List["a", "+", "n"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "m"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "n"]], ",", "k"]], "]"]]]]], SuperscriptBox["z", RowBox[List["-", "k"]]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "m"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], RowBox[List["Gamma", "[", RowBox[List["a", "+", "n"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "a"]], "-", "m"]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", RowBox[List["a", "+", "m"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "m"]], ",", RowBox[List["1", "+", "m", "-", "n"]]]], "}"]], ",", FractionBox["1", "z"]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "n"]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]]]

 MathML Form

 2 F ~ 1 ( a , a + n ; a + m ; 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[RowBox[List["a", "+", "n"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "+", "m"]], 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 ( n ) ( - z ) - a Γ ( m ) Γ ( a + n ) k = 0 n - 1 ( a ) k TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "k"], Pochhammer] ( 1 - m ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], "k"], Pochhammer] z - k k ! ( 1 - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "n"]], ")"]], "k"], Pochhammer] + ( - 1 ) n Γ ( a + m ) Γ ( a ) Γ ( a + n ) ( - z ) - a - m 3 F ~ 2 ( 1 , 1 , a + m ; m + 1 , m - n + 1 ; 1 z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "m"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["m", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "-", "n", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[FractionBox["1", "z"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] + ( - 1 ) n Γ ( a ) ( m - n - 1 ) ! ( - z ) - a - n k = 0 m - n - 1 ( a + n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "n"]], ")"]], "k"], Pochhammer] ( 1 - m + n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "m", "+", "n"]], ")"]], "k"], Pochhammer] k ! ( k + n ) ! ( log ( - z ) - ψ TagBox["\[Psi]", PolyGamma] ( m - n - k ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + n + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( a + k + n ) ) z - k /; n m m n z ( 0 , 1 ) 2 F ~ 1 ( a , a + n ; a + m ; 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[RowBox[List["a", "+", "n"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "+", "m"]], 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 ( n ) ( - z ) - a Γ ( m ) Γ ( a + n ) k = 0 n - 1 ( a ) k TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "k"], Pochhammer] ( 1 - m ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], "k"], Pochhammer] z - k k ! ( 1 - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "n"]], ")"]], "k"], Pochhammer] + ( - 1 ) n Γ ( a + m ) Γ ( a ) Γ ( a + n ) ( - z ) - a - m 3 F ~ 2 ( 1 , 1 , a + m ; m + 1 , m - n + 1 ; 1 z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "m"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["m", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "-", "n", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[FractionBox["1", "z"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] + ( - 1 ) n Γ ( a ) ( m - n - 1 ) ! ( - z ) - a - n k = 0 m - n - 1 ( a + n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "+", "n"]], ")"]], "k"], Pochhammer] ( 1 - m + n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "m", "+", "n"]], ")"]], "k"], Pochhammer] k ! ( k + n ) ! ( log ( - z ) - ψ TagBox["\[Psi]", PolyGamma] ( m - n - k ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + n + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( a + k + n ) ) z - k /; n m m n z ( 0 , 1 ) [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a_", ",", RowBox[List["a_", "+", "n_"]], ",", RowBox[List["a_", "+", "m_"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "a"]], "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "-", "n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "m", "+", "n"]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["m", "-", "n", "-", "k"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "n", "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "n", "+", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["-", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "n", "-", "1"]], ")"]], "!"]]]]], "+", FractionBox[RowBox[List[RowBox[List["Gamma", "[", "n", "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "m"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["-", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "n"]], ",", "k"]], "]"]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", "m", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "n"]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "m"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "a"]], "-", "m"]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", RowBox[List["a", "+", "m"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "m"]], ",", RowBox[List["1", "+", "m", "-", "n"]]]], "}"]], ",", FractionBox["1", "z"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "n"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[GreaterEqual]", "n"]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]]]

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

 2001-10-29