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 HypergeometricPFQ

 http://functions.wolfram.com/07.27.03.0034.01

 Input Form

 HypergeometricPFQ[{1, b, c}, {m - b, m - c}, 1] == (-(1/2)) Sum[(Pochhammer[1 + b - m, k] Pochhammer[1 + c - m, k])/ (Pochhammer[1 - b, k] Pochhammer[1 - c, k]), {k, 1, m - 2}] + ((Sqrt[Pi] Pochhammer[m - 2 b, m] Pochhammer[m - 2 c, m] Pochhammer[m - b - c, m])/(2^(2 m + 1) Pochhammer[m - b - c - 1/2, m])) ((Gamma[1 - b] Gamma[1 - c] Gamma[2 m - b - c - 1/2])/ (Gamma[1/2 - b + m] Gamma[1/2 - c + m] Gamma[2 m - b - c])) /; Re[b + c] < m - 1/2 && Element[m - 2, Integers] && m - 2 >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "b", ",", "c"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["m", "-", "b"]], ",", RowBox[List["m", "-", "c"]]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["m", "-", "2"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "b", "-", "m"]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "c", "-", "m"]], ",", "k"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "c"]], ",", "k"]], "]"]]]], ")"]]]]]]]], "+", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", RowBox[List["2", "b"]]]], ",", "m"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", RowBox[List["2", "c"]]]], ",", "m"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "b", "-", "c"]], ",", "m"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", "m"]], "+", "1"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "b", "-", "c", "-", FractionBox["1", "2"]]], ",", "m"]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "b"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "c"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", "m"]], "-", "b", "-", "c", "-", FractionBox["1", "2"]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "b", "+", "m"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "c", "+", "m"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", "m"]], "-", "b", "-", "c"]], "]"]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", RowBox[List["b", "+", "c"]], "]"]], "<", RowBox[List["m", "-", FractionBox["1", "2"]]]]], "\[And]", RowBox[List[RowBox[List["m", "-", "2"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["m", "-", "2"]], "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 3 F 2 ( 1 , b , c ; m - b , m - c ; 1 ) 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["b", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["c", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["m", "-", "b"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "-", "c"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ( π ( m - 2 b ) m TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "b"]]]], ")"]], "m"], Pochhammer] ( m - 2 c ) m TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "c"]]]], ")"]], "m"], Pochhammer] ( m - b - c ) m TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "-", "b", "-", "c"]], ")"]], "m"], Pochhammer] Γ ( 1 - b ) Γ ( 1 - c ) Γ ( 2 m - b - c - 1 2 ) ) / ( 2 2 m + 1 ( m - b - c - 1 2 ) m TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "-", "b", "-", "c", "-", FractionBox["1", "2"]]], ")"]], "m"], Pochhammer] Γ ( m - b + 1 2 ) Γ ( m - c + 1 2 ) Γ ( 2 m - b - c ) ) - 1 2 k = 1 m - 2 ( b - m + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["b", "-", "m", "+", "1"]], ")"]], "k"], Pochhammer] ( c - m + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["c", "-", "m", "+", "1"]], ")"]], "k"], Pochhammer] ( 1 - b ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], "k"], Pochhammer] ( 1 - c ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "c"]], ")"]], "k"], Pochhammer] /; Re ( b + c ) < m - 1 2 m - 2 Condition HypergeometricPFQ 1 b c m -1 b m -1 c 1 1 2 Pochhammer m -1 2 b m Pochhammer m -1 2 c m Pochhammer m -1 b -1 c m Gamma 1 -1 b Gamma 1 -1 c Gamma 2 m -1 b -1 c -1 1 2 2 2 m 1 Pochhammer m -1 b -1 c -1 1 2 m Gamma m -1 b 1 2 Gamma m -1 c 1 2 Gamma 2 m -1 b -1 c -1 -1 1 2 k 1 m -2 Pochhammer b -1 m 1 k Pochhammer c -1 m 1 k Pochhammer 1 -1 b k Pochhammer 1 -1 c k -1 b c m -1 1 2 m -2 [/itex]

 Rule Form

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

 2001-10-29