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 HypergeometricPFQ

 http://functions.wolfram.com/07.29.03.0007.01

 Input Form

 HypergeometricPFQ[{1, b, c, b + 1/2, c + 1/2}, {1/2 - b + n, 1 - b + n, 1/2 - c + n, 1 - c + n}, 1] == 1 - ((1 + (-1)^n) Pochhammer[2 b - 2 n, n] Pochhammer[2 c - 2 n, n])/ (4 Pochhammer[1 - 2 b, n] Pochhammer[1 - 2 c, n]) + Gamma[1 - 2 b] Gamma[1 - 2 c] (((Sqrt[Pi] Gamma[1/2 - 2 b - 2 c + 2 n] Pochhammer[1 - 4 b + 2 n, 2 n] Pochhammer[1 - 4 c + 2 n, 2 n])/(2^(4 n) Gamma[1/2 - 2 b + 2 n] Gamma[1/2 - 2 c + 2 n]) + (-1)^n Pochhammer[1 - 2 b + n, n] Pochhammer[1 - 2 c + n, n])/(4 Gamma[1 - 2 b - 2 c + 2 n])) - Sum[(Pochhammer[2 b - 2 n, 2 k] Pochhammer[2 c - 2 n, 2 k])/ (Pochhammer[1 - 2 b, 2 k] Pochhammer[1 - 2 c, 2 k]), {k, 0, Floor[(n - 1)/2]}] /; Re[b + c] < n + 1/4 && Element[n, Integers] && n > 0

 Standard Form

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 MathML Form

 5 F 4 ( 1 , b , c , b + 1 2 , c + 1 2 ; 1 2 - b + n , - b + n + 1 , 1 2 - c + n , 1 - c + n ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["5", TraditionalForm]], SubscriptBox["F", FormBox["4", 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]], ",", TagBox[RowBox[List["b", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["c", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "2"], "-", "b", "+", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "b"]], "+", "n", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], "-", "c", "+", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "c", "+", "n"]], 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] 1 - ( 1 + ( - 1 ) n ) ( 2 b - 2 n ) n TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", RowBox[List["2", " ", "n"]]]], ")"]], "n"], Pochhammer] ( 2 c - 2 n ) n TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "-", RowBox[List["2", " ", "n"]]]], ")"]], "n"], Pochhammer] 4 ( 1 - 2 b ) n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "b"]]]], ")"]], "n"], Pochhammer] ( 1 - 2 c ) n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "c"]]]], ")"]], "n"], Pochhammer] + Γ ( 1 - 2 b ) Γ ( 1 - 2 c ) 4 Γ ( 1 - 2 b - 2 c + 2 n ) ( ( π Γ ( 1 2 - 2 b - 2 c + 2 n ) ( 1 - 4 b + 2 n ) 2 n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "b"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], RowBox[List["2", " ", "n"]]], Pochhammer] ( 1 - 4 c + 2 n ) 2 n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "c"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], RowBox[List["2", " ", "n"]]], Pochhammer] ) / ( 2 4 n Γ ( 1 2 - 2 b + 2 n ) Γ ( 1 2 - 2 c + 2 n ) ) + ( - 1 ) n ( 1 - 2 b + n ) n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "b"]], "+", "n"]], ")"]], "n"], Pochhammer] ( 1 - 2 c + n ) n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "c"]], "+", "n"]], ")"]], "n"], Pochhammer] ) - k = 0 n - 1 2 ( 2 b - 2 n ) 2 k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", RowBox[List["2", " ", "n"]]]], ")"]], RowBox[List["2", " ", "k"]]], Pochhammer] ( 2 c - 2 n ) 2 k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c"]], "-", RowBox[List["2", " ", "n"]]]], ")"]], RowBox[List["2", " ", "k"]]], Pochhammer] ( 1 - 2 b ) 2 k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "b"]]]], ")"]], RowBox[List["2", " ", "k"]]], Pochhammer] ( 1 - 2 c ) 2 k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "c"]]]], ")"]], RowBox[List["2", " ", "k"]]], Pochhammer] /; Re ( b + c ) < n + 1 4 n + Condition HypergeometricPFQ 1 b c b 1 2 c 1 2 1 2 -1 b n -1 b n 1 1 2 -1 c n 1 -1 c n 1 1 -1 1 -1 n Pochhammer 2 b -1 2 n n Pochhammer 2 c -1 2 n n 4 Pochhammer 1 -1 2 b n Pochhammer 1 -1 2 c n -1 Gamma 1 -1 2 b Gamma 1 -1 2 c 4 Gamma 1 -1 2 b -1 2 c 2 n -1 1 2 Gamma 1 2 -1 2 b -1 2 c 2 n Pochhammer 1 -1 4 b 2 n 2 n Pochhammer 1 -1 4 c 2 n 2 n 2 4 n Gamma 1 2 -1 2 b 2 n Gamma 1 2 -1 2 c 2 n -1 -1 n Pochhammer 1 -1 2 b n n Pochhammer 1 -1 2 c n n -1 k 0 n -1 2 -1 Pochhammer 2 b -1 2 n 2 k Pochhammer 2 c -1 2 n 2 k Pochhammer 1 -1 2 b 2 k Pochhammer 1 -1 2 c 2 k -1 b c n 1 4 n SuperPlus [/itex]

 Rule Form

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

 2001-10-29