html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 HypergeometricPFQ

 http://functions.wolfram.com/07.27.06.0020.02

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z] == ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/ (Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]])) ((((Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2] - Subscript[a, 1]] Gamma[Subscript[a, 3] - Subscript[a, 1]])/ (Gamma[Subscript[b, 1] - Subscript[a, 1]] Gamma[Subscript[b, 2] - Subscript[a, 1]])) Sum[(Pochhammer[Subscript[a, 1], k] Pochhammer[1 + Subscript[a, 1] - Subscript[b, 1], k] Pochhammer[1 + Subscript[a, 1] - Subscript[b, 2], k])/z^k/(k! Pochhammer[1 + Subscript[a, 1] - Subscript[a, 2], k] Pochhammer[1 + Subscript[a, 1] - Subscript[a, 3], k]), {k, 0, Infinity}])/(-z)^Subscript[a, 1] + (((Gamma[Subscript[a, 2]] Gamma[Subscript[a, 1] - Subscript[a, 2]] Gamma[Subscript[a, 3] - Subscript[a, 2]])/ (Gamma[Subscript[b, 1] - Subscript[a, 2]] Gamma[Subscript[b, 2] - Subscript[a, 2]])) Sum[(Pochhammer[Subscript[a, 2], k] Pochhammer[1 + Subscript[a, 2] - Subscript[b, 1], k] Pochhammer[1 + Subscript[a, 2] - Subscript[b, 2], k])/z^k/(k! Pochhammer[1 + Subscript[a, 2] - Subscript[a, 1], k] Pochhammer[1 + Subscript[a, 2] - Subscript[a, 3], k]), {k, 0, Infinity}])/(-z)^Subscript[a, 2] + (((Gamma[Subscript[a, 3]] Gamma[Subscript[a, 1] - Subscript[a, 3]] Gamma[Subscript[a, 2] - Subscript[a, 3]])/ (Gamma[Subscript[b, 1] - Subscript[a, 3]] Gamma[Subscript[b, 2] - Subscript[a, 3]])) Sum[(Pochhammer[Subscript[a, 3], k] Pochhammer[1 + Subscript[a, 3] - Subscript[b, 1], k] Pochhammer[1 + Subscript[a, 3] - Subscript[b, 2], k])/z^k/(k! Pochhammer[1 + Subscript[a, 3] - Subscript[a, 1], k] Pochhammer[1 + Subscript[a, 3] - Subscript[a, 2], k]), {k, 0, Infinity}])/(-z)^Subscript[a, 3]) /; Abs[z] > 1 && !Element[Subscript[a, 1] - Subscript[a, 2], Integers] && !Element[Subscript[a, 1] - Subscript[a, 3], Integers] && !Element[Subscript[a, 2] - Subscript[a, 3], Integers]

 Standard Form

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 MathML Form

 3 F 2 ( a 1 , a 2 , a 3 ; b 1 , b 2 ; z ) 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[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "3"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] Γ ( b 1 ) Γ ( b 2 ) Γ ( a 1 ) Γ ( a 2 ) Γ ( a 3 ) ( Γ ( a 1 ) Γ ( a 2 - a 1 ) Γ ( a 3 - a 1 ) Γ ( b 1 - a 1 ) Γ ( b 2 - a 1 ) ( - z ) - a 1 k = 0 ( a 1 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "1"], ")"]], "k"], Pochhammer] ( a 1 - b 1 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["b", "1"], "+", "1"]], ")"]], "k"], Pochhammer] ( a 1 - b 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["b", "2"], "+", "1"]], ")"]], "k"], Pochhammer] z - k k ! ( a 1 - a 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"], "+", "1"]], ")"]], "k"], Pochhammer] ( a 1 - a 3 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "3"], "+", "1"]], ")"]], "k"], Pochhammer] + Γ ( a 2 ) Γ ( a 1 - a 2 ) Γ ( a 3 - a 2 ) Γ ( b 1 - a 2 ) Γ ( b 2 - a 2 ) ( - z ) - a 2 k = 0 ( a 2 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "2"], ")"]], "k"], Pochhammer] ( a 2 - b 1 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["b", "1"], "+", "1"]], ")"]], "k"], Pochhammer] ( a 2 - b 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["b", "2"], "+", "1"]], ")"]], "k"], Pochhammer] z - k k ! ( - a 1 + a 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SubscriptBox["a", "1"]]], "+", SubscriptBox["a", "2"], "+", "1"]], ")"]], "k"], Pochhammer] ( a 2 - a 3 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"], "+", "1"]], ")"]], "k"], Pochhammer] + Γ ( a 3 ) Γ ( a 1 - a 3 ) Γ ( a 2 - a 3 ) Γ ( b 1 - a 3 ) Γ ( b 2 - a 3 ) ( - z ) - a 3 k = 0 ( a 3 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "3"], ")"]], "k"], Pochhammer] ( a 3 - b 1 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["b", "1"], "+", "1"]], ")"]], "k"], Pochhammer] ( a 3 - b 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["b", "2"], "+", "1"]], ")"]], "k"], Pochhammer] z - k k ! ( - a 1 + a 3 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SubscriptBox["a", "1"]]], "+", SubscriptBox["a", "3"], "+", "1"]], ")"]], "k"], Pochhammer] ( - a 2 + a 3 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SubscriptBox["a", "2"]]], "+", SubscriptBox["a", "3"], "+", "1"]], ")"]], "k"], Pochhammer] ) /; "\[LeftBracketingBar]" z "\[RightBracketingBar]" > 1 a 1 - a 2 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] a 1 - a 3 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] a 2 - a 3 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript a 3 Subscript b 1 Subscript b 2 z Gamma Subscript b 1 Gamma Subscript b 2 Gamma Subscript a 1 Gamma Subscript a 2 Gamma Subscript a 3 -1 Gamma Subscript a 1 Gamma Subscript a 2 -1 Subscript a 1 Gamma Subscript a 3 -1 Subscript a 1 Gamma Subscript b 1 -1 Subscript a 1 Gamma Subscript b 2 -1 Subscript a 1 -1 -1 z -1 Subscript a 1 k 0 Pochhammer Subscript a 1 k Pochhammer Subscript a 1 -1 Subscript b 1 1 k Pochhammer Subscript a 1 -1 Subscript b 2 1 k z -1 k k Pochhammer Subscript a 1 -1 Subscript a 2 1 k Pochhammer Subscript a 1 -1 Subscript a 3 1 k -1 Gamma Subscript a 2 Gamma Subscript a 1 -1 Subscript a 2 Gamma Subscript a 3 -1 Subscript a 2 Gamma Subscript b 1 -1 Subscript a 2 Gamma Subscript b 2 -1 Subscript a 2 -1 -1 z -1 Subscript a 2 k 0 Pochhammer Subscript a 2 k Pochhammer Subscript a 2 -1 Subscript b 1 1 k Pochhammer Subscript a 2 -1 Subscript b 2 1 k z -1 k k Pochhammer -1 Subscript a 1 Subscript a 2 1 k Pochhammer Subscript a 2 -1 Subscript a 3 1 k -1 Gamma Subscript a 3 Gamma Subscript a 1 -1 Subscript a 3 Gamma Subscript a 2 -1 Subscript a 3 Gamma Subscript b 1 -1 Subscript a 3 Gamma Subscript b 2 -1 Subscript a 3 -1 -1 z -1 Subscript a 3 k 0 Pochhammer Subscript a 3 k Pochhammer Subscript a 3 -1 Subscript b 1 1 k Pochhammer Subscript a 3 -1 Subscript b 2 1 k z -1 k k Pochhammer -1 Subscript a 1 Subscript a 3 1 k Pochhammer -1 Subscript a 2 Subscript a 3 1 k -1 z 1 Subscript a 1 -1 Subscript a 2 Subscript a 1 -1 Subscript a 3 Subscript a 2 -1 Subscript a 3 [/itex]

 Rule Form

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

 2001-10-29