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/07.31.16.0001.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, c z] HypergeometricPFQ[{Subscript[\[Alpha], 1], \[Ellipsis], Subscript[\[Alpha], r]}, {Subscript[\[Beta], 1], \[Ellipsis], Subscript[\[Beta], s]}, d z] == Sum[Subscript[c, k] z^k, {k, 0, Infinity}] /; Subscript[c, k] == ((d^k Product[Pochhammer[Subscript[\[Alpha], j], k], {j, 1, r}])/(k! Product[Pochhammer[Subscript[\[Beta], j], k], {j, 1, s}])) HypergeometricPFQ[{-k, 1 - Subscript[\[Beta], 1] - k, \[Ellipsis], 1 - Subscript[\[Beta], s] - k, Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {1 - Subscript[\[Alpha], 1] - k, \[Ellipsis], 1 - Subscript[\[Alpha], r] - k, Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, ((-1)^(r + s - 1) c)/d] || Subscript[c, k] == ((c^k Product[Pochhammer[Subscript[a, j], k], {j, 1, p}])/(k! Product[Pochhammer[Subscript[b, j], k], {j, 1, q}])) HypergeometricPFQ[{-k, 1 - Subscript[b, 1] - k, \[Ellipsis], 1 - Subscript[b, q] - k, Subscript[\[Alpha], 1], \[Ellipsis], Subscript[\[Alpha], r]}, {1 - Subscript[a, 1] - k, \[Ellipsis], 1 - Subscript[a, p] - k, Subscript[\[Beta], 1], \[Ellipsis], Subscript[\[Beta], s]}, ((-1)^(p + q - 1) d)/c]

 Standard Form

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 MathML Form

 p F q ( a 1 , , a p ; b 1 , , b q ; c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox["F", FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], ";", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], ";", RowBox[List["c", " ", "z"]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] r F s ( α 1 , , α r ; β 1 , , β s ; d z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["r", TraditionalForm]], SubscriptBox["F", FormBox["s", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[SubscriptBox["\[Alpha]", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["\[Alpha]", "r"], ";", SubscriptBox["\[Beta]", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["\[Beta]", "s"], ";", RowBox[List["d", " ", "z"]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] k = 0 c k z k /; c k d k j = 1 r ( α j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["\[Alpha]", "j"], ")"]], "k"], Pochhammer] k ! j = 1 s ( β j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["\[Beta]", "j"], ")"]], "k"], Pochhammer] p + s + 1 F q + r ( - k , 1 - β 1 - k , , 1 - β s - k , a 1 , , a p ; 1 - α 1 - k , , 1 - α r - k , b 1 , , b q ; ( - 1 ) r + s - 1 c d ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["p", "+", "s", "+", "1"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["q", "+", "r"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], ",", RowBox[List["1", "-", SubscriptBox["\[Beta]", "1"], "-", "k"]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["\[Beta]", "s"], "-", "k"]], ",", SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], ";", RowBox[List["1", "-", SubscriptBox["\[Alpha]", "1"], "-", "k"]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["\[Alpha]", "r"], "-", "k"]], ",", SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], ";", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["r", "+", "s", "-", "1"]]], "c"]], "d"]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] c k c k j = 1 p ( a j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "j"], ")"]], "k"], Pochhammer] k ! j = 1 q ( b j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "j"], ")"]], "k"], Pochhammer] q + r + 1 F p + s ( - k , 1 - b 1 - k , , 1 - b q - k , α 1 , , α r ; 1 - a 1 - k , , 1 - a r - k , β 1 , , β s ; ( - 1 ) p + q - 1 d c ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["q", "+", "r", "+", "1"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["p", "+", "s"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], ",", RowBox[List["1", "-", SubscriptBox["b", "1"], "-", "k"]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["b", "q"], "-", "k"]], ",", SubscriptBox["\[Alpha]", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["\[Alpha]", "r"], ";", RowBox[List["1", "-", SubscriptBox["a", "1"], "-", "k"]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["a", "r"], "-", "k"]], ",", SubscriptBox["\[Beta]", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["\[Beta]", "s"], ";", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "+", "q", "-", "1"]]], "d"]], "c"]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] p F q ( a 1 , , a p ; b 1 , , b q ; c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox["F", FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], ";", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], ";", RowBox[List["c", " ", "z"]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] r F s ( α 1 , , α r ; β 1 , , β s ; d z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["r", TraditionalForm]], SubscriptBox["F", FormBox["s", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[SubscriptBox["\[Alpha]", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["\[Alpha]", "r"], ";", SubscriptBox["\[Beta]", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["\[Beta]", "s"], ";", RowBox[List["d", " ", "z"]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] k = 0 c k z k /; c k d k j = 1 r ( α j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["\[Alpha]", "j"], ")"]], "k"], Pochhammer] k ! j = 1 s ( β j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["\[Beta]", "j"], ")"]], "k"], Pochhammer] p + s + 1 F q + r ( - k , 1 - β 1 - k , , 1 - β s - k , a 1 , , a p ; 1 - α 1 - k , , 1 - α r - k , b 1 , , b q ; ( - 1 ) r + s - 1 c d ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["p", "+", "s", "+", "1"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["q", "+", "r"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], ",", RowBox[List["1", "-", SubscriptBox["\[Beta]", "1"], "-", "k"]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["\[Beta]", "s"], "-", "k"]], ",", SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], ";", RowBox[List["1", "-", SubscriptBox["\[Alpha]", "1"], "-", "k"]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["\[Alpha]", "r"], "-", "k"]], ",", SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], ";", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["r", "+", "s", "-", "1"]]], "c"]], "d"]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] c k c k j = 1 p ( a j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "j"], ")"]], "k"], Pochhammer] k ! j = 1 q ( b j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "j"], ")"]], "k"], Pochhammer] q + r + 1 F p + s ( - k , 1 - b 1 - k , , 1 - b q - k , α 1 , , α r ; 1 - a 1 - k , , 1 - a r - k , β 1 , , β s ; ( - 1 ) p + q - 1 d c ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["q", "+", "r", "+", "1"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["p", "+", "s"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], ",", RowBox[List["1", "-", SubscriptBox["b", "1"], "-", "k"]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["b", "q"], "-", "k"]], ",", SubscriptBox["\[Alpha]", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["\[Alpha]", "r"], ";", RowBox[List["1", "-", SubscriptBox["a", "1"], "-", "k"]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["a", "r"], "-", "k"]], ",", SubscriptBox["\[Beta]", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["\[Beta]", "s"], ";", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "+", "q", "-", "1"]]], "d"]], "c"]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] [/itex]

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

 2001-10-29