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/07.26.17.0022.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] == Sum[(z^k/k!) ((Pochhammer[Subscript[a, 1], k] Pochhammer[Subscript[a, 2], k])/Product[Pochhammer[Subscript[b, j], k], {j, 1, 3}]) HypergeometricPFQ[{1, (Subscript[a, 1] + k)/n, \[Ellipsis], (Subscript[a, 1] + k + n - 1)/n, (Subscript[a, 2] + k)/n, \[Ellipsis], (Subscript[a, 2] + k + n - 1)/n}, {(k + 1)/n, \[Ellipsis], (k + n)/n, (Subscript[b, 1] + k)/n, \[Ellipsis], (Subscript[b, 1] + k + n - 1)/n, (Subscript[b, 2] + k)/n, \[Ellipsis], (Subscript[b, 2] + k + n - 1)/n, (Subscript[b, 3] + k)/n, \[Ellipsis], (Subscript[b, 3] + k + n - 1)/n}, z^n/n^(2 n)], {k, 0, n - 1}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[SuperscriptBox["z", "k"], RowBox[List["k", "!"]]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "1"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "2"], ",", "k"]], "]"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "3"], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "j"], ",", "k"]], "]"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k", "+", "n", "-", "1"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["a", "2"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "2"], "+", "k", "+", "n", "-", "1"]], "n"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["k", "+", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["k", "+", "n"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "k", "+", "n", "-", "1"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "2"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "2"], "+", "k", "+", "n", "-", "1"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "3"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "3"], "+", "k", "+", "n", "-", "1"]], "n"]]], "}"]], ",", RowBox[List[SuperscriptBox["n", RowBox[List[RowBox[List["-", "2"]], "n"]]], SuperscriptBox["z", "n"]]]]], "]"]]]]]]]]]]

 MathML Form

 2 F 3 ( a 1 , a 2 ; b 1 , b 2 , b 3 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["3", 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]]]], 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]], ",", TagBox[SubscriptBox["b", "3"], 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] k = 0 n - 1 z k ( a 1 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "1"], ")"]], "k"], Pochhammer] ( a 2 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "2"], ")"]], "k"], Pochhammer] k ! j = 1 3 ( b j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "j"], ")"]], "k"], Pochhammer] 2 n + 1 F 4 n ( 1 , k + a 1 n , , k + n + a 1 - 1 n , k + a 2 n , , k + n + a 2 - 1 n ; k + 1 n , , k + n n , k + b 1 n , , k + n + b 1 - 1 n , k + b 2 n , , k + n + b 2 - 1 n , k + b 3 n , , k + n + b 3 - 1 n ; n - 2 n z n ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List[RowBox[List["2", "n"]], "+", "1"]], TraditionalForm]], SubscriptBox["F", RowBox[List["4", "n"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["a", "1"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "n", "+", SubscriptBox["a", "1"], "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["a", "2"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List[TagBox[FractionBox[RowBox[List["k", "+", "n", "+", SubscriptBox["a", "2"], "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[FractionBox[RowBox[List["k", "+", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]]]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "n"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["b", "1"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "n", "+", SubscriptBox["b", "1"], "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["b", "2"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "n", "+", SubscriptBox["b", "2"], "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["b", "3"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List[TagBox[FractionBox[RowBox[List["k", "+", "n", "+", SubscriptBox["b", "3"], "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[RowBox[List[SuperscriptBox["n", RowBox[List[RowBox[List["-", "2"]], " ", "n"]]], " ", SuperscriptBox["z", "n"]]], HypergeometricPFQ, Rule[Editable, True]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript b 1 Subscript b 2 Subscript b 3 z k 0 n -1 z k Pochhammer Subscript a 1 k Pochhammer Subscript a 2 k k j 1 3 Pochhammer Subscript b j k -1 HypergeometricPFQ 1 k Subscript a 1 n -1 [/itex]

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

 2001-10-29