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 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

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", RowBox[List["c", " ", "z"]]]], "]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Alpha]", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["\[Alpha]", "r"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["\[Beta]", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["\[Beta]", "s"]]], "}"]], ",", RowBox[List["d", " ", "z"]]]], "]"]]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["c", "k"], SuperscriptBox["z", "k"]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["c", "k"], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["d", "k"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "r"], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["\[Alpha]", "j"], ",", "k"]], "]"]]]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "s"], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["\[Beta]", "j"], ",", "k"]], "]"]]]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[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]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", SubscriptBox["\[Alpha]", "1"], "-", "k"]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["\[Alpha]", "r"], "-", "k"]], ",", SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["r", "+", "s", "-", "1"]]], "c"]], "d"]]], "]"]]]]]], "\[Or]", RowBox[List[SubscriptBox["c", "k"], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["c", "k"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "j"], ",", "k"]], "]"]]]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "j"], ",", "k"]], "]"]]]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[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]", ",", SubscriptBox["\[Alpha]", "r"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", SubscriptBox["a", "1"], "-", "k"]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["a", "p"], "-", "k"]], ",", SubscriptBox["\[Beta]", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["\[Beta]", "s"]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "+", "q", "-", "1"]]], "d"]], "c"]]], "]"]]]]]]]]]]]]

 MathML Form

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

 2001-10-29