Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
HypergeometricPFQRegularized






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricPFQRegularized[{a1,...,ap},{b1,...,bq},z] > Transformations > Products, sums, and powers of the direct function > Products of the direct function





http://functions.wolfram.com/07.32.16.0001.01









  


  










Input Form





HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, c z] HypergeometricPFQRegularized[{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] == (((-1)^(k t) d^k Product[Gamma[1 - Subscript[\[Alpha], j]], {j, 1, r}])/ (k! Product[Gamma[Subscript[\[Beta], j] + k], {j, 1, s}])) HypergeometricPFQRegularized[{-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] == (((-1)^(k p) c^k Product[Gamma[1 - Subscript[a, j]], {j, 1, p}])/(k! Product[Gamma[Subscript[b, j] + k], {j, 1, q}])) HypergeometricPFQRegularized[{-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["HypergeometricPFQRegularized", "[", 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["HypergeometricPFQRegularized", "[", 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[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", " ", "t"]]], SuperscriptBox["d", "k"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "r"], RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["\[Alpha]", "j"]]], "]"]]]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "s"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["\[Beta]", "j"], "+", "k"]], "]"]]]]]]], RowBox[List["HypergeometricPFQRegularized", "[", 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[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", " ", "p"]]], SuperscriptBox["c", "k"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["a", "j"]]], "]"]]]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "j"], "+", "k"]], "]"]]]]]]], RowBox[List["HypergeometricPFQRegularized", "[", 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







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mi> p </mi> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mi> q </mi> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> ; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;p&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;q&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;;&quot;, SubscriptBox[&quot;b&quot;, &quot;1&quot;]]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;;&quot;, RowBox[List[&quot;c&quot;, &quot; &quot;, &quot;z&quot;]]]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mi> r </mi> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mi> s </mi> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> &#945; </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> &#945; </mi> <mi> r </mi> </msub> <mo> ; </mo> <msub> <mi> &#946; </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> &#946; </mi> <mi> s </mi> </msub> <mo> ; </mo> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;r&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;s&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;\[Alpha]&quot;, &quot;1&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;\[Alpha]&quot;, &quot;r&quot;], &quot;;&quot;, SubscriptBox[&quot;\[Beta]&quot;, &quot;1&quot;]]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;\[Beta]&quot;, &quot;s&quot;], &quot;;&quot;, RowBox[List[&quot;d&quot;, &quot; &quot;, &quot;z&quot;]]]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> t </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> d </mi> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> &#945; </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> s </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> &#946; </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mrow> <mi> p </mi> <mo> + </mo> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mrow> <mi> q </mi> <mo> + </mo> <mi> r </mi> </mrow> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> &#946; </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> &#946; </mi> <mi> s </mi> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> &#945; </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> k </mi> </mrow> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> &#945; </mi> <mi> r </mi> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> r </mi> <mo> + </mo> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mi> d </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[RowBox[List[&quot;p&quot;, &quot;+&quot;, &quot;s&quot;, &quot;+&quot;, &quot;1&quot;]], TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[RowBox[List[&quot;q&quot;, &quot;+&quot;, &quot;r&quot;]], TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;-&quot;, &quot;k&quot;]], &quot;,&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;\[Beta]&quot;, &quot;1&quot;], &quot;-&quot;, &quot;k&quot;]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;\[Beta]&quot;, &quot;s&quot;], &quot;-&quot;, &quot;k&quot;]], &quot;,&quot;, SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;;&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;\[Alpha]&quot;, &quot;1&quot;], &quot;-&quot;, &quot;k&quot;]]]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;\[Alpha]&quot;, &quot;r&quot;], &quot;-&quot;, &quot;k&quot;]], &quot;,&quot;, SubscriptBox[&quot;b&quot;, &quot;1&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;;&quot;, FractionBox[RowBox[List[SuperscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[&quot;r&quot;, &quot;+&quot;, &quot;s&quot;, &quot;-&quot;, &quot;1&quot;]]], &quot;c&quot;]], &quot;d&quot;]]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> c </mi> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mrow> <mi> q </mi> <mo> + </mo> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mrow> <mi> p </mi> <mo> + </mo> <mi> s </mi> </mrow> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <msub> <mi> &#945; </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> &#945; </mi> <mi> r </mi> </msub> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> k </mi> </mrow> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> r </mi> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <msub> <mi> &#946; </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> &#946; </mi> <mi> s </mi> </msub> <mo> ; </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> p </mi> <mo> + </mo> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mi> c </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[RowBox[List[&quot;q&quot;, &quot;+&quot;, &quot;r&quot;, &quot;+&quot;, &quot;1&quot;]], TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[RowBox[List[&quot;p&quot;, &quot;+&quot;, &quot;s&quot;]], TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;-&quot;, &quot;k&quot;]], &quot;,&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;1&quot;], &quot;-&quot;, &quot;k&quot;]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;-&quot;, &quot;k&quot;]], &quot;,&quot;, SubscriptBox[&quot;\[Alpha]&quot;, &quot;1&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;\[Alpha]&quot;, &quot;r&quot;], &quot;;&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;-&quot;, &quot;k&quot;]]]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;a&quot;, &quot;r&quot;], &quot;-&quot;, &quot;k&quot;]], &quot;,&quot;, SubscriptBox[&quot;\[Beta]&quot;, &quot;1&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;\[Beta]&quot;, &quot;s&quot;], &quot;;&quot;, FractionBox[RowBox[List[SuperscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[&quot;p&quot;, &quot;+&quot;, &quot;q&quot;, &quot;-&quot;, &quot;1&quot;]]], &quot;d&quot;]], &quot;c&quot;]]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mi> p </mi> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mi> q </mi> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> ; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;p&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;q&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;;&quot;, SubscriptBox[&quot;b&quot;, &quot;1&quot;]]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;;&quot;, RowBox[List[&quot;c&quot;, &quot; &quot;, &quot;z&quot;]]]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mi> r </mi> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mi> s </mi> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> &#945; </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> &#945; </mi> <mi> r </mi> </msub> <mo> ; </mo> <msub> <mi> &#946; </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> &#946; </mi> <mi> s </mi> </msub> <mo> ; </mo> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;r&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;s&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[SubscriptBox[&quot;\[Alpha]&quot;, &quot;1&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;\[Alpha]&quot;, &quot;r&quot;], &quot;;&quot;, SubscriptBox[&quot;\[Beta]&quot;, &quot;1&quot;]]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;\[Beta]&quot;, &quot;s&quot;], &quot;;&quot;, RowBox[List[&quot;d&quot;, &quot; &quot;, &quot;z&quot;]]]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> t </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> d </mi> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> &#945; </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> s </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> &#946; </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mrow> <mi> p </mi> <mo> + </mo> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mrow> <mi> q </mi> <mo> + </mo> <mi> r </mi> </mrow> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> &#946; </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> &#946; </mi> <mi> s </mi> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> &#945; </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> k </mi> </mrow> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> &#945; </mi> <mi> r </mi> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> r </mi> <mo> + </mo> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mi> d </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[RowBox[List[&quot;p&quot;, &quot;+&quot;, &quot;s&quot;, &quot;+&quot;, &quot;1&quot;]], TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[RowBox[List[&quot;q&quot;, &quot;+&quot;, &quot;r&quot;]], TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;-&quot;, &quot;k&quot;]], &quot;,&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;\[Beta]&quot;, &quot;1&quot;], &quot;-&quot;, &quot;k&quot;]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;\[Beta]&quot;, &quot;s&quot;], &quot;-&quot;, &quot;k&quot;]], &quot;,&quot;, SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;;&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;\[Alpha]&quot;, &quot;1&quot;], &quot;-&quot;, &quot;k&quot;]]]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;\[Alpha]&quot;, &quot;r&quot;], &quot;-&quot;, &quot;k&quot;]], &quot;,&quot;, SubscriptBox[&quot;b&quot;, &quot;1&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;;&quot;, FractionBox[RowBox[List[SuperscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[&quot;r&quot;, &quot;+&quot;, &quot;s&quot;, &quot;-&quot;, &quot;1&quot;]]], &quot;c&quot;]], &quot;d&quot;]]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> c </mi> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mrow> <mi> q </mi> <mo> + </mo> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mrow> <mi> p </mi> <mo> + </mo> <mi> s </mi> </mrow> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <msub> <mi> &#945; </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> &#945; </mi> <mi> r </mi> </msub> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> k </mi> </mrow> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> r </mi> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <msub> <mi> &#946; </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> &#946; </mi> <mi> s </mi> </msub> <mo> ; </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> p </mi> <mo> + </mo> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mi> c </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[RowBox[List[&quot;q&quot;, &quot;+&quot;, &quot;r&quot;, &quot;+&quot;, &quot;1&quot;]], TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[RowBox[List[&quot;p&quot;, &quot;+&quot;, &quot;s&quot;]], TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;-&quot;, &quot;k&quot;]], &quot;,&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;1&quot;], &quot;-&quot;, &quot;k&quot;]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;-&quot;, &quot;k&quot;]], &quot;,&quot;, SubscriptBox[&quot;\[Alpha]&quot;, &quot;1&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;\[Alpha]&quot;, &quot;r&quot;], &quot;;&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;-&quot;, &quot;k&quot;]]]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;a&quot;, &quot;r&quot;], &quot;-&quot;, &quot;k&quot;]], &quot;,&quot;, SubscriptBox[&quot;\[Beta]&quot;, &quot;1&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;\[Beta]&quot;, &quot;s&quot;], &quot;;&quot;, FractionBox[RowBox[List[SuperscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[&quot;p&quot;, &quot;+&quot;, &quot;q&quot;, &quot;-&quot;, &quot;1&quot;]]], &quot;d&quot;]], &quot;c&quot;]]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", 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["HypergeometricPFQRegularized", "[", 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_"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[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]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", " ", "t"]]], " ", SuperscriptBox["d", "k"], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "r"], RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["\[Alpha]", "j"]]], "]"]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", 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"]]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "s"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["\[Beta]", "j"], "+", "k"]], "]"]]]]]]]]], "||", RowBox[List[SubscriptBox["c", "k"], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", " ", "p"]]], " ", SuperscriptBox["c", "k"], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["a", "j"]]], "]"]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", 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"]]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "j"], "+", "k"]], "]"]]]]]]]]]]]]]]]]]










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





2001-10-29





© 1998- Wolfram Research, Inc.