Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

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] > Identities > Functional identities > Relations between contiguous functions





http://functions.wolfram.com/07.32.17.0013.01









  


  








Input Form





HypergeometricPFQRegularized[{a, b, c, Subscript[a, 4], \[Ellipsis], Subscript[a, p]}, {d, e, Subscript[b, 3], \[Ellipsis], Subscript[b, q]}, z] - ((a b (d - c) (e - c))/((a - c) (b - c))) HypergeometricPFQRegularized[{a + 1, b + 1, c, Subscript[a, 4], \[Ellipsis], Subscript[a, p]}, {d + 1, e + 1, Subscript[b, 3], \[Ellipsis], Subscript[b, q]}, z] - ((a c (d - b) (e - b))/((a - b) (c - b))) HypergeometricPFQRegularized[ {a + 1, b, c + 1, Subscript[a, 4], \[Ellipsis], Subscript[a, p]}, {d + 1, e + 1, Subscript[b, 3], \[Ellipsis], Subscript[b, q]}, z] - ((b c (d - a) (e - a))/((b - a) (c - a))) HypergeometricPFQRegularized[ {a, b + 1, c + 1, Subscript[a, 4], \[Ellipsis], Subscript[a, p]}, {d + 1, e + 1, Subscript[b, 3], \[Ellipsis], Subscript[b, q]}, z] == 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "b", ",", "c", ",", SubscriptBox["a", "4"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List["d", ",", "e", ",", SubscriptBox["b", "3"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]], "-", RowBox[List[FractionBox[RowBox[List["a", " ", "b", RowBox[List["(", RowBox[List["d", "-", "c"]], ")"]], RowBox[List["(", RowBox[List["e", "-", "c"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "-", "c"]], ")"]], RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]]]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["a", "+", "1"]], ",", RowBox[List["b", "+", "1"]], ",", "c", ",", SubscriptBox["a", "4"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["d", "+", "1"]], ",", RowBox[List["e", "+", "1"]], ",", SubscriptBox["b", "3"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List["a", " ", "c", RowBox[List["(", RowBox[List["d", "-", "b"]], ")"]], RowBox[List["(", RowBox[List["e", "-", "b"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], RowBox[List["(", RowBox[List["c", "-", "b"]], ")"]]]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["a", "+", "1"]], ",", "b", ",", RowBox[List["c", "+", "1"]], ",", SubscriptBox["a", "4"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["d", "+", "1"]], ",", RowBox[List["e", "+", "1"]], ",", SubscriptBox["b", "3"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List["b", " ", "c", RowBox[List["(", RowBox[List["d", "-", "a"]], ")"]], RowBox[List["(", RowBox[List["e", "-", "a"]], ")"]]]], RowBox[List[" ", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "a"]], ")"]], RowBox[List["(", RowBox[List["c", "-", "a"]], ")"]]]]]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", RowBox[List["b", "+", "1"]], ",", RowBox[List["c", "+", "1"]], ",", SubscriptBox["a", "4"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["d", "+", "1"]], ",", RowBox[List["e", "+", "1"]], ",", SubscriptBox["b", "3"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]]]]]], "\[Equal]", "0"]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mi> p </mi> </msub> <msub> <mi> F </mi> <mi> q </mi> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> c </mi> <mo> , </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> ; </mo> <mi> d </mi> </mrow> <mo> , </mo> <mi> e </mi> <mo> , </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mi> z </mi> </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[&quot;F&quot;, FormBox[&quot;q&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;a&quot;, &quot;,&quot;, &quot;b&quot;, &quot;,&quot;, &quot;c&quot;, &quot;,&quot;, SubscriptBox[&quot;a&quot;, &quot;4&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;;&quot;, &quot;d&quot;]], &quot;,&quot;, &quot;e&quot;, &quot;,&quot;, SubscriptBox[&quot;b&quot;, &quot;3&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;;&quot;, &quot;z&quot;]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> <mo> - </mo> <mtext> </mtext> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> e </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <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> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> c </mi> <mo> , </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> ; </mo> <mrow> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> , </mo> <mrow> <mi> e </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mi> z </mi> </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[RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, RowBox[List[&quot;b&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, &quot;c&quot;, &quot;,&quot;, SubscriptBox[&quot;a&quot;, &quot;4&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;;&quot;, RowBox[List[&quot;d&quot;, &quot;+&quot;, &quot;1&quot;]]]], &quot;,&quot;, RowBox[List[&quot;e&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, SubscriptBox[&quot;b&quot;, &quot;3&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;b&quot;, &quot;q&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> - </mo> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> e </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <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> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mrow> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> ; </mo> <mrow> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> , </mo> <mrow> <mi> e </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mi> z </mi> </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[RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, &quot;b&quot;, &quot;,&quot;, RowBox[List[&quot;c&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, SubscriptBox[&quot;a&quot;, &quot;4&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;;&quot;, RowBox[List[&quot;d&quot;, &quot;+&quot;, &quot;1&quot;]]]], &quot;,&quot;, RowBox[List[&quot;e&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, SubscriptBox[&quot;b&quot;, &quot;3&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;b&quot;, &quot;q&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> - </mo> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> e </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <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> <mi> a </mi> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> ; </mo> <mrow> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> , </mo> <mrow> <mi> e </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mi> z </mi> </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[&quot;a&quot;, &quot;,&quot;, RowBox[List[&quot;b&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, RowBox[List[&quot;c&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, SubscriptBox[&quot;a&quot;, &quot;4&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;;&quot;, RowBox[List[&quot;d&quot;, &quot;+&quot;, &quot;1&quot;]]]], &quot;,&quot;, RowBox[List[&quot;e&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, SubscriptBox[&quot;b&quot;, &quot;3&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;;&quot;, &quot;z&quot;]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mi> p </mi> </msub> <msub> <mi> F </mi> <mi> q </mi> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> c </mi> <mo> , </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> ; </mo> <mi> d </mi> </mrow> <mo> , </mo> <mi> e </mi> <mo> , </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mi> z </mi> </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[&quot;F&quot;, FormBox[&quot;q&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;a&quot;, &quot;,&quot;, &quot;b&quot;, &quot;,&quot;, &quot;c&quot;, &quot;,&quot;, SubscriptBox[&quot;a&quot;, &quot;4&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;;&quot;, &quot;d&quot;]], &quot;,&quot;, &quot;e&quot;, &quot;,&quot;, SubscriptBox[&quot;b&quot;, &quot;3&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;;&quot;, &quot;z&quot;]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> <mo> - </mo> <mtext> </mtext> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> e </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <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> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> c </mi> <mo> , </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> ; </mo> <mrow> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> , </mo> <mrow> <mi> e </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mi> z </mi> </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[RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, RowBox[List[&quot;b&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, &quot;c&quot;, &quot;,&quot;, SubscriptBox[&quot;a&quot;, &quot;4&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;;&quot;, RowBox[List[&quot;d&quot;, &quot;+&quot;, &quot;1&quot;]]]], &quot;,&quot;, RowBox[List[&quot;e&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, SubscriptBox[&quot;b&quot;, &quot;3&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;b&quot;, &quot;q&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> - </mo> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> e </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <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> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mrow> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> ; </mo> <mrow> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> , </mo> <mrow> <mi> e </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mi> z </mi> </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[RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, &quot;b&quot;, &quot;,&quot;, RowBox[List[&quot;c&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, SubscriptBox[&quot;a&quot;, &quot;4&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;;&quot;, RowBox[List[&quot;d&quot;, &quot;+&quot;, &quot;1&quot;]]]], &quot;,&quot;, RowBox[List[&quot;e&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, SubscriptBox[&quot;b&quot;, &quot;3&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;b&quot;, &quot;q&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> - </mo> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> e </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <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> <mi> a </mi> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> ; </mo> <mrow> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> , </mo> <mrow> <mi> e </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mi> z </mi> </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[&quot;a&quot;, &quot;,&quot;, RowBox[List[&quot;b&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, RowBox[List[&quot;c&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, SubscriptBox[&quot;a&quot;, &quot;4&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;;&quot;, RowBox[List[&quot;d&quot;, &quot;+&quot;, &quot;1&quot;]]]], &quot;,&quot;, RowBox[List[&quot;e&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, SubscriptBox[&quot;b&quot;, &quot;3&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;;&quot;, &quot;z&quot;]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </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["a_", ",", "b_", ",", "c_", ",", SubscriptBox["a_", "4"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "p_"]]], "}"]], ",", RowBox[List["{", RowBox[List["d_", ",", "e_", ",", SubscriptBox["b_", "3"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]], ",", "z_"]], "]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a_", " ", "b_", " ", RowBox[List["(", RowBox[List["d_", "-", "c_"]], ")"]], " ", RowBox[List["(", RowBox[List["e_", "-", "c_"]], ")"]]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["a_", "+", "1"]], ",", RowBox[List["b_", "+", "1"]], ",", "c_", ",", SubscriptBox["a_", "4"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "p_"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["d_", "+", "1"]], ",", RowBox[List["e_", "+", "1"]], ",", SubscriptBox["b_", "3"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]], ",", "z_"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["a_", "-", "c_"]], ")"]], " ", RowBox[List["(", RowBox[List["b_", "-", "c_"]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a_", " ", "c_", " ", RowBox[List["(", RowBox[List["d_", "-", "b_"]], ")"]], " ", RowBox[List["(", RowBox[List["e_", "-", "b_"]], ")"]]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["a_", "+", "1"]], ",", "b_", ",", RowBox[List["c_", "+", "1"]], ",", SubscriptBox["a_", "4"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "p_"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["d_", "+", "1"]], ",", RowBox[List["e_", "+", "1"]], ",", SubscriptBox["b_", "3"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]], ",", "z_"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["a_", "-", "b_"]], ")"]], " ", RowBox[List["(", RowBox[List["c_", "-", "b_"]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b_", " ", "c_", " ", RowBox[List["(", RowBox[List["d_", "-", "a_"]], ")"]], " ", RowBox[List["(", RowBox[List["e_", "-", "a_"]], ")"]]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["a_", ",", RowBox[List["b_", "+", "1"]], ",", RowBox[List["c_", "+", "1"]], ",", SubscriptBox["a_", "4"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "p_"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["d_", "+", "1"]], ",", RowBox[List["e_", "+", "1"]], ",", SubscriptBox["b_", "3"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]], ",", "z_"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["b_", "-", "a_"]], ")"]], " ", RowBox[List["(", RowBox[List["c_", "-", "a_"]], ")"]]]]]]], "]"]], "\[RuleDelayed]", "0"]]]]


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





2001-10-29