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variants of this functions
HypergeometricPFQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] > Identities > Functional identities > Division on even and odd parts and generalization





http://functions.wolfram.com/07.31.17.0021.01









  


  










Input Form





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










Standard Form





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MathML Form







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</mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mi> F </mi> <mrow> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> q </mi> </mrow> <mo> + </mo> <mi> n </mi> </mrow> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> + </mo> <mi> k </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mfrac> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> ; </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> + </mo> <mi> k </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> ; </mo> <mrow> <msup> <mi> n </mi> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> p </mi> <mo> - </mo> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[RowBox[List[RowBox[List[&quot;n&quot;, &quot; &quot;, &quot;p&quot;]], &quot;+&quot;, &quot;1&quot;]], TraditionalForm]], SubscriptBox[&quot;F&quot;, RowBox[List[RowBox[List[&quot;n&quot;, &quot; &quot;, &quot;q&quot;]], &quot;+&quot;, &quot;n&quot;]]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;,&quot;, FractionBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;+&quot;, &quot;k&quot;]], &quot;n&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, FractionBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;+&quot;, &quot;k&quot;, &quot;+&quot;, &quot;n&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;n&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, FractionBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;+&quot;, &quot;k&quot;]], &quot;n&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[FractionBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;+&quot;, &quot;k&quot;, &quot;+&quot;, &quot;n&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;n&quot;], &quot;;&quot;, FractionBox[RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;n&quot;]]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, FractionBox[RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;n&quot;]], &quot;n&quot;], &quot;,&quot;, FractionBox[RowBox[List[SubscriptBox[&quot;b&quot;, &quot;1&quot;], &quot;+&quot;, &quot;k&quot;]], &quot;n&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, FractionBox[RowBox[List[SubscriptBox[&quot;b&quot;, &quot;1&quot;], &quot;+&quot;, &quot;k&quot;, &quot;+&quot;, &quot;n&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;n&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, FractionBox[RowBox[List[SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;+&quot;, &quot;k&quot;]], &quot;n&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[FractionBox[RowBox[List[SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;+&quot;, &quot;k&quot;, &quot;+&quot;, &quot;n&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;n&quot;], &quot;;&quot;, RowBox[List[SuperscriptBox[&quot;n&quot;, RowBox[List[&quot;n&quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;p&quot;, &quot;-&quot;, &quot;q&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;)&quot;]]]]], SuperscriptBox[&quot;z&quot;, &quot;n&quot;]]]]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










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





2001-10-29