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variants of this functions
HypergeometricPFQRegularized






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/07.32.17.0022.01









  


  










Input Form





HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] == (2 Pi)^(((n - 1)/2) (q + 1)) Sum[n^(-\[Eta] - (q + 1) k) z^k Product[Pochhammer[Subscript[a, j], k], {j, 1, p}] HypergeometricPFQRegularized[{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}] /; \[Eta] == Sum[Subscript[b, j], {j, 1, q}] + (q - 1)/2










Standard Form





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MathML Form







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</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[OverscriptBox[&quot;F&quot;, &quot;~&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> </mrow> <mo> /; </mo> <mrow> <mi> &#951; </mi> <mo> &#10869; </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", 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_"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["q", "+", "1"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[SuperscriptBox["n", RowBox[List[RowBox[List["-", "\[Eta]"]], "-", RowBox[List[RowBox[List["(", RowBox[List["q", "+", "1"]], ")"]], " ", "k"]]]]], " ", SuperscriptBox["z", "k"], " ", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "j"], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List[SubscriptBox["aa", "1"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["aa", "1"], "+", "k", "+", "n", "-", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["aa", "p"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["aa", "p"], "+", "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["bb", "1"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["bb", "1"], "+", "k", "+", "n", "-", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["bb", "q"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["bb", "q"], "+", "k", "+", "n", "-", "1"]], "n"]]], "}"]], ",", RowBox[List[SuperscriptBox["n", RowBox[List["n", " ", RowBox[List["(", RowBox[List["p", "-", "q", "-", "1"]], ")"]]]]], " ", SuperscriptBox["z", "n"]]]]], "]"]]]]]]]], "/;", RowBox[List["\[Eta]", "\[Equal]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "q"], SubscriptBox["b", "j"]]], "+", FractionBox[RowBox[List["q", "-", "1"]], "2"]]]]]]]]]]]










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





2001-10-29





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