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] > Specific values > Specialized values > Case q+1F~qat z==-1





http://functions.wolfram.com/07.32.03.0059.01









  


  










Input Form





HypergeometricPFQRegularized[{1, a, a, Subscript[a, 4], \[Ellipsis], Subscript[a, q + 1]}, {a + 1, a + 1, Subscript[a, 4] + 1, \[Ellipsis], Subscript[a, q + 1] + 1}, -1] == (1/(2 Gamma[a]^2 Product[Gamma[Subscript[a, j]], {j, 4, q + 1}])) (Sum[Product[If[l == k, 1, (PolyGamma[(Subscript[a, k] + 1)/2] - PolyGamma[Subscript[a, k]/2])/((Subscript[a, l] - Subscript[a, k]) (a - Subscript[a, k])^2)], {l, 4, q + 1}] + ((PolyGamma[(a + 1)/2] - PolyGamma[a/2])/(a - Subscript[a, k])) Product[1/(Subscript[a, l] - a), {l, 4, q + 1}], {k, 4, q + 1}] - (1/2) (PolyGamma[1, (a + 1)/2] - PolyGamma[1, a/2]) Product[1/(Subscript[a, l] - a), {l, 4, q + 1}]) /; Subscript[a, l] != Subscript[a, k] && 4 <= l <= q + 1 && 4 <= k <= q + 1 && l != k










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "a", ",", "a", ",", SubscriptBox["a", "4"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["a", "+", "1"]], ",", RowBox[List["a", "+", "1"]], ",", RowBox[List[SubscriptBox["a", "4"], "+", "1"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", "1"]]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], "2"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "4"]], RowBox[List["q", "+", "1"]]], RowBox[List["Gamma", "[", SubscriptBox["a", "j"], "]"]]]]]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "4"]], RowBox[List["q", "+", "1"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "4"]], RowBox[List["q", "+", "1"]]], RowBox[List["If", "[", RowBox[List[RowBox[List["l", "\[Equal]", "k"]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["PolyGamma", "[", FractionBox[RowBox[List[SubscriptBox["a", "k"], "+", "1"]], "2"], "]"]], "-", RowBox[List["PolyGamma", "[", FractionBox[SubscriptBox["a", "k"], "2"], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["a", "l"], "-", SubscriptBox["a", "k"]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", SubscriptBox["a", "k"]]], ")"]], "2"]]]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["PolyGamma", "[", FractionBox[RowBox[List["a", "+", "1"]], "2"], "]"]], "-", RowBox[List["PolyGamma", "[", FractionBox["a", "2"], "]"]]]], RowBox[List["a", "-", SubscriptBox["a", "k"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "4"]], RowBox[List["q", "+", "1"]]], FractionBox["1", RowBox[List[SubscriptBox["a", "l"], "-", "a"]]]]]]]]], ")"]]]], "-", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", FractionBox[RowBox[List["a", "+", "1"]], "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", FractionBox["a", "2"]]], "]"]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "4"]], RowBox[List["q", "+", "1"]]], FractionBox["1", RowBox[List[SubscriptBox["a", "l"], "-", "a"]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "l"], "\[NotEqual]", SubscriptBox["a", "k"]]], "\[And]", RowBox[List["4", "\[LessEqual]", "l", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "\[And]", RowBox[List["4", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "\[And]", RowBox[List["l", "\[NotEqual]", "k"]]]]]]]]










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> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mi> q </mi> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> a </mi> <mo> , </mo> <mi> a </mi> <mo> , </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ; </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </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;1&quot;]], TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;q&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[&quot;1&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;a&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;a&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, &quot;4&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, RowBox[List[TagBox[SubscriptBox[&quot;a&quot;, RowBox[List[&quot;q&quot;, &quot;+&quot;, &quot;1&quot;]]], HypergeometricPFQ, Rule[Editable, True]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;4&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, RowBox[List[&quot;q&quot;, &quot;+&quot;, &quot;1&quot;]]], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]]]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 4 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 4 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mfrac> <mi> a </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> l </mi> <mo> = </mo> <mn> 4 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mn> 1 </mn> <mrow> <msub> <mi> a </mi> <mi> l </mi> </msub> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8719; </mo> <munder> <mrow> <mi> l </mi> <mo> = </mo> <mn> 4 </mn> </mrow> <mrow> <mi> l </mi> <mo> &#8800; </mo> <mi> k </mi> </mrow> </munder> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mfrac> <msub> <mi> a </mi> <mi> k </mi> </msub> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> l </mi> </msub> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> a </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> l </mi> <mo> = </mo> <mn> 4 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mn> 1 </mn> <mrow> <msub> <mi> a </mi> <mi> l </mi> </msub> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mi> l </mi> </msub> <mo> &#8800; </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> &#8743; </mo> <mrow> <mn> 4 </mn> <mo> &#8804; </mo> <mi> l </mi> <mo> &#8804; </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mn> 4 </mn> <mo> &#8804; </mo> <mi> k </mi> <mo> &#8804; </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> l </mi> <mo> &#8800; </mo> <mi> k </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#62387; </ms> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ci> TraditionalForm </ci> </apply> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> OverscriptBox </ci> <ms> F </ms> <ms> ~ </ms> </apply> <apply> <ci> FormBox </ci> <ms> q </ms> <ci> TraditionalForm </ci> </apply> </apply> </list> </apply> <ms> &#8289; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> 1 </ms> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <ms> a </ms> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <ms> a </ms> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 4 </ms> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <ms> &#8230; </ms> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 4 </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <ms> &#8230; </ms> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ci> HypergeometricPFQ </ci> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <ms> a </ms> <ms> ) </ms> </list> </apply> <ms> 2 </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 4 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> j </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 4 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> a </ms> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> RowBox </ci> <list> <ms> l </ms> <ms> = </ms> <ms> 4 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> l </ms> </apply> <ms> - </ms> <ms> a </ms> </list> </apply> </apply> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> UnderscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> l </ms> <ms> = </ms> <ms> 4 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> l </ms> <ms> &#8800; </ms> <ms> k </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> l </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <ms> 1 </ms> <ms> ) </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <ms> 1 </ms> <ms> ) </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> a </ms> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> RowBox </ci> <list> <ms> l </ms> <ms> = </ms> <ms> 4 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> l </ms> </apply> <ms> - </ms> <ms> a </ms> </list> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> l </ms> </apply> <ms> &#8800; </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <ms> &#8804; </ms> <ms> l </ms> <ms> &#8804; </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <ms> &#8804; </ms> <ms> k </ms> <ms> &#8804; </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> l </ms> <ms> &#8800; </ms> <ms> k </ms> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "a_", ",", "a_", ",", SubscriptBox["a_", "4"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", RowBox[List["q_", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["a_", "+", "1"]], ",", RowBox[List["a_", "+", "1"]], ",", RowBox[List[SubscriptBox["a_", "4"], "+", "1"]], ",", "\[Ellipsis]_", ",", RowBox[List[SubscriptBox["a_", RowBox[List["q_", "+", "1"]]], "+", "1"]]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "4"]], RowBox[List["q", "+", "1"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "4"]], RowBox[List["q", "+", "1"]]], RowBox[List["If", "[", RowBox[List[RowBox[List["l", "\[Equal]", "k"]], ",", "1", ",", FractionBox[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SubscriptBox["a", "k"], "+", "1"]], ")"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", FractionBox[SubscriptBox["a", "k"], "2"], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["aa", "l"], "-", SubscriptBox["a", "k"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", SubscriptBox["a", "k"]]], ")"]], "2"]]]]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", FractionBox[RowBox[List["a", "+", "1"]], "2"], "]"]], "-", RowBox[List["PolyGamma", "[", FractionBox["a", "2"], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "4"]], RowBox[List["q", "+", "1"]]], FractionBox["1", RowBox[List[SubscriptBox["aa", "l"], "-", "a"]]]]]]], RowBox[List["a", "-", SubscriptBox["a", "k"]]]]]], ")"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", FractionBox[RowBox[List["a", "+", "1"]], "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", FractionBox["a", "2"]]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "4"]], RowBox[List["q", "+", "1"]]], FractionBox["1", RowBox[List[SubscriptBox["aa", "l"], "-", "a"]]]]]]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], "2"], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "4"]], RowBox[List["q", "+", "1"]]], RowBox[List["Gamma", "[", SubscriptBox["a", "j"], "]"]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["aa", "l"], "\[NotEqual]", SubscriptBox["a", "k"]]], "&&", RowBox[List["4", "\[LessEqual]", "l", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "&&", RowBox[List["4", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "&&", RowBox[List["l", "\[NotEqual]", "k"]]]]]]]]]]










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





2001-10-29