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
HypergeometricPFQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricPFQ[{a1,a2,a3,a4,a5},{b1,b2,b3,b4},z] > Specific values > Values at z==-1 > For fixed a4





http://functions.wolfram.com/07.29.03.0024.01









  


  










Input Form





HypergeometricPFQ[{1, 1, 1, d, 2 - d}, {2, 2, 1 + d, 3 - d}, -1] == (1/(24 (-1 + d)^4)) (2 (-2 + d) d Csc[d Pi] (6 (-1 + d) Pi + (6 + (-1 + d)^2 Pi^2) Sin[d Pi]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", "1", ",", "d", ",", RowBox[List["2", "-", "d"]]]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", "2", ",", RowBox[List["1", "+", "d"]], ",", RowBox[List["3", "-", "d"]]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["24", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "d"]], ")"]], "4"]]]], RowBox[List["(", RowBox[List["2", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "d"]], ")"]], " ", "d", " ", RowBox[List["Csc", "[", RowBox[List["d", " ", "\[Pi]"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["6", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "d"]], ")"]], " ", "\[Pi]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["6", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "d"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["d", " ", "\[Pi]"]], "]"]]]]]], ")"]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 5 </mn> </msub> <msub> <mi> F </mi> <mn> 4 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mi> d </mi> <mo> , </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> d </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mrow> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mn> 3 </mn> <mo> - </mo> <mi> d </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;5&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;4&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;1&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;1&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;1&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;d&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;2&quot;, &quot;-&quot;, &quot;d&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[&quot;2&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;2&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;d&quot;, &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;3&quot;, &quot;-&quot;, &quot;d&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, False]], HypergeometricPFQ] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 1 </cn> <cn type='integer'> 1 </cn> <ci> d </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> </list> <list> <cn type='integer'> 2 </cn> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> d </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> </list> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <power /> <apply> <plus /> <ci> d </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> d </ci> <cn type='integer'> -2 </cn> </apply> <ci> d </ci> <apply> <csc /> <apply> <times /> <ci> d </ci> <pi /> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <pi /> <apply> <plus /> <ci> d </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> d </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 6 </cn> </apply> <apply> <sin /> <apply> <times /> <ci> d </ci> <pi /> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", "1", ",", "d_", ",", RowBox[List["2", "-", "d_"]]]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", "2", ",", RowBox[List["1", "+", "d_"]], ",", RowBox[List["3", "-", "d_"]]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "d"]], ")"]], " ", "d", " ", RowBox[List["Csc", "[", RowBox[List["d", " ", "\[Pi]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "d"]], ")"]], " ", "\[Pi]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["6", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "d"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["d", " ", "\[Pi]"]], "]"]]]]]], ")"]]]], RowBox[List["24", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "d"]], ")"]], "4"]]]]]]]]










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





2001-10-29