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},{b1,b2},z] > Specific values > Values at z=1 > For fixed a1, a2, b1





http://functions.wolfram.com/07.27.03.0022.01









  


  








Input Form





HypergeometricPFQ[{1, a, b}, {3, d}, 1] == ((2 (d - 2) (d - 1))/((a - 2) (a - 1) (b - 2) (b - 1))) ((Gamma[d - 2] Gamma[2 + d - a - b])/(Gamma[d - a] Gamma[d - b]) - 1) - (2 (d - 1))/((a - 1) (b - 1)) /; Re[d - a - b] > -2 && a != 1 && a != 2 && b != 1 && b != 2 && c != 1 && c != 2








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "a", ",", "b"]], "}"]], ",", RowBox[List["{", RowBox[List["3", ",", "d"]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["2", RowBox[List["(", RowBox[List["d", "-", "2"]], ")"]], RowBox[List["(", RowBox[List["d", "-", "1"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "-", "2"]], ")"]], RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], RowBox[List["(", RowBox[List["b", "-", "2"]], ")"]], RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]]]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["d", "-", "2"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["2", "+", "d", "-", "a", "-", "b"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["d", "-", "a"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["d", "-", "b"]], "]"]]]]], "-", "1"]], ")"]]]], "-", FractionBox[RowBox[List["2", RowBox[List["(", RowBox[List["d", "-", "1"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", RowBox[List["d", "-", "a", "-", "b"]], "]"]], ">", RowBox[List["-", "2"]]]], "\[And]", RowBox[List["a", "\[NotEqual]", "1"]], "\[And]", RowBox[List["a", "\[NotEqual]", "2"]], "\[And]", RowBox[List["b", "\[NotEqual]", "1"]], "\[And]", RowBox[List["b", "\[NotEqual]", "2"]], "\[And]", RowBox[List["c", "\[NotEqual]", "1"]], "\[And]", RowBox[List["c", "\[NotEqual]", "2"]]]]]]]]








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> <mn> 3 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ; </mo> <mrow> <mn> 3 </mn> <mo> , </mo> <mi> d </mi> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;3&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;2&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;a&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;b&quot;, HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[&quot;3&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;d&quot;, HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[&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> <mrow> <mfrac> <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> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> a </mi> <mo> &#8800; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> a </mi> <mo> &#8800; </mo> <mn> 2 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> b </mi> <mo> &#8800; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> b </mi> <mo> &#8800; </mo> <mn> 2 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> c </mi> <mo> &#8800; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> c </mi> <mo> &#8800; </mo> <mn> 2 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> <ci> a </ci> <ci> b </ci> </list> <list> <cn type='integer'> 3 </cn> <ci> d </ci> </list> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> d </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> d </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> d </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> d </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <real /> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> -2 </cn> </apply> <apply> <neq /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <neq /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <neq /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <neq /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <neq /> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <apply> <neq /> <ci> c </ci> <cn type='integer'> 2 </cn> </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", ",", "a_", ",", "b_"]], "}"]], ",", RowBox[List["{", RowBox[List["3", ",", "d_"]], "}"]], ",", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["d", "-", "1"]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["d", "-", "2"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "d", "-", "a", "-", "b"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["d", "-", "a"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["d", "-", "b"]], "]"]]]]], "-", "1"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]]]]], "-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "-", "1"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", RowBox[List["d", "-", "a", "-", "b"]], "]"]], ">", RowBox[List["-", "2"]]]], "&&", RowBox[List["a", "\[NotEqual]", "1"]], "&&", RowBox[List["a", "\[NotEqual]", "2"]], "&&", RowBox[List["b", "\[NotEqual]", "1"]], "&&", RowBox[List["b", "\[NotEqual]", "2"]], "&&", RowBox[List["c", "\[NotEqual]", "1"]], "&&", RowBox[List["c", "\[NotEqual]", "2"]]]]]]]]]]


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





2001-10-29