|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/07.28.03.0058.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
HypergeometricPFQ[{1, b, b, b}, {b + 1, b + 1, b + 1}, -1] ==
(b^3/16) (PolyGamma[2, (b + 1)/2] - PolyGamma[2, b/2])
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "b", ",", "b", ",", "b"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["b", "+", "1"]], ",", RowBox[List["b", "+", "1"]], ",", RowBox[List["b", "+", "1"]]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["b", "3"], "16"], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["2", ",", FractionBox[RowBox[List["b", "+", "1"]], "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", FractionBox["b", "2"]]], "]"]]]], ")"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<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>   </mo> <mn> 4 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ; </mo> <mrow> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </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["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["b", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["b", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["b", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["b", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <msup> <mi> b </mi> <mn> 3 </mn> </msup> <mn> 16 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> b </mi> <mn> 2 </mn> </mfrac> <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> <ci> b </ci> <ci> b </ci> <ci> b </ci> </list> <list> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </list> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 16 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "b_", ",", "b_", ",", "b_"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["b_", "+", "1"]], ",", RowBox[List["b_", "+", "1"]], ",", RowBox[List["b_", "+", "1"]]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "16"], " ", SuperscriptBox["b", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["2", ",", FractionBox[RowBox[List["b", "+", "1"]], "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", FractionBox["b", "2"]]], "]"]]]], ")"]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
HypergeometricPFQ[{},{},z] | HypergeometricPFQ[{},{b},z] | HypergeometricPFQ[{a},{},z] | HypergeometricPFQ[{a},{b},z] | HypergeometricPFQ[{a1},{b1,b2},z] | HypergeometricPFQ[{a1,a2},{b1},z] | HypergeometricPFQ[{a1,a2},{b1,b2},z] | HypergeometricPFQ[{a1,a2},{b1,b2,b3},z] | HypergeometricPFQ[{a1,a2,a3},{b1,b2},z] | HypergeometricPFQ[{a1,a2,a3,a4,a5},{b1,b2,b3,b4},z] | HypergeometricPFQ[{a1,a2,a3,a4,a5,a6},{b1,b2,b3,b4,b5},z] | HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] | |
|
|
|