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] > Differential equations > Ordinary linear differential equations and wronskians > For the direct function itself > Representation of fundamental system solutions near point z==0 for p<=q+1 in the general case





http://functions.wolfram.com/07.32.13.0001.01









  


  










Input Form





D[Fold[Function[{f, k}, z D[f, z] + (Subscript[b, k] - 1) f], w[z], {1, \[Ellipsis], q}], z] - Fold[Function[{f, l}, z D[f, z] + Subscript[a, l] f], w[z], {1, \[Ellipsis], p}] == 0 /; (w[z] == Subscript[c, 1] HypergeometricPFQRegularized[ {Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] + Sum[Subscript[c, k + 1] z^(1 - Subscript[b, k]) HypergeometricPFQRegularized[{1 + Subscript[a, 1] - Subscript[b, k], \[Ellipsis], 1 + Subscript[a, p] - Subscript[b, k]}, {2 - Subscript[b, k], 1 + Subscript[b, 1] - Subscript[b, k], \[Ellipsis], 1 + Subscript[b, k - 1] - Subscript[b, k], 1 + Subscript[b, k + 1] - Subscript[b, k], \[Ellipsis], 1 + Subscript[b, q] - Subscript[b, k]}, z], {k, 1, q}] /; ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= q && 1 <= k <= q, !Element[Subscript[b, j] - Subscript[b, k], Integers]] && !Element[Subscript[b, k], Integers])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "z"], RowBox[List["Fold", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List[RowBox[List["{", RowBox[List["f", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List["z", " ", RowBox[List[SubscriptBox["\[PartialD]", "z"], "f"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["b", "k"], "-", "1"]], ")"]], " ", "f"]]]]]], "]"]], ",", RowBox[List["w", "[", "z", "]"]], ",", RowBox[List["{", RowBox[List["1", ",", "\[Ellipsis]", ",", "q"]], "}"]]]], "]"]]]], "-", " ", RowBox[List["Fold", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List[RowBox[List["{", RowBox[List["f", ",", "l"]], "}"]], ",", RowBox[List[RowBox[List["z", " ", RowBox[List[SubscriptBox["\[PartialD]", "z"], "f"]]]], "+", RowBox[List[SubscriptBox["a", "l"], " ", "f"]]]]]], "]"]], ",", RowBox[List["w", "[", "z", "]"]], ",", RowBox[List["{", RowBox[List["1", ",", "\[Ellipsis]", ",", "p"]], "}"]]]], "]"]]]], "\[Equal]", "0"]], "/;", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], 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"]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], RowBox[List[SubscriptBox["c", RowBox[List["k", "+", "1"]]], " ", SuperscriptBox["z", RowBox[List["1", "-", SubscriptBox["b", "k"]]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "k"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", "p"], "-", SubscriptBox["b", "k"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "-", SubscriptBox["b", "k"]]], ",", RowBox[List["1", "+", SubscriptBox["b", "1"], "-", SubscriptBox["b", "k"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", RowBox[List["k", "-", "1"]]], "-", SubscriptBox["b", "k"]]], ",", RowBox[List["1", "+", SubscriptBox["b", RowBox[List["k", "+", "1"]]], "-", SubscriptBox["b", "k"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "q"], "-", SubscriptBox["b", "k"]]]]], "}"]], ",", "z"]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["\[ForAll]", RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], "\[Element]", "Integers"]], "\[And]", RowBox[List["j", "\[NotEqual]", "k"]], "\[And]", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "q"]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "q"]]]]]]], RowBox[List["(", "\[InvisibleSpace]", RowBox[List["Not", "[", RowBox[List[RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["b", "k"]]], "\[Element]", "Integers"]], "]"]], ")"]]]], "\[And]", RowBox[List["Not", "[", RowBox[List[SubscriptBox["b", "k"], "\[Element]", "Integers"]], "]"]]]]]], ")"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mtext> </mtext> <mrow> <mrow> <mstyle scriptlevel='0'> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> d </mi> <mtext> </mtext> </mrow> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> l </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <msub> <mi> a </mi> <mi> l </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mstyle> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mi> p </mi> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mi> q </mi> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> p </mi> </msub> </mrow> <mo> ; </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;p&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;q&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox[&quot;a&quot;, &quot;1&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, &quot;p&quot;], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[SubscriptBox[&quot;b&quot;, &quot;1&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;b&quot;, &quot;q&quot;], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], &quot;;&quot;, TagBox[&quot;z&quot;, HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <mrow> <msub> <mi> c </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mi> p </mi> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mi> q </mi> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;p&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;q&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;k&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;, &quot;p&quot;], &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;k&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;2&quot;, &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;k&quot;]]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;b&quot;, &quot;1&quot;], &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;k&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;b&quot;, RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;1&quot;]]], &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;k&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;b&quot;, RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;1&quot;]]], &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;k&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;b&quot;, &quot;q&quot;], &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;k&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;z&quot;, HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mo> &#8704; </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> } </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> j </mi> <mo> &#8800; </mo> <mi> k </mi> </mrow> <mo> &#8743; </mo> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <mi> j </mi> <mo> &#8804; </mo> <mi> q </mi> </mrow> <mo> &#8743; </mo> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <mi> k </mi> <mo> &#8804; </mo> <mi> q </mi> </mrow> </mrow> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> b </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> q </ci> </uplimit> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <product /> <bvar> <ci> l </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> p </ci> </uplimit> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> l </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> w </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> p </ci> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> q </ci> </apply> </list> <ci> z </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> q </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <ci> &#8230; </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> p </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <ci> &#8230; </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <ci> &#8230; </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> q </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <forall /> <bvar> <list> <ci> j </ci> <ci> k </ci> </list> </bvar> <bvar> <apply> <and /> <apply> <in /> <list> <ci> j </ci> <ci> k </ci> </list> <integers /> </apply> <apply> <neq /> <ci> j </ci> <ci> k </ci> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> j </ci> <ci> q </ci> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> k </ci> <ci> q </ci> </apply> </apply> </bvar> <apply> <notin /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> </apply> <integers /> </apply> </apply> <apply> <notin /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> <integers /> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], RowBox[List["Fold", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List[RowBox[List["{", RowBox[List["f_", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List["z_", " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], "f_"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["b_", "k"], "-", "1"]], ")"]], " ", "f_"]]]]]], "]"]], ",", RowBox[List["w", "[", "z_", "]"]], ",", RowBox[List["{", RowBox[List["1", ",", "\[Ellipsis]_", ",", "q_"]], "}"]]]], "]"]]]], "-", RowBox[List["Fold", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List[RowBox[List["{", RowBox[List["f_", ",", "l_"]], "}"]], ",", RowBox[List[RowBox[List["z_", " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], "f_"]]]], "+", RowBox[List[SubscriptBox["a_", "l_"], " ", "f_"]]]]]], "]"]], ",", RowBox[List["w", "[", "z_", "]"]], ",", RowBox[List["{", RowBox[List["1", ",", "\[Ellipsis]_", ",", "p_"]], "}"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", 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"]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], RowBox[List[SubscriptBox["c", RowBox[List["k", "+", "1"]]], " ", SuperscriptBox["z", RowBox[List["1", "-", SubscriptBox["b", "k"]]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "k"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", "p"], "-", SubscriptBox["b", "k"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "-", SubscriptBox["b", "k"]]], ",", RowBox[List["1", "+", SubscriptBox["b", "1"], "-", SubscriptBox["b", "k"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", RowBox[List["k", "-", "1"]]], "-", SubscriptBox["b", "k"]]], ",", RowBox[List["1", "+", SubscriptBox["b", RowBox[List["k", "+", "1"]]], "-", SubscriptBox["b", "k"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "q"], "-", SubscriptBox["b", "k"]]]]], "}"]], ",", "z"]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["\[ForAll]", RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], "\[Element]", "Integers"]], "&&", RowBox[List["j", "\[NotEqual]", "k"]], "&&", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "q"]], "&&", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "q"]]]]]]], RowBox[List["(", RowBox[List["!", RowBox[List[RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["b", "k"]]], "\[Element]", "Integers"]]]], ")"]]]], "&&", RowBox[List["!", RowBox[List[SubscriptBox["b", "k"], "\[Element]", "Integers"]]]]]]]], ")"]]]]]]]]










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





2001-10-29