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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/07.32.03.0069.01









  


  








Input Form





HypergeometricPFQRegularized[{1, Subscript[a, 2], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[a, 2] + 1, \[Ellipsis], Subscript[a, n + 1] + 1, Subscript[a, n + 2] + 2, \[Ellipsis], Subscript[a, q + 1] + 2}, -1] == ((-1)^(q - 1) (Sum[Binomial[q + k - 1, k] 2^(k - q + n) (Zeta[q - n - k, 1/4] - Zeta[q - n - k, 3/4]), {k, 0, q - n - 2}] - Sum[(-1)^(q - k) Binomial[q - n + k - 1, k] 2^(k - q) (Zeta[q - k, 1/4] - Zeta[q - k, 3/4]), {k, 0, q - 2}] - Sum[Binomial[q + k - 1, k] 2^(q - n - k), {k, 0, q - n - 1}] + Pi Binomial[2 q - n - 2, q - 1]))/Pi^(q/2) /; Subscript[a, 2] == Subscript[a, 3] == \[Ellipsis] == Subscript[a, q + 1] == 1/2 && q > 1








Standard Form





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MathML Form







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</ms> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> 2 </ms> </list> </apply> </apply> <ms> + </ms> <ms> 2 </ms> </list> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <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> 2 </ms> </list> </apply> </list> </apply> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <apply> <ci> Function </ci> <ci> Null </ci> <apply> <ci> HoldComplete </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> <list> <ci> HoldAllComplete </ci> </list> </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> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> Function </ci> <ci> Null </ci> <apply> <ci> HoldComplete </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> <list> <ci> HoldAllComplete </ci> </list> </apply> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> &#960; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> q </ms> </list> </apply> <ms> / </ms> <ms> 2 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#960; </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> </ms> <ms> q </ms> </list> </apply> <ms> - </ms> <ms> n </ms> <ms> - </ms> <ms> 2 </ms> </list> </apply> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> Binomial </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> n </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> q </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> <list> <apply> <ci> TagBox </ci> <ms> k </ms> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> Binomial </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> n </ms> <ms> - </ms> <ms> k </ms> </list> </apply> </apply> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> n </ms> <ms> - </ms> <ms> 2 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> q </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> <list> <apply> <ci> TagBox </ci> <ms> k </ms> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> Binomial </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> n </ms> <ms> - </ms> <ms> q </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#950; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> n </ms> <ms> - </ms> <ms> k </ms> </list> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 4 </ms> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <lambda> <bvar> <ci> $CellContext`e1 </ci> </bvar> <bvar> <ci> $CellContext`e2 </ci> </bvar> <apply> <ci> Zeta </ci> <ci> $CellContext`e1 </ci> <ci> $CellContext`e2 </ci> </apply> </lambda> </apply> </apply> <ms> - </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#950; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> n </ms> <ms> - </ms> <ms> k </ms> </list> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 4 </ms> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <lambda> <bvar> <ci> $CellContext`e1 </ci> </bvar> <bvar> <ci> $CellContext`e2 </ci> </bvar> <apply> <ci> Zeta </ci> <ci> $CellContext`e1 </ci> <ci> $CellContext`e2 </ci> </apply> </lambda> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> 2 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> k </ms> </list> </apply> </apply> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> - </ms> <ms> n </ms> <ms> + </ms> <ms> q </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> <list> <apply> <ci> TagBox </ci> <ms> k </ms> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> Binomial </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> - </ms> <ms> q </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#950; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> k </ms> </list> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 4 </ms> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <lambda> <bvar> <ci> $CellContext`e1 </ci> </bvar> <bvar> <ci> $CellContext`e2 </ci> </bvar> <apply> <ci> Zeta </ci> <ci> $CellContext`e1 </ci> <ci> $CellContext`e2 </ci> </apply> </lambda> </apply> </apply> <ms> - </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#950; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> k </ms> </list> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 4 </ms> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <lambda> <bvar> <ci> $CellContext`e1 </ci> </bvar> <bvar> <ci> $CellContext`e2 </ci> </bvar> <apply> <ci> Zeta </ci> <ci> $CellContext`e1 </ci> <ci> $CellContext`e2 </ci> </apply> </lambda> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </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> 2 </ms> </apply> <ms> &#10869; </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 3 </ms> </apply> <ms> &#10869; </ms> <ms> &#8230; </ms> <ms> &#10869; </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> &#10869; </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> &gt; </ms> <ms> 1 </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", ",", SubscriptBox["a_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", RowBox[List["q_", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a_", "2"], "+", "1"]], ",", "\[Ellipsis]_", ",", RowBox[List[SubscriptBox["a_", RowBox[List["n_", "+", "1"]]], "+", "1"]], ",", RowBox[List[SubscriptBox["a_", RowBox[List["n_", "+", "2"]]], "+", "2"]], ",", "\[Ellipsis]_", ",", RowBox[List[SubscriptBox["a_", RowBox[List["q_", "+", "1"]]], "+", "2"]]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "-", "1"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["-", FractionBox["q", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["q", "-", "n", "-", "2"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["q", "+", "k", "-", "1"]], ",", "k"]], "]"]], " ", SuperscriptBox["2", RowBox[List["k", "-", "q", "+", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["q", "-", "n", "-", "k"]], ",", FractionBox["1", "4"]]], "]"]], "-", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["q", "-", "n", "-", "k"]], ",", FractionBox["3", "4"]]], "]"]]]], ")"]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["q", "-", "2"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "-", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["q", "-", "n", "+", "k", "-", "1"]], ",", "k"]], "]"]], " ", SuperscriptBox["2", RowBox[List["k", "-", "q"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["q", "-", "k"]], ",", FractionBox["1", "4"]]], "]"]], "-", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["q", "-", "k"]], ",", FractionBox["3", "4"]]], "]"]]]], ")"]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["q", "-", "n", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["q", "+", "k", "-", "1"]], ",", "k"]], "]"]], " ", SuperscriptBox["2", RowBox[List["q", "-", "n", "-", "k"]]]]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "q"]], "-", "n", "-", "2"]], ",", RowBox[List["q", "-", "1"]]]], "]"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "2"], "\[Equal]", SubscriptBox["a", "3"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["q", "+", "1"]]], "\[Equal]", FractionBox["1", "2"]]], "&&", RowBox[List["q", ">", "1"]]]]]]]]]]


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





2001-10-29