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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/07.32.03.0027.01









  


  










Input Form





HypergeometricPFQRegularized[{1, a, a, Subscript[a, 4], \[Ellipsis], Subscript[a, q + 1]}, {a + 1, a + 1, Subscript[a, 4] + 1, \[Ellipsis], Subscript[a, q + 1] + 1}, 1] == (-(1/(Gamma[a]^2 Product[Gamma[Subscript[a, j]], {j, 4, q + 1}]))) (Sum[Product[If[l == k, 1, PolyGamma[Subscript[a, k]]/ ((Subscript[a, l] - Subscript[a, k]) (a - Subscript[a, k])^2)], {l, 4, q + 1}] + (PolyGamma[a]/(a - Subscript[a, k])) Product[1/(Subscript[a, l] - a), {l, 4, q + 1}], {k, 4, q + 1}] - PolyGamma[1, a] Product[1/(Subscript[a, l] - a), {l, 4, q + 1}]) /; Subscript[a, l] != Subscript[a, k] && 4 <= l <= q + 1 && 4 <= k <= q + 1 && l != k










Standard Form





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MathML Form







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</ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <ms> a </ms> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> RowBox </ci> <list> <ms> l </ms> <ms> = </ms> <ms> 4 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> l </ms> </apply> <ms> - </ms> <ms> a </ms> </list> </apply> </apply> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> UnderscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> l </ms> <ms> = </ms> <ms> 4 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> l </ms> <ms> &#8800; </ms> <ms> k </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> l </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <ms> 1 </ms> <ms> ) </ms> </list> </apply> </apply> <ms> ( </ms> <ms> a </ms> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> RowBox </ci> <list> <ms> l </ms> <ms> = </ms> <ms> 4 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> l </ms> </apply> <ms> - </ms> <ms> a </ms> </list> </apply> </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> l </ms> </apply> <ms> &#8800; </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <ms> &#8804; </ms> <ms> l </ms> <ms> &#8804; </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <ms> &#8804; </ms> <ms> k </ms> <ms> &#8804; </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> l </ms> <ms> &#8800; </ms> <ms> k </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", ",", "a_", ",", "a_", ",", SubscriptBox["a_", "4"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", RowBox[List["q_", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["a_", "+", "1"]], ",", RowBox[List["a_", "+", "1"]], ",", RowBox[List[SubscriptBox["a_", "4"], "+", "1"]], ",", "\[Ellipsis]_", ",", RowBox[List[SubscriptBox["a_", RowBox[List["q_", "+", "1"]]], "+", "1"]]]], "}"]], ",", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "4"]], RowBox[List["q", "+", "1"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "4"]], RowBox[List["q", "+", "1"]]], RowBox[List["If", "[", RowBox[List[RowBox[List["l", "\[Equal]", "k"]], ",", "1", ",", FractionBox[RowBox[List["PolyGamma", "[", SubscriptBox["a", "k"], "]"]], RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["aa", "l"], "-", SubscriptBox["a", "k"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", SubscriptBox["a", "k"]]], ")"]], "2"]]]]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["PolyGamma", "[", "a", "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "4"]], RowBox[List["q", "+", "1"]]], FractionBox["1", RowBox[List[SubscriptBox["aa", "l"], "-", "a"]]]]]]], RowBox[List["a", "-", SubscriptBox["a", "k"]]]]]], ")"]]]], "-", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", "a"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "4"]], RowBox[List["q", "+", "1"]]], FractionBox["1", RowBox[List[SubscriptBox["aa", "l"], "-", "a"]]]]]]]]], RowBox[List[SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], "2"], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "4"]], RowBox[List["q", "+", "1"]]], RowBox[List["Gamma", "[", SubscriptBox["a", "j"], "]"]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["aa", "l"], "\[NotEqual]", SubscriptBox["a", "k"]]], "&&", RowBox[List["4", "\[LessEqual]", "l", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "&&", RowBox[List["4", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "&&", RowBox[List["l", "\[NotEqual]", "k"]]]]]]]]]]










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





2001-10-29