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http://functions.wolfram.com/07.32.03.0030.01
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HypergeometricPFQRegularized[{-n, Subscript[a, 2], \[Ellipsis],
Subscript[a, q + 1]}, {Subscript[a, 2] - Subscript[n, 2], \[Ellipsis],
Subscript[a, q + 1] - Subscript[n, q + 1]}, 1] ==
(-1)^\[Sigma] (n!/Product[Gamma[Subscript[a, j]], {j, 2, q + 1}])
((-(Subscript[a, 2] - Subscript[n, 2] + n)) \[Sigma] +
Sum[Subscript[n, k] Sum[Subscript[a, j] - Subscript[a, j + 1] +
Subscript[n, j + 1], {j, 2, k - 1}], {k, 3, q + 1}]) /;
\[Sigma] == n + 1 && Element[Subscript[n, j], Integers] &&
Subscript[n, j] >= 0 && 2 <= j <= q + 1 &&
\[Sigma] == Sum[Subscript[n, j], {j, 2, q + 1}]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["n", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "-", SubscriptBox["n", RowBox[List["q", "+", "1"]]]]]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Sigma]", " "]]], FractionBox[RowBox[List["n", "!"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List["Gamma", "[", SubscriptBox["a", "j"], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["n", "2"], "+", "n"]], ")"]]]], "\[Sigma]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "3"]], RowBox[List["q", "+", "1"]]], RowBox[List[SubscriptBox["n", "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "2"]], RowBox[List["k", "-", "1"]]], RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", RowBox[List["j", "+", "1"]]], "+", SubscriptBox["n", RowBox[List["j", "+", "1"]]]]], ")"]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["\[Sigma]", "\[Equal]", RowBox[List["n", "+", "1"]]]], "\[And]", RowBox[List[SubscriptBox["n", "j"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["n", "j"], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["2", "\[LessEqual]", "j", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "\[And]", RowBox[List["\[Sigma]", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "2"]], RowBox[List["q", "+", "1"]]], SubscriptBox["n", "j"]]]]]]]]]]]
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<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>   </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mi> q </mi> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> - </mo> <msub> <mi> n </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["q", "+", "1"]], TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List[TagBox[SubscriptBox["a", RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["n", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "-", SubscriptBox["n", RowBox[List["q", "+", "1"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> σ </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mrow> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 3 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msub> <mi> n </mi> <mi> k </mi> </msub> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> a </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> n </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> σ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> σ </mi> <mo> ⩵ </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> n </mi> <mi> j </mi> </msub> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mn> 2 </mn> <mo> ≤ </mo> <mi> j </mi> <mo> ≤ </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> σ </mi> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <msub> <mi> n </mi> <mi> j </mi> </msub> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <ci> … </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> σ </ci> </apply> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 2 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> j </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 3 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 2 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <plus /> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> σ </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> σ </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> j </ci> </apply> <ci> ℕ </ci> </apply> <apply> <leq /> <cn type='integer'> 2 </cn> <ci> j </ci> <apply> <plus /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <eq /> <ci> σ </ci> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 2 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n_"]], ",", SubscriptBox["a_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", RowBox[List["q_", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a_", "2"], "-", SubscriptBox["n_", "2"]]], ",", "\[Ellipsis]_", ",", RowBox[List[SubscriptBox["a_", RowBox[List["q_", "+", "1"]]], "-", SubscriptBox["n_", RowBox[List["q_", "+", "1"]]]]]]], "}"]], ",", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Sigma]"], " ", RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[SubscriptBox["aa", "2"], "-", SubscriptBox["nn", "2"], "+", "n"]], ")"]]]], " ", "\[Sigma]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "3"]], RowBox[List["q", "+", "1"]]], RowBox[List[SubscriptBox["n", "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "2"]], RowBox[List["k", "-", "1"]]], RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", RowBox[List["j", "+", "1"]]], "+", SubscriptBox["n", RowBox[List["j", "+", "1"]]]]], ")"]]]]]]]]]], ")"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List["Gamma", "[", SubscriptBox["a", "j"], "]"]]]]], "/;", RowBox[List[RowBox[List["\[Sigma]", "\[Equal]", RowBox[List["n", "+", "1"]]]], "&&", RowBox[List[SubscriptBox["n", "j"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["n", "j"], "\[GreaterEqual]", "0"]], "&&", RowBox[List["2", "\[LessEqual]", "j", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "&&", RowBox[List["\[Sigma]", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "2"]], RowBox[List["q", "+", "1"]]], SubscriptBox["n", "j"]]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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