Regularized generalized hypergeometric function: Differentiation (formula 07.32.20.0016)

HypergeometricPFQRegularized

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQRegularized[{a₁,...,a_p},{b₁,...,b_q},z]

Differentiation

Symbolic differentiation

With respect to z

http://functions.wolfram.com/07.32.20.0016.01

Input Form

D[z^\[Alpha] HypergeometricPFQRegularized[{-(n/r), (-n + 1)/r, \[Ellipsis], (-n + r - 1)/r, Subscript[a, r + 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z^m], {z, n}] == m^(n - \[Alpha] - 1/2) (2 Pi)^((m - 1)/2) Gamma[\[Alpha] + 1] z^(\[Alpha] - n) HypergeometricPFQRegularized[{-(n/r), (-n + 1)/r, \[Ellipsis], (-n + r - 1)/r, (\[Alpha] + 1)/m, (\[Alpha] + 2)/m, \[Ellipsis], (\[Alpha] + m)/m, Subscript[a, r + 1], \[Ellipsis], Subscript[a, p]}, {(\[Alpha] - n + 1)/m, (\[Alpha] - n + 2)/m, \[Ellipsis], (\[Alpha] - n + m)/m, Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z^m] /; Element[r, Integers] && r > 0 && Element[m, Integers] && m > 0 && Element[n, Integers] && n > 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["(", RowBox[List[SuperscriptBox["z", "\[Alpha]"], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["n", "r"]]], ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "1"]], "r"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "r", "-", "1"]], "r"], ",", SubscriptBox["a", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", SuperscriptBox["z", "m"]]], "]"]]]], ")"]]]], "\[Equal]", " ", RowBox[List[SuperscriptBox["m", RowBox[List["n", "-", "\[Alpha]", "-", FractionBox["1", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "\[Pi]"]], ")"]], FractionBox[RowBox[List["m", "-", "1"]], "2"]], RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", "+", "1"]], "]"]], SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "n"]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["n", "r"]]], ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "1"]], "r"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "r", "-", "1"]], "r"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "m"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "2"]], "m"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["\[Alpha]", "+", "m"]], "m"], ",", SubscriptBox["a", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "-", "n", "+", "1"]], "m"], ",", FractionBox[RowBox[List["\[Alpha]", "-", "n", "+", "2"]], "m"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["\[Alpha]", "-", "n", "+", "m"]], "m"], ",", SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", SuperscriptBox["z", "m"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["r", "\[Element]", "Integers"]], "\[And]", RowBox[List["r", ">", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> n </mi> </msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mi> α </mi> </msup> <mo> ⁢ </mo> <mrow> <mrow> <msub> <mo>   </mo> <mi> p </mi> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mi> q </mi> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> n </mi> <mi> r </mi> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> + </mo> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </mfrac> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> p </mi> </msub> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List["-", FractionBox["n", "r"]]], ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "1"]], "r"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "r", "-", "1"]], "r"], ",", SubscriptBox["a", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] </annotation> </semantics> <mo> ; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <semantics> <msup> <mi> z </mi> <mi> m </mi> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["z", "m"], HypergeometricPFQ, Rule[Editable, True]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <msup> <mi> m </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> α </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> α </mi> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mrow> <msub> <mo>   </mo> <mrow> <mi> p </mi> <mo> + </mo> <mi> m </mi> </mrow> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mrow> <mi> q </mi> <mo> + </mo> <mi> m </mi> </mrow> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> n </mi> <mi> r </mi> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> + </mo> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> α </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mi> m </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> α </mi> <mo> + </mo> <mi> m </mi> </mrow> <mi> m </mi> </mfrac> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> p </mi> </msub> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List["-", FractionBox["n", "r"]]], ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "1"]], "r"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "r", "-", "1"]], "r"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "m"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "2"]], "m"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["\[Alpha]", "+", "m"]], "m"], ",", SubscriptBox["a", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] </annotation> </semantics> <mo> ; </mo> <mfrac> <mrow> <mi> α </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> α </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mi> m </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> α </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mi> m </mi> </mrow> <mi> m </mi> </mfrac> <mo> , </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <semantics> <msup> <mi> z </mi> <mi> m </mi> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["z", "m"], HypergeometricPFQ, Rule[Editable, True]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> r </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> D </ms> <ms> [ </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> α </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms>  </ms> </apply> <ms> p </ms> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> OverscriptBox </ci> <ms> F </ms> <ms> ~ </ms> </apply> <apply> <ci> FormBox </ci> <ms> q </ms> <ci> TraditionalForm </ci> </apply> </apply> </list> </apply> <ms> ⁡ </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> n </ms> <ms> r </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> n </ms> </list> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> r </ms> </apply> <ms> , </ms> <ms> … </ms> <ms> , </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> n </ms> </list> </apply> <ms> + </ms> <ms> r </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> r </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <apply> <ci> RowBox </ci> <list> <ms> r </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> , </ms> <ms> … </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> p </ms> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ; </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> , </ms> <ms> … </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> q </ms> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> m </ms> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <ms> z </ms> <ms> , </ms> <ms> n </ms> <ms> } </ms> </list> </apply> <ms> ] </ms> </list> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> m </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> - </ms> <ms> α </ms> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> π </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> 2 </ms> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> Γ </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> α </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <apply> <ci> RowBox </ci> <list> <ms> α </ms> <ms> - </ms> <ms> n </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms>  </ms> <apply> <ci> RowBox </ci> <list> <ms> p </ms> <ms> + </ms> <ms> m </ms> </list> </apply> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> OverscriptBox </ci> <ms> F </ms> <ms> ~ </ms> </apply> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> m </ms> </list> </apply> <ci> TraditionalForm </ci> </apply> </apply> </list> </apply> <ms> ⁡ </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> n </ms> <ms> r </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> n </ms> </list> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> r </ms> </apply> <ms> , </ms> <ms> … </ms> <ms> , </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> n </ms> </list> </apply> <ms> + </ms> <ms> r </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> r </ms> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> α </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> m </ms> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> α </ms> <ms> + </ms> <ms> 2 </ms> </list> </apply> <ms> m </ms> </apply> <ms> , </ms> <ms> … </ms> <ms> , </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> α </ms> <ms> + </ms> <ms> m </ms> </list> </apply> <ms> m </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <apply> <ci> RowBox </ci> <list> <ms> r </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> , </ms> <ms> … </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> p </ms> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ; </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> α </ms> <ms> - </ms> <ms> n </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> m </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> α </ms> <ms> - </ms> <ms> n </ms> <ms> + </ms> <ms> 2 </ms> </list> </apply> <ms> m </ms> </apply> <ms> , </ms> <ms> … </ms> <ms> , </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> α </ms> <ms> - </ms> <ms> n </ms> <ms> + </ms> <ms> m </ms> </list> </apply> <ms> m </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> <ms> , </ms> <ms> … </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> q </ms> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> m </ms> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> r </ms> <ms> ∈ </ms> <apply> <ci> SuperscriptBox </ci> <ms> ℕ </ms> <ms> + </ms> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> ∈ </ms> <apply> <ci> SuperscriptBox </ci> <ms> ℕ </ms> <ms> + </ms> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> ∈ </ms> <apply> <ci> SuperscriptBox </ci> <ms> ℕ </ms> <ms> + </ms> </apply> </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[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["z_", "\[Alpha]_"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["n_", "r_"]]], ",", FractionBox[RowBox[List[RowBox[List["-", "n_"]], "+", "1"]], "r_"], ",", "\[Ellipsis]_", ",", FractionBox[RowBox[List[RowBox[List["-", "n_"]], "+", "r_", "-", "1"]], "r_"], ",", SubscriptBox["a_", RowBox[List["r_", "+", "1"]]], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "p_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]], ",", SuperscriptBox["z_", "m_"]]], "]"]]]], ")"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["m", RowBox[List["n", "-", "\[Alpha]", "-", FractionBox["1", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], FractionBox[RowBox[List["m", "-", "1"]], "2"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", "+", "1"]], "]"]], " ", SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "n"]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["n", "r"]]], ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "1"]], "r"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "r", "-", "1"]], "r"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "m"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "2"]], "m"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["\[Alpha]", "+", "m"]], "m"], ",", SubscriptBox["a", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["aa", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "-", "n", "+", "1"]], "m"], ",", FractionBox[RowBox[List["\[Alpha]", "-", "n", "+", "2"]], "m"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["\[Alpha]", "-", "n", "+", "m"]], "m"], ",", SubscriptBox["bb", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["bb", "q"]]], "}"]], ",", SuperscriptBox["z", "m"]]], "]"]]]], "/;", RowBox[List[RowBox[List["r", "\[Element]", "Integers"]], "&&", RowBox[List["r", ">", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

2001-10-29

HypergeometricPFQRegularized[{},{b},z]
HypergeometricPFQRegularized[{a₁},{b₁},z]
HypergeometricPFQRegularized[{a₁,a₂},{b₁},z]