Generalized hypergeometric function: Specific values (formula 07.31.03.0010)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

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

Specific values

Specialized values

Case _pF_qwith fixed a₁, ..., a_p

http://functions.wolfram.com/07.31.03.0010.01

Input Form

HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[a, 1] + Subscript[m, 1], \[Ellipsis], Subscript[a, n] + Subscript[m, n], Subscript[b, n + 1], \[Ellipsis], Subscript[b, q]}, z] == Product[(Pochhammer[Subscript[a, j], Subscript[m, j]]/ (Subscript[m, j] - 1)!) Sum[\[Ellipsis] Sum[(1/(Subscript[a, k] + Subscript[j, k])) Product[(Pochhammer[1 - Subscript[m, l], Subscript[j, l]]/ Subscript[j, l]!) Product[If[i == k, 1, 1/(Subscript[a, i] + Subscript[j, i] - Subscript[a, k] - Subscript[j, k])] HypergeometricPFQ[{Subscript[a, k] + Subscript[j, k], Subscript[a, n + 1], \[Ellipsis], Subscript[a, p]}, {Subscript[a, k] + Subscript[j, k] + 1, Subscript[b, n + 1], \[Ellipsis], Subscript[b, q]}, z], {i, 1, n}], {l, 1, n}], {Subscript[j, n], 0, Subscript[m, n] - 1}], {k, 1, n}, {Subscript[j, 1], 0, Subscript[m, 1] - 1}], {j, 1, n}] /; Element[Subscript[m, n], Integers] && Subscript[m, n] > 0 && n <= q

Standard Form

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MathML Form

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<ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> i </ms> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> i </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> k </ms> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms>  </ms> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <ms> p </ms> <ms> - </ms> <ms> n </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ci> TraditionalForm </ci> </apply> </apply> <apply> <ci> SubscriptBox </ci> <ms> F </ms> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> n </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> </list> </apply> <ms> ⁡ </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> k </ms> </apply> </list> </apply> <ms> , </ms> <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> … </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> <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> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> k </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> , </ms> <ms> … </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> q </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> <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> TagBox </ci> <ms> z </ms> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

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

2001-10-29

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{},{b},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅},{b₁,b₂,b₃,b₄},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅,a₆},{b₁,b₂,b₃,b₄,b₅},z]