Generalized hypergeometric function 2F3: Transformations (formula 07.26.16.0001)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},z]

Transformations

Products, sums, and powers of the direct function

Products of the direct function

http://functions.wolfram.com/07.26.16.0001.01

Input Form

HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, c z] HypergeometricPFQ[{Subscript[\[Alpha], 1], Subscript[\[Alpha], 2]}, {Subscript[\[Beta], 1], Subscript[\[Beta], 2], Subscript[\[Beta], 3]}, d z] == Sum[Subscript[c, k] z^k, {k, 0, Infinity}] /; Subscript[c, k] == ((d^k Pochhammer[Subscript[\[Alpha], 1], k] Pochhammer[Subscript[\[Alpha], 2], k])/ (k! Product[Pochhammer[Subscript[\[Beta], j], k], {j, 1, 3}])) HypergeometricPFQ[{-k, 1 - Subscript[\[Beta], 1] - k, 1 - Subscript[\[Beta], 2] - k, 1 - Subscript[\[Beta], 3] - k, Subscript[a, 1], Subscript[a, 2]}, {1 - Subscript[\[Alpha], 1] - k, 1 - Subscript[\[Alpha], 2] - k, Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, c/d] || Subscript[c, k] == ((c^k Pochhammer[Subscript[a, 1], k] Pochhammer[Subscript[a, 2], k])/ (k! Product[Pochhammer[Subscript[b, j], k], {j, 1, 3}])) HypergeometricPFQ[{-k, 1 - Subscript[b, 1] - k, 1 - Subscript[b, 2] - k, 1 - Subscript[b, 3] - k, Subscript[\[Alpha], 1], Subscript[\[Alpha], 2]}, {1 - Subscript[a, 1] - k, 1 - Subscript[a, 2] - k, Subscript[\[Beta], 1], Subscript[\[Beta], 2], Subscript[\[Beta], 3]}, d/c]

Standard Form

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MathML Form

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<ci> HypergeometricPFQ </ci> <list> <apply> <ci> Subscript </ci> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> α </ci> <cn type='integer'> 2 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> β </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> β </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> β </ci> <cn type='integer'> 3 </cn> </apply> </list> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> k </ci> </apply> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> d </ci> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <ci> Subscript </ci> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <ci> Subscript </ci> <ci> α </ci> <cn type='integer'> 2 </cn> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <cn type='integer'> 3 </cn> </uplimit> <apply> <ci> Pochhammer </ci> <apply> <ci> Subscript </ci> <ci> β </ci> <ci> j </ci> </apply> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> β </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> β </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> β </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </list> <list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> α </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> </list> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> d </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> c </ci> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <cn type='integer'> 3 </cn> </uplimit> <apply> <ci> Pochhammer </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> j </ci> </apply> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> α </ci> <cn type='integer'> 2 </cn> </apply> </list> <list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> β </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> β </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> β </ci> <cn type='integer'> 3 </cn> </apply> </list> <apply> <times /> <ci> d </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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