Gauss hypergeometric function 2F1: Series representations (formula 07.23.06.0044)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Series representations

Generalized power series

Expansions on branch cuts

For the function itself

http://functions.wolfram.com/07.23.06.0044.01

Input Form

Hypergeometric2F1[a, b, c, z] == ((Pi Gamma[c])/(Gamma[a] Gamma[b] Gamma[c - a] Gamma[c - b])) E^(2 I (c - a - b) Pi Floor[Arg[x - z]/(2 Pi)]) Sum[((-1)^k/k!) ((1 - x)^(c - a - b - k) Csc[(a + b - c) Pi] Gamma[-a + c] Gamma[-b + c] Hypergeometric2F1Regularized[-a + c, -b + c, 1 - a - b + c - k, 1 - x] - (Csc[(a + b - c) Pi] + 2 I E^(I (a + b - c) Pi Floor[Arg[x - z]/(2 Pi)]) Floor[Arg[x - z]/(2 Pi)]) Gamma[a + k] Gamma[b + k] Hypergeometric2F1Regularized[a + k, b + k, 1 + a + b - c + k, 1 - x]) (z - x)^k, {k, 0, Infinity}] /; !Element[c - a - b, Integers] && Element[x, Reals] && x > 1

Standard Form

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MathML Form

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</apply> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <notin /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <integers /> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <gt /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{},{b},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},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]
HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]