Gauss hypergeometric function 2F1: Series representations (formula 07.23.06.0051)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Series representations

Generalized power series

Expansions at z==1

For the function itself

General case

http://functions.wolfram.com/07.23.06.0051.01

Input Form

Hypergeometric2F1[a, b, c, z] == Subscript[F, Infinity][z, a, b, c] /; (Subscript[F, n][z, a, b, c] == ((Gamma[c] Gamma[c - a - b])/(Gamma[c - a] Gamma[c - b])) Sum[((Pochhammer[a, k] Pochhammer[b, k])/(Pochhammer[a + b - c + 1, k] k!)) (1 - z)^k, {k, 0, n}] + ((Gamma[c] Gamma[a + b - c])/ (Gamma[a] Gamma[b])) (1 - z)^(c - a - b) Sum[((Pochhammer[c - a, k] Pochhammer[c - b, k])/ (Pochhammer[1 - a - b + c, k] k!)) (1 - z)^k, {k, 0, n}] == Hypergeometric2F1[a, b, c, z] - (1 - z)^(1 + n) ((Gamma[c] Gamma[-a - b + c] Pochhammer[a, 1 + n] Pochhammer[b, 1 + n])/ ((1 + n)! Gamma[-a + c] Gamma[-b + c] Pochhammer[1 + a + b - c, 1 + n])) HypergeometricPFQ[{1, 1 + a + n, 1 + b + n}, {2 + n, 2 + a + b - c + n}, 1 - z] - (1 - z)^(1 - a - b + c + n) ((Gamma[a + b - c] Gamma[c] Pochhammer[-a + c, 1 + n] Pochhammer[-b + c, 1 + n])/((1 + n)! Gamma[a] Gamma[b] Pochhammer[1 - a - b + c, 1 + n])) HypergeometricPFQ[ {1, 1 - a + c + n, 1 - b + c + n}, {2 + n, 2 - a - b + c + n}, 1 - z] && Element[n, Integers] && n >= 0) && !Element[c - a - b, Integers]

Standard Form

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MathML 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]