Regularized confluent hypergeometric function 0F1: Specific values (formula 07.18.03.0004)

Hypergeometric0F1Regularized

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric0F1Regularized[b,z]

Specific values

Specialized values

For fixed z and symbolic parameter

http://functions.wolfram.com/07.18.03.0004.01

Input Form

Hypergeometric0F1Regularized[b, z] == ((z^((1/2) (1 - b - Abs[b - 1])) Gamma[-(2/3)] Sign[b - 1])/ (6 3^(5/6) Gamma[1 - Abs[b - 1]])) (-2 (3 3^(1/6) AiryAiPrime[3^(2/3) z^(1/3)] + 3^(2/3) Sign[b - 1] AiryBiPrime[3^(2/3) z^(1/3)]) Sum[((-(2/3) - k + Abs[b - 1])! (-1)^k z^k)/ (k! (Abs[b - 1] - 2/3 - 2 k)! Pochhammer[2/3, k] Pochhammer[1 - Abs[b - 1], k]), {k, 0, Abs[b - 1] - 2/3}] + 9 z^(2/3) (Sqrt[3] AiryAi[3^(2/3) z^(1/3)] + Sign[b - 1] AiryBi[3^(2/3) z^(1/3)]) Sum[((Abs[b - 1] - 5/3 - k)! (-1)^k z^k)/(k! (Abs[b - 1] - 5/3 - 2 k)! Pochhammer[5/3, k] Pochhammer[1 - Abs[b - 1], k]), {k, 0, Abs[b - 1] - 5/3}]) /; Element[Abs[b - 1] - 2/3, Integers]

Standard Form

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MathML Form

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Rule Form

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

2001-10-29

HypergeometricPFQRegularized[{a},{b},z]
HypergeometricPFQRegularized[{a₁,a₂},{b₁},z]
HypergeometricPFQRegularized[{a₁,...,a_p},{b₁,...,b_q},z]