Confluent hypergeometric function 0F1: Representations through more general functions (formula 07.17.26.0106)

Hypergeometric0F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric0F1[b,z]

Representations through more general functions

Through Meijer G

Classical cases for products of ₀F₁

http://functions.wolfram.com/07.17.26.0106.01

Input Form

Hypergeometric0F1[b, z] Hypergeometric0F1[1 - b - n, z] == ((Gamma[b] Gamma[1 - b - n])/(2^(n + 1) Sqrt[Pi])) (2 Sum[(((-1)^Floor[(n + 1)/2] Gamma[1/2 - n + Floor[n/2] + k])/ (k! Gamma[b + k] Gamma[1 - b - n + k])) Pochhammer[1 - k + Floor[n/2], n - Floor[n/2]] (4 z)^k, {k, 0, Floor[n/2]}] - (-1)^n Pi MeijerG[{{(1 + n)/2, (2 + n)/2}, {1/2}}, {{1 + n}, {b + n, 0, 1 - b, 1/2}}, 4 z]) /; Element[n, Integers] && n >= 0

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)

2007-05-02

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