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16 Generalized Hypergeometric Functions & Meijer G-FunctionMeijer G-Function

§16.18 Special Cases

The F11 and F12 functions introduced in Chapters 13 and 15, as well as the more general Fqp functions introduced in the present chapter, are all special cases of the Meijer G-function. This is a consequence of the following relations:

16.18.1 Fqp(a1,,apb1,,bq;z)=(k=1qΓ(bk)/k=1pΓ(ak))Gp,q+11,p(-z;1-a1,,1-ap0,1-b1,,1-bq)=(k=1qΓ(bk)/k=1pΓ(ak))Gq+1,pp,1(-1z;1,b1,,bqa1,,ap).

As a corollary, special cases of the F11 and F12 functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer G-function. Representations of special functions in terms of the Meijer G-function are given in Erdélyi et al. (1953a, §5.6), Luke (1969a, §§6.4–6.5), and Mathai (1993, §3.10).