§16.18 Special Cases

Notes:: See Luke (1969a, Chapter 5).
Keywords:: Meijer -function, generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/16.18

The $\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits$ and $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits$ functions introduced in Chapters 13 and 15, as well as the more general $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits$ functions introduced in the present chapter, are all special cases of the Meijer -function. This is a consequence of the following relations:

16.18.1 $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)=\left({\textstyle\ifrac{\prod\limits_{{k=1}}^{q}\mathop{% \Gamma\/}\nolimits\!\left(b_{k}\right)}{\prod\limits_{{k=1}}^{p}\mathop{\Gamma% \/}\nolimits\!\left(a_{k}\right)}}\right)\mathop{{G^{{1,p}}_{{p,q+1}}}\/}% \nolimits\!\left(-z;{1-a_{1},\dots,1-a_{p}\atop 0,1-b_{1},\dots,1-b_{q}}\right% )=\left({\textstyle\ifrac{\prod\limits_{{k=1}}^{q}\mathop{\Gamma\/}\nolimits\!% \left(b_{k}\right)}{\prod\limits_{{k=1}}^{p}\mathop{\Gamma\/}\nolimits\!\left(% a_{k}\right)}}\right)\mathop{{G^{{p,1}}_{{q+1,p}}}\/}\nolimits\!\left(-\frac{1% }{z};{1,b_{1},\dots,b_{q}\atop a_{1},\dots,a_{p}}\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{{G^{{m,n}}_{{p,q}}}\/}\nolimits\!\left(z;{a_{1},\dots,a_{p}\atop b_{1}% ,\dots,b_{q}}\right)$ : Meijer -function, : nonnegative integer, : nonnegative integer, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.18.E1
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As a corollary, special cases of the $\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits$ and $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits$ 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 -function. Representations of special functions in terms of the Meijer -function are given in Erdélyi et al. (1953a, §5.6), Luke (1969a, §§6.4–6.5), and Mathai (1993, §3.10).