# §16.18 Special Cases

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 $G$-function. This is a consequence of the following relations:

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

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 $G$-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).