Table of Integrals, Series, and Products, 8th Edition

9.33 Series of G-functions

$\begin{array}{ccc}{G}_{pq}^{mn}\left(\lambda x|{b_1,\mathrm{\dots },b_q}_{a_1,\mathrm{\dots },a_p}^{}\right)& =\hfill & {\lambda }^{b_1}{\displaystyle \sum _{r=0}^{\infty }\frac{1}{r!}{\left(1-\lambda \right)}^r{G}_{pq}^{mn}\left(x|{b_1+r,b_2,\mathrm{\dots },b_q}_{a_1,\mathrm{\dots },a_p}^{}\right)\left[\left|\lambda -1\right|<1,m\ge 1,\text{ if }m=1\text{ and }p<q,\lambda \text{ may be arbitrary}\right]}\hfill \\ =\hfill & {\lambda }^{b_q}{\displaystyle \sum _{r=0}^{\infty }\frac{1}{r!}{\left(\lambda -1\right)}^r{G}_{pq}^{mn}\left(x|{b_1,\mathrm{\dots },b_{q-1},b_q+r}_{a_1,\mathrm{\dots },a_p}^{}\right)}[m<q,\left|\lambda -1\right|<1]\hfill \\ =\hfill & {\lambda }^{a_1-1}{{\displaystyle \sum _{r=0}^{\infty }\frac{1}{r!}\left(\lambda -\frac{1}{\lambda }\right)}}^r{G}_{pq}^{mn}\left(x|{b_1,\mathrm{\dots }{b}_q}_{a_1-r,a_2,\mathrm{\dots },a_p}^{}\right)\left[n\ge 1,\text{ Re}\lambda \text{<}\frac{1}{2},\left(\text{if }n=1\text{ and }p>q,\text{ then }\lambda \text{ may be arbitrary}\right)\right]\hfill \\ =\hfill & {\lambda }^{a_p-1}{\displaystyle \sum _{r=0}^{\infty }\frac{1}{r!}{\left(\frac{1}{\lambda }-1\right)}^r{G}_{pq}^{mn}\left(x|{b_1,\mathrm{\dots },b_q}_{a_1,\mathrm{\dots },a_{p-1},a_p-r}^{}\right)}\left[n<p,\text{ Re}\gamma \text{<}\frac{1}{2}\right]\hfill \end{array}$

  EH I 213(1) EH I 213(2) EH I 213(3) EH I 213(4)

For integrals of the G-function, see 7.8.

9.34 Connections with other special functions

1. 

$J_v\left(x\right)x^{\mu }=2^{\mu }{G}_{02}^{10}\left(\frac{1}{4}{x}^2|{\frac{1}{2}v+\frac{1}{2}\mu ,\frac{1}{2}\mu -\frac{1}{2}v}_{}\right)$

  EH I 219(44)

2. 

$Y_v\left(x\right)x^{\mu }=2^{\mu }=2^{\mu }{G}_{13}^{20}(\frac{1}{4}{x}^2|{\frac{1}{2}\mu -\frac{1}{2}v,\frac{1}{2}\mu +\frac{1}{2}v,\frac{1}{2}\mu -\frac{1}{2}v-\frac{1}{2}}_{\frac{1}{2}\mu -\frac{1}{}}^{}$