# §16.10 Expansions in Series of $\mathop{{{}_{p}F_{q}}\/}\nolimits$ Functions

The following expansion, with appropriate conditions and together with similar results, is given in Fields and Wimp (1961):

 16.10.1 $\mathop{{{}_{p+r}F_{q+s}}\/}\nolimits\!\left({a_{1},\dots,a_{p},c_{1},\dots,c_% {r}\atop b_{1},\dots,b_{q},d_{1},\dots,d_{s}};z\zeta\right)=\sum_{k=0}^{\infty% }\frac{{\left(\mathbf{a}\right)_{k}}{\left(\alpha\right)_{k}}{\left(\beta% \right)_{k}}(-z)^{k}}{{\left(\mathbf{b}\right)_{k}}{\left(\gamma+k\right)_{k}}% k!}\mathop{{{}_{p+2}F_{q+1}}\/}\nolimits\!\left({\alpha+k,\beta+k,a_{1}+k,% \dots,a_{p}+k\atop\gamma+2k+1,b_{1}+k,\dots,b_{q}+k};z\right)\mathop{{{}_{r+2}% F_{s+2}}\/}\nolimits\!\left({-k,\gamma+k,c_{1},\dots,c_{r}\atop\alpha,\beta,d_% {1},\dots,d_{s}};\zeta\right).$

Here $\alpha$, $\beta$, and $\gamma$ are free real or complex parameters.

The next expansion is given in Nørlund (1955, equation (1.21)):

 16.10.2 $\mathop{{{}_{p+1}F_{p}}\/}\nolimits\!\left({a_{1},\dots,a_{p+1}\atop b_{1},% \dots,b_{p}};z\zeta\right)=(1-z)^{-a_{1}}\sum_{k=0}^{\infty}\frac{{\left(a_{1}% \right)_{k}}}{k!}\mathop{{{}_{p+1}F_{p}}\/}\nolimits\!\left({-k,a_{2},\dots,a_% {p+1}\atop b_{1},\dots,b_{p}};\zeta\right)\left(\frac{z}{z-1}\right)^{k}.$

When $|\zeta-1|<1$ the series on the right-hand side converges in the half-plane $\Re{z}<\frac{1}{2}$.

Expansions of the form $\sum_{n=1}^{\infty}(\pm 1)^{n}\mathop{{{}_{p}F_{p+1}}\/}\nolimits\!\left(% \mathbf{a};\mathbf{b};-n^{2}z^{2}\right)$ are discussed in Miller (1997), and further series of generalized hypergeometric functions are given in Luke (1969b, Chapter 9), Luke (1975, §§5.10.2 and 5.11), and Prudnikov et al. (1990, §§5.3, 6.8–6.9).