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

Keywords:: generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/16.10

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

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

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : 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/16.10.E1
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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 $\mathop{{{}_{{p+1}}F_{{p}}}\/}\nolimits\!\left({a_{1},\dots,a_{{p+1}}\atop b_{1},\dots,b_{p}};z\zeta\right)=(1-z)^{{-a_{1}}}\sum _{{k=0}}^{{\infty}}\frac{\left(a_{1}\right)_{{k}}}{k!}\mathop{{{}_{{p+1}}F_{{p}}}\/}\nolimits\!\left({-k,a_{2},\dots,a_{{p+1}}\atop b_{1},\dots,b_{p}};\zeta\right)\left(\frac{z}{z-1}\right)^{k}.$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $p$ : nonnegative integer, $z$ : 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/16.10.E2
Encodings:: TeX, pMML, png

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

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).

§16.10 Expansions in Series of Functions

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