§15.15 Sums

ⓘ

Keywords:: compendia, hypergeometric function, sums
Permalink:: http://dlmf.nist.gov/15.15
See also:: Annotations for Ch.15

15.15.1		$\mathbf{F}\left({a,b\atop c};\frac{1}{z}\right)=\left(1-\frac{z_{0}}{z}\right)% ^{-a}\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}}{s!}\*\mathbf{F}\left({-s,b% \atop c};\frac{1}{z_{0}}\right)\left(1-\frac{z}{z_{0}}\right)^{-s}.$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §15.19(i) Permalink: http://dlmf.nist.gov/15.15.E1 Encodings: TeX, pMML, png Correction (effective with 1.1.2): The Pochhammer symbol now links to its definition. See also: Annotations for §15.15 and Ch.15

Here $z_{0}$ ( ${\neq 0}$ ) is an arbitrary complex constant and the expansion converges when $|z-z_{0}|>\max(|z_{0}|,|z_{0}-1|)$ . For further information see Bühring (1987a) and Kalla (1992).

For compendia of finite sums and infinite series involving hypergeometric functions see Prudnikov et al. (1990, §§5.3 and 6.7) and Hansen (1975).