§19.19 Taylor and Related Series

For $N=0,1,2,\mathrm{\dots }$ define the homogeneous hypergeometric polynomial

where the summation extends over all nonnegative integers $m_1,\mathrm{\dots },m_n$ whose sum is $N$. The following two multivariate hypergeometric series apply to each of the integrals (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23):

If $n=2$, then (19.19.3) is a Gauss hypergeometric series (see (19.25.43) and (15.2.1)).

Define the elementary symmetric function $E_s(𝐳)$ by

and define the $n$-tuple $\frac{\mathrm{𝟏}}{\mathrm{𝟐}}=(\frac{1}{2},\mathrm{\dots },\frac{1}{2})$. Then

where $M={\sum }_{j=1}^n{m}_j$ and the summation extends over all nonnegative integers $m_1,\mathrm{\dots },m_n$ such that ${\sum }_{j=1}^{n}j{m}_j=N$.

This form of $T_N$ can be applied to (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) if we use

as well as (19.16.5) and (19.16.6). The number of terms in $T_N$ can be greatly reduced by using variables $𝐙=\mathrm{𝟏}-(𝐳/A)$ with $A$ chosen to make $E_1(𝐙)=0$. Then $T_N$ has at most one term if $N\le 5$ in the series for $R_F$. For $R_J$ and $R_D$, $T_N$ has at most one term if $N\le 3$, and two terms if $N=4$ or 5.

where

Special cases are given in (19.36.1) and (19.36.2).