# §19.19 Taylor and Related Series

For define the homogeneous hypergeometric polynomial

where the summation extends over all nonnegative integers whose sum is . 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 , then (19.19.3) is a Gauss hypergeometric series (see (19.25.43) and (15.2.1)).

Define the elementary symmetric function by

19.19.4

and define the -tuple . Then

where and the summation extends over all nonnegative integers such that .

This form of 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 can be greatly reduced by using variables with chosen to make . Then has at most one term if in the series for . For and , has at most one term if , and two terms if or 5.

where

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