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):
Define the elementary symmetric function by
and define the -tuple . Then
where and the summation extends over all nonnegative integers such that .
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.