  We give a recursion for the multivariate Rogers-Szeg\"o polynomials, along
with another recursive functional equation, and apply them to compute special
values. We also consider the sum of all $q$-multinomial coefficients of some
fixed degree and length, and give a recursion for this sum which follows from
the recursion of the multivariate Rogers-Szeg\"o polynomials, and generalizes
the recursion for the Galois numbers. The sum of all $q$-multinomial
coefficients of degree $n$ and length $m$ is the number of flags of length
$m-1$ of subspaces of an $n$-dimensional vector space over a field with $q$
elements. We give a combinatorial proof of the recursion for this sum of
$q$-multinomial coefficients in terms of finite vector spaces.
