Multivariate Rogers-Szegö polynomials and flags in finite vector spaces

We give a recursion for the multivariate Rogers-Szegö 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-Szegö 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. 2010 Mathematics Subject Classification: 05A19, 05A15, 05A30