Multivariate Rogers-Szegö polynomials and flags in finite vector spaces

الأصل · EN

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.