A partially ordered set and q-Krawtchouk polynomials

A derivation is given for the analytic expression of the polynomials, which uses a lowering operator on a partially ordered set, similar to a technique which has been applied to other partially ordered sets.

Abstract

A family of q-Krawtchouk polynomials are the eigenvalues of the association schemes of bilinear, alternating, symmetric, and hermitian forms over a finite field. Another derivation is given for the analytic expression of the polynomials, which uses a lowering operator on a partially ordered set. It does not rely upon the associated additive characters. It is similar to a technique which has been applied to other partially ordered sets.