q-Laguerre polynomials and partitions with inequality conditions

We introduce the partition function pm(n), which counts the number of partitions of n having more parts larger than m compared to parts less than or equal to m. We show that the generating function for pm(n) is related to the q-Laguerre polynomial. Using relations among q-Laguerre polynomials, we investigate partition identities and inequalities for the partition function pd,α,β(n), which includes pm(n) for a special case.