Mixed q-analogues, lacunary polynomials and Truesdellian partial differential equations

Multiparameter families of possibly lacunary polynomials in x are constructed that have remarkably explicit (or “almost” explicit) expansions about both x=0 and x=1. They are placed in the framework of questions about mixed q-analogues with fewest terms. The properties of the most tractable of these surprisingly tractable polynomials are established with the aid of solutions to partial differential equations of a type much studied by Truesdell.