Properties of Multivariate b-Ary Stern Polynomials

Abstract

Given an integer base b ≥ 2, we investigate a multivariate b-ary polynomial analogue of Stern’s diatomic sequence which arose in the study of hyper b-ary representations of integers. We derive various properties of these polynomials, including a generating function and identities that lead to factorizations of the polynomials. We use some of these results to extend an identity of Courtright and Sellers on the b-ary Stern numbers $\mathit s_{b(n)}$. We also extend a result of Defant and a result of Coons and Spiegelhofer on the maximal values of $\mathit s_{b(n)}$ within certain intervals.