On arithmetic properties of binary partition polynomials

Abstract

Let the polynomial bn(t) be defined as the n-th coefficient in the power series expansion (in variable x) of the functionB(t,x)=∏n=0∞11−tx2n=∑n=0∞bn(t)xn. The polynomial bn(t)=∑i=0na(i,n)ti has the following combinatorial interpretation: the i-th coefficient a(i,n) counts the number of representations of n as sums of exactly i powers of 2. The bn(t) can be seen as a polynomial analogue of the number bn(1), which counts the number of binary partitions of n. In the present paper we obtain several results concerning arithmetic properties of the polynomials bn(t) as well as its coefficients. Moreover, we show an interesting connection between coefficients of bn(t) and the number counting so called s-partitions, i.e., the representations of n as sums of numbers of the form 2k−1.