Some formulae for coefficients in restricted $q$-products
Abstract
In this paper, we derive some formulae involving coefficients of polynomials which occur quite naturally in the study of restricted partitions. Our method involves a recently discovered sieve technique by Li and Wan (Sci. China. Math. 2010). Based on this method, by considering cyclic groups of different orders we obtain some new results for these coefficients. The general result holds for any group of the form where and expresses certain partial sums of coefficients in terms of expressions involving roots of unity. By specializing to different values, we see that these expressions simplify in some cases and we obtain several nice identities involving these coefficients. We also use a result of Sudler (QJMAAT 1964) to obtain an asymptotic formula for the maximum absolute value of these coefficients.
Cite
@article{arxiv.2005.01067,
title = {Some formulae for coefficients in restricted $q$-products},
author = {Ankush Goswami and Venkata Raghu Tej Pantangi},
journal= {arXiv preprint arXiv:2005.01067},
year = {2020}
}
Comments
This is the revised version of arXiv:2005.01067 accepted for publication in J. Number Theory. arXiv admin note: text overlap with arXiv:2004.08954