Determinants with Bernoulli polynomials and the restricted partition function

Abstract

Let $r\ge 1$ be an integer, $\mathbf{a}=(a_1,\dots ,a_r)$ a vector of positive integers and let $D\ge 1$ be a common multiple of $a_1,\dots ,a_r$. We study two natural determinants of order $rD$ with Bernoulli polynomials and we present connections with the restricted partition function $p_{\mathbf{a}}(n):=$ the number of integer solutions $(x_1,\dots ,x_r)$ to ${\sum }_{j=1}^r{a}_j{x}_j=n$ with $x_1\ge 0,\dots ,x_r\ge 0$.

14 pages; This is the second part of the paper arXiv:1806.08996

References in corpus (2)

• On the restricted partition function via determinants with Bernoulli polynomials
• On the restricted partition function via determinants with Bernoulli polynomials. II