On the restricted partition function via determinants with Bernoulli polynomials

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 prove that, if a determinant ${\Delta }_{r,D}$, which depends only on $r$ and $D$, with entries consisting in values of Bernoulli polynomials is nonzero, then 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 $x_1\ge 0,\dots ,x_r\ge 0$ can be computed in terms of values of Bernoulli polynomials and Bernoulli Barnes numbers.

18 pages. Minor changes to the previous version

References in corpus (1)

• On the restricted partition function

Cited by in corpus (3)

• On the restricted partition function via determinants with Bernoulli polynomials. II
• A note on the number of partitions of $n$ into $k$ parts
• Determinants with Bernoulli polynomials and the restricted partition function