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On the restricted partition function via determinants with Bernoulli polynomials

Mircea Cimpoeas

Abstract

Let $r\geq 1$ be an integer, $\mathbf a=(a_1,\ldots,a_r)$ a vector of positive integers and let $D\geq 1$ be a common multiple of $a_1,\ldots,a_r$. We prove that, if a determinant $Δ_{r,D}$, 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 $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$ can be computed in terms of values of Bernoulli polynomials and Bernoulli Barnes numbers.

On the restricted partition function via determinants with Bernoulli polynomials

Abstract

Let be an integer, a vector of positive integers and let be a common multiple of . We prove that, if a determinant , which depends only on and , with entries consisting in values of Bernoulli polynomials is nonzero, then the restricted partition function the number of integer solutions to with can be computed in terms of values of Bernoulli polynomials and Bernoulli Barnes numbers.

Paper Structure

This paper contains 3 sections, 17 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. The case $D=1$

Key Result

Proposition 2.1

With the above notations, if $\Delta_{r,D}\neq 0$, then where $\Delta_{r,D}^{m,v}$ is the determinant obtained from $\Delta_{r,D}$, as defined in pista, by replacing the $(mD+v)$-th column with the column $(\frac{(-1)^{r-1} n!}{(n+r)!}B_{n+r}(\mathbf a)-\delta_{n0})_{0\leq n\leq rD-1}$.

Theorems & Definitions (37)

  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Proposition 2.3
  • proof
  • Conjecture 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • ...and 27 more