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

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 $a_1,\ldots,a_r$. In a continuation of a previous paper we prove that, if $D=1$ or $D$ is a prime number, 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 by solving a system of linear equations with coefficients which are values of Bernoulli polynomials and Bernoulli Barnes numbers.

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

Abstract

Let be an integer, a vector of positive integers and let be a common multiple of . In a continuation of a previous paper we prove that, if or is a prime number, the restricted partition function the number of integer solutions to with can be computed by solving a system of linear equations with coefficients which are values of Bernoulli polynomials and Bernoulli Barnes numbers.

Paper Structure

This paper contains 4 sections, 8 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Properties of Bernoulli polynomials
  3. Preliminary results
  4. Proof of Theorem 1.2

Key Result

Proposition 1.1

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

Theorems & Definitions (15)

  • Proposition 1.1
  • proof
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 5 more