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Determinants with Bernoulli polynomials and the restricted partition function

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 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 $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$.

Determinants with Bernoulli polynomials and the restricted partition function

Abstract

Let be an integer, a vector of positive integers and let be a common multiple of . We study two natural determinants of order with Bernoulli polynomials and we present connections with the restricted partition function the number of integer solutions to with .

Paper Structure

This paper contains 4 sections, 8 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Determinants with Bernoulli polynomials
  4. An approach to compute $p_{\mathbf a}(n)$

Key Result

Proposition 2.1

(See lucrare2 and lucrare2) 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 $(\frac{(-1)^{r-1} n!}{(n+r)!}B_{n+r}(\mathbf a)-\delta_{n0})_{0\leq n\leq rD-1}$. Consequently,

Theorems & Definitions (21)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • ...and 11 more