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

Mircea Cimpoeaş

doi:10.33044/revuma.v61n2a15

Abstract

Let r ≥ 1 be an integer, a = (a1, . . . , ar) a vector of positive integers, and let D ≥ 1 be a common multiple of a1, . . . , ar. We prove that if D = 1 or D is a prime number then the restricted partition function pa(n) := the number of integer solutions (x1, . . . , xr) to Pr j=1 ajxj = n, with x1 ≥ 0, . . . , xr ≥ 0, can be computed by solving a system of linear equations with coefficients that are values of Bernoulli polynomials and Bernoulli–Barnes numbers.

Highlights

Let a := (a1, a2, . . . , ar) be a sequence of positive integers, r ≥ 1
We prove that if D = 1 or D is a prime number the restricted partition function pa(n) := the number of integer solutions (x1, . . . , xr) to r j=1 aj xj n, with x1 ≥ 0, . . . , xr ≥ 0, can be computed by solving a system of linear equations with coefficients that are values of Bernoulli polynomials and Bernoulli–Barnes numbers
The restricted partition function associated to a is pa : N → N, pa(n) := the number of integer solutions (x1, . . . , xr) of r i=1 aixi

Summary

Introduction

Let a := (a1, a2, . . . , ar) be a sequence of positive integers, r ≥ 1. The restricted partition function associated to a is pa : N → N, pa(n) := the number of integer solutions Restricted partition function; Bernoulli polynomial; Bernoulli–Barnes numbers. It is well known (see for instance [2, Theorem 12.13]) that ζ(−n, w) = − Bn+1(w) , n+1 for all n ∈ N, w > 0. Let α : α1 < α2 < · · · < αrD be a sequence of integers with α1 ≥ 2. It follows from (1.8) and (1.9) by Cramer’s rule. We can compute pa(n) in terms of values of Bernoulli polynomials and Bernoulli–Barnes numbers. Our method based on p-adic valuations and congruences for Bernoulli numbers and for the values of Bernoulli polynomials, is not refined enough to prove it

Properties of Bernoulli polynomials

Preliminary results