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

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.


Introduction
Let a := (a 1 , a 2 , . . ., a r ) be a sequence of positive integers, r ≥ 1.The restricted partition function associated to a is p a : N → N, p a (n) := the number of integer solutions (x 1 , . . ., x r ) of r i=1 a i x i = n with x i ≥ 0. Let D be a common multiple of a 1 , . . ., a r .According to [5], p a (n) is a quasi-polynomial of degree r − 1, with the period D, i.e. p a (n) = d a,r−1 (n)n r−1 + • • • + d a,1 (n)n + d a,0 (n), (∀)n ≥ 0, (1.1) where d a,m (n+D) = d a,m (n), (∀)0 ≤ m ≤ r −1, n ≥ 0, and d a,r−1 (n) is not identically zero.The restricted partition function p a (n) was studied extensively in the literature, starting with the works of Sylvester [14] and Bell [5].Popoviciu [11] gave a precise formula for r = 2. Recently, Bayad and Beck [4, Theorem 3.1] proved an explicit expression of p a (n) in terms of Bernoulli-Barnes polynomials and the Fourier Dedekind sums, in the case that a 1 , . . ., a r are are pairwise coprime.In [7] we proved that the computation of p a (n) can be reduced to solving the linear congruency a 1 j 1 + • • • + a r j r ≡ n(mod D) in the range 0 ≤ j 1 ≤ D a 1 , . . ., 0 ≤ j r ≤ D ar .In [9] we proved that if a determinant ∆ r,D , which depends only on r and D, with entries consisting in values of Bernoulli polynomials is nonzero, then p a (n) can be computed in terms of values of Bernoulli polynomials and Bernoulli Barnes numbers.The aim of our paper is to tackle the same problem, from another perspective which relays on the arithmetic properties of Bernoulli polynomials.
First we recall some definitions.The Barnes zeta function associated to a and w > 0 is see [3] and [13] for further details.It is well known that ζ a (s, w) is meromorphic on C with poles at most in the set {1. . . ., r}.We consider the function In [7,Lemma 2.6] we proved that where is the Hurwitz zeta function.See also [8].The Bernoulli numbers B j are defined by They are related with the Bernoulli numbers by It is well know, see for instance [2,Theorem 12.13], that The Bernoulli-Barnes polynomials are defined by The Bernoulli-Barnes numbers are defined by According to [12,Formula (3.10)], it holds that From (1.2) and (1.6) it follows that From (1.3), (1.5) and (1.7) it follows that Let α : ) and multiplying with D, we obtain the system of linear equations which has the determinant Note that, with the notation given in [9, (2.10)], we have that ∆ r,D = ∆ r,D (0, 1, . . ., rD − 1), here ommiting the condition α 1 ≥ 2.
Our main theorem is the following: There exists a sequence of integers α : In particular, we can compute p a (n) in terms of values of Bernoulli polynomials and Bernoulli-Barnes numbers.
We believe that the result holds for any integer D ≥ 1.Unfortunately, our method based on p-adic value and congruences for Bernoulli numbers and for values of Bernoulli polynomials, is not refined enough to prove it.

Properties of Bernoulli polynomials
We several properties of the Bernoulli polynomials.We have that: (2.1) For any integers n ≥ 1 and 1 ≤ v ≤ D, using (1.4), we let According to the a result of T. Clausen and C. von Staudt (see [6], [15]), we have that where A 2n ∈ Z and the sum is over the all primes p such that p − 1|2n.
Let p be a prime.For any integer a, the p-adic order of a is v p (a) := max{k : Lemma 2.1.For any integer n ≥ 1, it holds that: (1) Bn ( 12 ) = 0, if n is odd, and Bn ( 1 2 ) ≡ 1(mod 2), if n is even. ( Assume n is even.According to (2.2), we have that Since 2|2nB 1 = −n and v 2 (2 j B j ) ≥ 1 for any j ≥ 2, the conclusion follows immediately.
(2) According to (2.2), we have that From (2.5), we have that v p (p j B j ) ≥ 1 for j ≥ 1, hence the conclusion follows immediately.
Proof.We have that On the other hand, from (2.4), it follows that hence we get the required result.

Proof of Theorem 1.2
The D = 1 was proved in Proposition 3.1.Also, the case D = 2 was proved in Proposition 3.3.Assume that D := p > 2 is a prime number.Let k := ⌊ p−1 2 ⌋.According to Proposition 3.4, it is enough to prove that ∆′ r,p (α) = 0 and ∆′′ r,p (α) = 0. Let be a sequence of positive integers.We define From (4.1) and (4.2) it follows that On the other hand, from Lemma 2.1(2) it follows that B α j ( v p ) ≡ v α j (mod p), (∀)1 ≤ j ≤ rp.Aknowledgment: I would like to express my gratitude to Florin Nicolae for the valuable discussions regarding this paper.