On the Restricted Partition Function via Determinants with Bernoulli Polynomials

Determinants with Bernoulli polynomials and the restricted partition function

“…Remark 2.2. In [8] it was conjectured that ∆ r,D = 0 for any r, D ≥ 1. An affirmative answer was given in the case r = 1, r = 2 and D = 1.…”

“…In [8], we proved that if a determinant ∆ r,D , see (2.5), 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. In the second section, we outline several construction and results from [8]. In the third section, we study the polynomial F r,D (x 1 , .…”

“…, 0) = 0. In the last section, we propose another approach to the initial problem, studied in [8], of computing p a (n) in terms of values of Bernoulli polynomials and Bernoulli Barnes numbers. In formula (4.3) we show that…”

A note on the number of partitions of $n$ into $k$ parts

“…In Proposition 3.3, we give another formulas for p(n, 3) and q(n, 3). In [5] we proved that if a certain determinant is nonzero, then the restricted partition function p a (n) can be computed by solving a system of linear equations with coefficients which are values of Bernoulli polynomials and Bernoulli Barnes numbers. Using a similar method, we prove that if a certain determinant ∆(k), which depends only on k, is nonzero, then p(n, k) and q(n, k) can be expressed in terms of values of Bernoulli polynomials and Bernoulli Barnes numbers; see Theorem 3.4.…”

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

“…, 0 ≤ j r ≤ D ar . In [8], 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 this paper is to tackle the same problem, from another perspective that relays on the arithmetic properties of Bernoulli polynomials.…”