Determinants with Bernoulli polynomials and the restricted partition function

Abstract

Let r ≥ 1 be an integer, a = (a 1 , . . . , a r ) a vector of positive integers and let D ≥ 1 be a common multiple of a 1 , . . . , a r . We study two natural determinants of order rD with Bernoulli polynomials and we present connections with the restricted partition function p a (n) := the number of integer solutions (x 1 , . . . , x r ) to r j=1 a j x j = n with x 1 ≥ 0, . . . , x r ≥ 0