Restricted partition functions as Bernoulli and Eulerian polynomials of higher order

Restricted partition functions as Bernoulli and Eulerian polynomials of higher order

Abstract Explicit expressions for restricted partition function W (s, d) and its quasiperiodic components W j (s, dm) (called Sylvester waves) for a set of positive integers dm = {d1, d2, . . . , dm} are derived. The formulas are represented in a form of a finite sum over Bernoulli and Eulerian polynomials of higher order with periodic coefficients. A novel recursive relation for the Sylvester ...

Restricted Partition Functions as Bernoulli and Euler Polynomials of Higher Order

Explicit expressions for restricted partition function W (s,d) and its quasiperiodic components Wj(s,d ) (called Sylvester waves) for a set of positive integers d = {d1, d2, . . . , dm} are derived. The formulas are represented in a form of a finite sum over Bernoulli and Euler polynomials of higher order with periodic coefficients. A novel recursive relation for the Sylvester waves is establis...

Extension of the Bernoulli and Eulerian Polynomials of Higher Order and Vector Partition Function

Following the ideas of L. Carlitz we introduce a generalization of the Bernoulli and Eulerian polynomials of higher order to vectorial index and argument. These polynomials are used for computation of the vector partition function W (s,D), i.e., a number of integer solutions to a linear system x ≥ 0, Dx = s. It is shown that W (s,D) can be expressed through the vector Bernoulli polynomials of h...

Higher Order Bernoulli Polynomials and Newton Polygons

Symmetry Properties of Higher-Order Bernoulli Polynomials

Let p be a fixed prime number. Throughout this paper Zp, Qp, and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp. For x ∈ Cp, we use the notation x q 1 − q / 1 − q . Let UD Zp be the space of uniformly differentiable functions on Zp, and let vp be the normalized exponential valuation of Cp wi...