Restricted partition functions as Bernoulli and Eulerian polynomials of higher order

Abstract

Explicit expressions for restricted partition function W (s, d m ) and its quasiperiodic components W j (s, d m ) (called Sylvester waves) for a set of positive integers d m = {d 1 , d 2 , . . . , d m } 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 waves is established. Application to counting algebraically independent homogeneous polynomial invariants of finite groups is discussed.

An explicit form of the polynomial part of a restricted partition function

“…A version of this result, given in two different forms, was earlier obtained by Beck, Gessel and Komatsu [3], as mentioned in [20]. Similarly, for j = 2 we have ρ j = −1, and the right-hand side of (4) will typically lead to a convolution sum of higher-order Bernoulli and higher-order Euler polynomials; this was also done in [20]. Furthermore, Rubinstein and Fel extended this approach and expressed W j (s, d) for arbitrary j in terms of generalized Eulerian polynomials of higher order, in addition to the expected higher-order Bernoulli polynomials.…”

“…There is a more general concept of a higher-order Bernoulli polynomial, which allowed Rubinstein and Fel [20] to express the first Sylvester wave in a very compact form. It can be defined as follows (see, e.g., [9, p. 39] …”

“…This fact was used by Rubinstein and Fel [20] to write W 1 (s, d) in a very compact form in terms of a single higher-order Bernoulli polynomial [see (33) below]. A version of this result, given in two different forms, was earlier obtained by Beck, Gessel and Komatsu [3], as mentioned in [20].…”

“…In other words, the sum in (4) is taken over all primitive jth roots of unity ρ ν j . These Sylvester waves have been studied in great detail in recent years; see, e.g., [10,19,20]; see also [7,8,14] for a broader perspective, and [21] for computations related to restricted partitions.…”

“…This fact was used by Rubinstein and Fel [20] to write W 1 (s, d) in a very compact form in terms of a single higher-order Bernoulli polynomial [see (33) below]. A version of this result, given in two different forms, was earlier obtained by Beck, Gessel and Komatsu [3], as mentioned in [20]. Similarly, for j = 2 we have ρ j = −1, and the right-hand side of (4) will typically lead to a convolution sum of higher-order Bernoulli and higher-order Euler polynomials; this was also done in [20].…”