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

Abstract

We prove an explicit formula for the polynomial part of a restricted partition function, also known as the first Sylvester wave. This is achieved by way of some identities for higher-order Bernoulli polynomials, one of which is analogous to Raabe's well-known multiplication formula for the ordinary Bernoulli polynomials. As a consequence of our main result we obtain an asymptotic expression of the first Sylvester wave as the coefficients of the restricted partition grow arbitrarily large. in nonnegative integers x 1 , . . . , x m . For a history of this problem, see [5, p. 119ff.]. A standard method of dealing with questions of this type goes back to Euler and involves a generating function, which in our case isA major advance was made by Sylvester [22,23] who wrote the restricted partition function W (s, d) as a sum of "waves",where the sum is taken over all distinct divisors j of the components of d. Sylvester [23] showed that for each such j, W j (s, d) is the coefficient of t −1 , i.e., the residue, of the function

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

“…, a k ) is a sequence of positive integers and p a (n) is the restricted partition function associated to a, we denote P a (n), the polynomial part of p a (n). Several formulas of P a (n) were proved in [2], [7] and [4].…”

An Algebraic Approach to q-Partial Fractions and Sylvester Denumerants

“…Given a tuple of positive integers A = (a 1 , • • • , a r ), our first main result is deriving an explicit formula for the polynomial part (the first wave, W 1 (t; A)). For other explicit formulas for W 1 (t; A) see [4,12,10].…”

“…Formulas for the denumerants using degenerate Bernoulli number are not known in the literature. For formulas based on Bernoulli numbers see [6,12,18] and a formula without using Bernoulli number see [10]. Sills and Zeilberger [20] gave an algorithm which they called 'rigorous guessing' to obtain a q-partial fractions.…”

“…Recently, obtaining explicit formulas of the waves received a lot of attention. An explicit expression for W 1 (t; A) using Bernoulli numbers was given by Beck, Gessel and Komatsu [4] and without Bernoulli numbers by Dilcher and Vignat [12]. Rubinstein and Fel [18] derived formulas for W j (t; A) using Bernoulli and Euler polynomials of higher order.…”

Partitions and Sylvester waves

“…(46)] the waves are written in terms of Bernoulli and Eulerian polynomials of higher order. An interesting expression for the first wave W 1 that does not involve Bernoulli polynomials has recently appeared in [DV17]. A variation of a result of Glaisher using the Apostol coefficients (2.15) is the following, given in [O'S15, Eq.…”