The Restricted Partition and q-Partial Fractions

TL;DR

This paper explores the structure of the restricted partition function using q-partial fractions, revealing new connections with Ramanujan sums, Bernoulli and Euler numbers, and providing improved approximation and bounds.

Contribution

It introduces a novel expression of q-partial fraction coefficients as linear combinations of Ramanujan sums, including new special functions and combinatorial interpretations.

Findings

Coefficients expressed via Ramanujan sums and special numbers

New bounds and approximation methods for p_N(n)

Combinatorial interpretation of the sums

Abstract

The restricted partition function $p_{N}(n)$ counts the partitions of $n$ into at most $N$ parts. In the nineteenth century Sylvester showed that these partitions can be expressed as a sum of $k$-periodic quasi-polynomials ($1\leq k\leq N$) which he termed as Waves. It is now well-known that one can easily perform a wave decomposition using a special type of partial fraction decomposition (the so-called $q$-partial fractions) of the generating function of $p_{N}(n)$. In this paper we show that the coefficients of these $q$-partial fractions can be expressed as a linear combination of the Ramanujan sums. In particular, we show, for the first time, an appearance of the degenerate Bernoulli numbers, the degenerate Euler numbers and a special generalization of the Ramanujan sums, which we term as a Gaussian-Ramanujan sum, in the formulae for certain waves. These coefficients not only provide a good approximation of $p_{N}(n)$ but they can also be used for obtaining good bounds. Further, we provide a combinatorial meaning to these sums. Our approach for partial fractions is based on a projection operator on the $I$-adic completion of the ring of polynomials, where $I$ is an ideal generated by the Cyclotomic polynomial.