An Alternative Generating Function for $k$-Regular Partitions

TL;DR

This paper introduces a new combinatorial generating function for $k$-regular partitions, generalizing Euler's identity and revealing connections to Bessel polynomials for specific cases.

Contribution

It constructs a novel $k$-fold $q$-series for $k$-regular partitions, extending classical results and exploring new combinatorial and algebraic connections.

Findings

For $k=1$, recovers Euler's partition identity.

Establishes a connection to Bessel polynomials at $k=2$.

Provides a combinatorial construction for the generating function.

Abstract

We construct a $k$-fold $q$-series as a generating function of $k$-regular partitions for each positive integer $k$. The $k=1$ case is one of Euler's $q$-series identities pertaining to the partitions into distinct parts. The construction is combinatorial. Although we find a connection to Bessel polynomials in the $k=2$ case, this note is certainly not a study of Bessel polynomials and their $q$-analogs.