Special classes of $q$-bracket operators

TL;DR

This paper explores special classes of $q$-bracket operators related to partition functions, deriving new convolution identities and linking them through divisor sums, thereby generalizing classical results like Euler's and Stanley's theorems.

Contribution

It introduces new classes of $q$-bracket operators, derives convolution identities, and generalizes classical theorems in partition theory and number theory.

Findings

Generalized Euler's convolution identity for the partition function.

Derived an analogous identity for the totient function.

Extended Stanley's theorem and provided new combinatorial results.

Abstract

We study the $q$-bracket operator of Bloch and Okounkov when applied to $f(\lambda)=\sum_{\lambda_i \in \lambda}g(\lambda_i)$ and $f(\lambda)=\sum_{\substack{\lambda_i \in \lambda \lambda_i \text{distinct} }}g(\lambda_i)$. We use these expansions to derive convolution identities for the functions $f$ and link both classes of $q$-brackets through divisor sums. As a result, we generalize Euler's classic convolution identity for the partition function and obtain an analogous identity for the totient function. As corollaries, we generalize Stanley's theorem as well as provide several new combinatorial results.