Generalized adjoint actions

TL;DR

This paper generalizes the classical adjoint action formula by replacing the exponential with arbitrary formal power series, enabling new combinatorial applications to q-series and symmetric functions.

Contribution

It introduces a generalized formula for adjoint actions using arbitrary formal power series, extending classical results and enabling new combinatorial applications.

Findings

Derived a generalized adjoint action formula for any formal power series.

Applied the generalized formula to q-exponentials and q-binomial coefficients.

Established connections to Hall-Littlewood polynomials.

Abstract

The aim of this paper is to generalize the classical formula $e^xye^{-x}=\sum\limits_{k\ge 0} \frac{1}{k!} (ad~x)^k(y)$ by replacing $e^x$ with any formal power series $\displaystyle {f(x)=1+\sum_{k\ge 1} a_kx^k}$. We also obtain combinatorial applications to $q$-exponentials, $q$-binomials, and Hall-Littlewood polynomials.