Weight-dependent commutation relations and combinatorial identities

TL;DR

This paper develops new combinatorial identities for variables with specific weight-dependent commutation relations, extending known results for q-commuting variables and connecting to elliptic functions and rook theory.

Contribution

It introduces weight-dependent binomial theorems and elliptic derivatives, extending classical identities to elliptic and weight-dependent settings.

Findings

Derived weight-dependent binomial theorems

Established functional equations for generalized exponential functions

Connected weight-dependent Weyl algebra extensions to rook theory

Abstract

We derive combinatorial identities for variables satisfying specific systems of commutation relations, in particular elliptic commutation relations. The identities thus obtained extend corresponding ones for $q$-commuting variables $x$ and $y$ satisfying $yx=qxy$. In particular, we obtain weight-dependent binomial theorems, functional equations for generalized exponential functions, we derive results for an elliptic derivative of elliptic commuting variables, and finally study weight-dependent extensions of the Weyl algebra which we connect to rook theory.