A Noncommutative Weight-Dependent Generalization of the Binomial Theorem

Michael J. Schlosser

Séminaire lotharingien de combinatoire, Tome 81 (2020)

Résumé

A weight-dependent generalization of the binomial theorem for noncommuting variables is presented. This result extends the well-known binomial theorem for q-commuting variables by a generic weight function depending on two integers. For two special cases of the weight function, in both cases restricting it to depend only on a single integer, the noncommutative binomial theorem involves an expansion involving complete symmetric functions, and elementary symmetric functions, respectively. Another special case concerns the weight function to be a suitably chosen elliptic (i.e., doubly-periodic meromorphic) function, in which case an elliptic generalization of the binomial theorem is obtained. The latter is utilized to quickly recover Frenkel and Turaev's elliptic hypergeometric ₁₀V₉ summation formula, an identity fundamental to the theory of elliptic hypergeometric series. Further specializations yield noncommutative binomial theorems of basic hypergeometric type.