Séminaire Lotharingien de Combinatoire, B81j (2020), 24 pp.
Michael J. Schlosser
A Noncommutative Weight-Dependent Generalization of the Binomial Theorem
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.
Received: April 29, 2019.
Accepted: May 7, 2019.
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