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Séminaire Lotharingien de Combinatoire, B81j (2020), 24 pp.

# Michael J. Schlosser

# A Noncommutative Weight-Dependent Generalization of the Binomial Theorem

**Abstract.**
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 _{10}*V*_{9} 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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