## Michael J. Schlosser

# A noncommutative weight-dependent generalization of the binomial theorem

### (24 pages)

**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 a special case of the weight function,
restricting it to depend only on a single integer,
the noncommutative binomial theorem involves an expansion of
complete symmetric functions. 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.

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