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 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 10V9 summation formula,
an identity fundamental to the theory of elliptic hypergeometric series.
Further specializations yield noncommutative binomial theorems of basic
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