  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 on only 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.

Quantum Algebra;Combinatorics