A noncommutative weight-dependent generalization of the binomial theorem

Details A noncommutative weight-dependent generalization of the binomial theorem A noncommutative weight-dependent generalization of the binomial theorem

2011-06-10

arxiv.org/abs/1106.2112v2

Mathematics

Quantum Algebra

23 pages; flaws in the definition of the algebra corrected

Scientific paper

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.

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