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 ₁₀V₉ 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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