Special commutation relations and combinatorial identities
Seminar Room 1, Newton Institute
We study commutation relations involving special weight functions, for which we obtain a weight-dependent generalization of the binomial theorem. When the weight functions are suitably chosen elliptic (i.e., doubly-periodic meromorphic) functions, we have the notable special case of so-called "elliptic-commuting" variables (that generalize the q-commuting variables yx=qxy) satisfying an elliptic generalization of the binomial theorem. The latter is utilized to quickly recover Frenkel and Turaev's 10V9 summation formula, an identity fundamental to the (young) theory of elliptic hypergeometric series. Furthermore, the combinatorial interpretation of our commutation relations in terms of weighted enumeration of lattice paths allows us to deduce other combinatorial identities as well.