Michael J. Schlosser
An elliptic extension of the multinomial theorem
(14 pages)
Abstract.
We present a multinomial theorem for elliptic commuting
variables. This result extends the author's previously obtained
elliptic binomial theorem to higher rank. Two essential
ingredients are a simple elliptic star-triangle relation,
ensuring the uniqueness of the normal form coefficients,
and, for the recursion of the closed form elliptic multinomial
coefficients, the Weierstraß type A elliptic
partial fraction decomposition. From our elliptic multinomial
theorem we obtain, by convolution, an identity that is equivalent
to Rosengren's type A extension of the
Frenkel-Turaev 10V9 summation, which in the
trigonometric or basic limiting case reduces to Milne's type
A extension of the Jackson 8Φ7 summation.
Interpreted in terms of a weighted counting of lattice paths in
the integer lattice ℤr, our derivation of the
Ar Frenkel-Turaev summation constitutes the first
combinatorial proof of that fundamental identity, and,
at the same time, of important special cases including
the Ar Jackson summation.
The following version is available:
Back to Michael Schlosser's
home page.