Well-poised Macdonald functions 𝑊_{𝜆} and Jackson coefficients 𝜔_{𝜆} on 𝐵𝐶_{𝑛}

Abstract

The very well-poised elliptic Macdonald functions W λ/µ in n independent variables are defined and their properties are investigated. The W λ/µ are generalized by introducing an extra parameter to the elliptic Jackson coefficients ω λ/µ and their properties are studied. BCn multivariable Jackson sums in terms of both W λ and ω λ functions are proved.

An elliptic $BC_n$ Bailey Lemma, multiple Rogers–Ramanujan identities and Euler’s Pentagonal Number Theorems

“…, x n ) ∈ C n . That ω λ/µ (z; r; a, b) is symmetric is also proved in [14] using a remarkable elliptic BC n 10 ϕ 9 transformation identity.…”

“…As in [14], the Z-space V denotes the space of infinite lower-triangular matrices whose entries are rational functions in complex parameters ρ i , σ i for i ∈ Z ≥ over the field F = C(q, p, t, r, a, b). The condition that u ∈ V is lower triangular with respect to the partial inclusion ordering (3.1) can be stated in the form…”

“…The definitions of M (a, b) and S(b) involve the symmetric elliptic Macdonald functions W λ/µ and Jackson coefficients ω λ/µ on BC n defined and investigated in [13] and [14]. A brief review of the definitions and basic properties of these functions is in order.…”

“…The operator characterization [14] of ω λ/µ yields a recursion formula for Jackson coefficients in the form…”

Essays on the theory of elliptic hypergeometric functions

“…There are several known summation formulae for multiple elliptic hypergeometric series representing multivariate extensions of the Frenkel-Turaev sum [11,34,53,86,90,101,102]. A direct connection with the sums of residues of particular pole sequences of the kernels of elliptic beta integrals on root systems was established for some of them.…”