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.
Abstract. An elliptic BC n generalization of the classical two parameter Bailey Lemma is proved, and a basic one parameter BC n Bailey Lemma is obtained as a limiting case. Several summation and transformation formulas associated with the root system BC n are proved as applications, including a 6 ϕ 5 summation formula, a generalized Watson transformation and an unspecialized Rogers-Selberg identity. The last identity is specialized to give an infinite family of multilateral Rogers-Selberg identities. Standard determinant evaluations are then used to compute B n and D n generalizations of the Rogers-Ramanujan identities in terms of determinants of theta functions. Starting with the BC n 6 ϕ 5 summation formula, a similar program is followed to prove an infinite family of D n Euler Pentagonal Number Theorems.
We construct multiple qt-binomial coefficients and related multiple analogues of several celebrated families of special numbers in this paper. These multidimensional generalizations include the first and the second kind of qt-Stirling numbers, qt-Bell numbers, qt-Bernoulli numbers, qt-Catalan numbers and the qt-Fibonacci numbers. In the course of developing main properties of these extensions, we prove results that are significant in their own rights such as certain probability measures on the set of integer partitions.
The author has constructed multiple analogues of several families of combinatorial numbers in a recent article, including the bracket symbol, and the Stirling numbers of the first and second kind. In the present paper, a multiple analogue of another sequence, the Lah numbers, is developed, and certain associated identities and significant properties of all these sequences are constructed.
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