Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating, balanced, very-well-poised, elliptic hypergeometric series.

We compute the inverse of a specific infinite r-dimensional matrix, thus unifying multidimensional matrix inversions recently found by Milne, Lilly, and Bhatnagar. Our inversion is an r-dimensional extension of a matrix inversion previously found by Krattenthaler. We also compute the inverse of another infinite r-dimensional matrix. As applications of our matrix inversions, we derive new summation formulas for multidimensional basic hypergeometric series.

We give the explicit analytic development of Macdonald polynomials in terms of “modified complete ” and elementary symmetric functions. These expansions are obtained by inverting the Pieri formula. Specialization yields similar developments for monomial, Jack and Hall–Littlewood symmetric functions.

We compute the inverse of a specific infinite r-dimensional matrix, extending a matrix inverse of Krattenthaler. Our inversion is different from the r-dimensional matrix inversion recently found by Schlosser but generalizes a multidimensional matrix inversion previously found by Chu. As applications of our matrix inversion we derive some multidimensional q-series identities. Among these are q-analogues of Carlitz' multidimensional Abel-type expansion formulas. Furthermore, we derive a q-analogue of MacMahon's Master Theorem.

We invert a specic innite r-dimensional matrix, thus giving an extension of our previous matrix inversion result. As applications, we derive new summation formulas for series in Ar . 1.

Abstract. In this paper we study the factors of some alternating sums of products of binomial and q-binomial coefficients. We prove that for all positive integers n1,...,nm, nm+1 = n1, and 0 ≤ j ≤ m − 1, [ ] −1 n1 n1 + nm (−1) k q jk2 + ( k 2) m ∏ ni + ni+1 ∈ N[q], ni + k n1 k=−n1 which generalizes a result of Calkin [Acta Arith. 86 (1998), 17–26]. Moreover, we show that for all positive integers n, r and j, [ ] −1 [] 2n 2j ∑n (−1) n j n−k A 1 − q2k+1 q 1 − qn+k+1 [] [ ] r 2n k + j ∈ N[q], n − k k − j k=j where A = (r − 1) ( n) ( j+1) ( k) 2 + r

We give the explicit analytic development of any Jack or Macdonald polynomial in terms of elementary (resp. modified complete) symmetric functions. These two developments are obtained by inverting the Pieri formula. Résumé Nous donnons le développement analytique explicite de tout polynôme de Jack ou de Macdonald sur les fonctions symétriques élémentaires (resp. complètes modifiées). Nous obtenons ces deux développements par inversion de la formule de Pieri. Version française abrégée Au milieu des années cinquante, Hua introduisait les polynômes zonaux et posait le problème d’en obtenir un développement analytique explicite [1]. En dépit de nombreuses recherches, cette question est demeurée ouverte. Elle est désormais formulée dans le cadre plus général des polynômes de Macdonald. Les polynômes zonaux sont en effet un cas particulier des polynômes de Jack, qui sont eux-mêmes un cas limite des polynômes de Macdonald.

Abstract. We present a new matrix inverse with applications in the theory of bilateral basic hypergeometric series. Our matrix inversion result is directly extracted from an instance of Bailey’s very-well-poised 6ψ6 summation theorem, and involves two infinite matrices which are not lower-triangular. We combine our bilateral matrix inverse with known basic hypergeometric summation theorems to derive, via inverse relations, several new identities for bilateral basic hypergeometric series.

Abstract. Several new transformations for q-binomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the group-like properties of these positivity preserving transformations, as well as their connection with the Bailey lemma, many new summation and transformation formulas for basic hypergeometric series are found. The new q-binomial transformations are also applied to obtain multisum Rogers–Ramanujan identities, to find new representations for the Rogers–Szegö polynomials, and to make some progress on Bressoud’s generalized Borwein conjecture. For the original Borwein conjecture we formulate a refinement based on new triple sum representations of the Borwein polynomials. 1.

Dedicated to Alain Lascoux on the occasion of his 60th birthday Abstract. We present conjectures giving formulas for the Macdonald polynomials of type B, C, D which are indexed by a multiple of the first fundamental weight. The transition matrices between two different types are explicitly given.