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42
SUMMATION AND TRANSFORMATION FORMULAS FOR ELLIPTIC HYPERGEOMETRIC SERIES
, 2000
"... Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating, balanced, verywellpoised, elliptic hypergeometric series. ..."
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Cited by 59 (6 self)
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Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating, balanced, verywellpoised, elliptic hypergeometric series.
Inversion of the Pieri formula for Macdonald polynomials
 Adv. Math
"... 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. ..."
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Cited by 35 (13 self)
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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.
Multidimensional Matrix Inversions and A_r and D_r Basic Hypergeometric Series
 The Ramanujan J
, 1997
"... We compute the inverse of a specific infinite rdimensional matrix, thus unifying multidimensional matrix inversions recently found by Milne, Lilly, and Bhatnagar. Our inversion is an rdimensional extension of a matrix inversion previously found by Krattenthaler. We also compute the inverse of anot ..."
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Cited by 28 (13 self)
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We compute the inverse of a specific infinite rdimensional matrix, thus unifying multidimensional matrix inversions recently found by Milne, Lilly, and Bhatnagar. Our inversion is an rdimensional extension of a matrix inversion previously found by Krattenthaler. We also compute the inverse of another infinite rdimensional matrix. As applications of our matrix inversions, we derive new summation formulas for multidimensional basic hypergeometric series.
A New Multidimensional Matrix Inverse With Applications To Multiple qSeries
 DISC. MATH
, 1999
"... We compute the inverse of a specific infinite rdimensional matrix, extending a matrix inverse of Krattenthaler. Our inversion is different from the rdimensional matrix inversion recently found by Schlosser but generalizes a multidimensional matrix inversion previously found by Chu. As applications ..."
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Cited by 17 (9 self)
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We compute the inverse of a specific infinite rdimensional matrix, extending a matrix inverse of Krattenthaler. Our inversion is different from the rdimensional 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 qseries identities. Among these are qanalogues of Carlitz' multidimensional Abeltype expansion formulas. Furthermore, we derive a qanalogue of MacMahon's Master Theorem.
Factors of alternating sums of products of binomial and qbinomial coefficients, preprint, arXiv: math.NT/0511635
, 2005
"... Abstract. In this paper we study the factors of some alternating sums of products of binomial and qbinomial 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 ..."
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Cited by 16 (5 self)
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Abstract. In this paper we study the factors of some alternating sums of products of binomial and qbinomial 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
Positivity preserving transformations for qbinomial coefficients
 Trans. Amer. Math. Soc
, 2005
"... Abstract. Several new transformations for qbinomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the grouplike properties of these positivity preserving transformations, as well as their connection with the Bailey ..."
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Cited by 14 (4 self)
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Abstract. Several new transformations for qbinomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the grouplike 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 qbinomial 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.
A New Multidimensional Matrix Inversion in ...
"... We invert a specic innite rdimensional matrix, thus giving an extension of our previous matrix inversion result. As applications, we derive new summation formulas for series in Ar . 1. ..."
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Cited by 13 (8 self)
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We invert a specic innite rdimensional matrix, thus giving an extension of our previous matrix inversion result. As applications, we derive new summation formulas for series in Ar . 1.
An analytic formula for Macdonald polynomials
, 2003
"... 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 o ..."
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Cited by 11 (6 self)
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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 euxmêmes un cas limite des polynômes de Macdonald.
Simultaneous generation of Koecher and AlmkvistGrainville’s Apérylike formulae, Experiment
 Math
"... Abstract. We prove a very general identity, conjectured by Henri Cohen, involving the generating function of the familly (ζ(2r + 4s + 3))r,s≥0: it unifies two identities, proved by Koecher in 1980 and Almkvist & Granville in 1999, for the generating functions of (ζ(2r+3))r≥0 and (ζ(4s+3))s≥0 res ..."
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Cited by 10 (0 self)
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Abstract. We prove a very general identity, conjectured by Henri Cohen, involving the generating function of the familly (ζ(2r + 4s + 3))r,s≥0: it unifies two identities, proved by Koecher in 1980 and Almkvist & Granville in 1999, for the generating functions of (ζ(2r+3))r≥0 and (ζ(4s+3))s≥0 respectively. As a consequence, we obtain that, for any integer j ≥ 0, there exist at least [j/2] + 1 Apérylike formulae for ζ(2j + 3). 1.