Michael J. Schlosser

Elliptic determinant evaluations and the Macdonald identities for affine root systems

We obtain several determinant evaluations, related to affine root systems, which provide elliptic extensions of Weyl denominator formulas. Some of these are new, also in the polynomial special case, while others yield new proofs of the Macdonald identities for the seven infinite families of irreducible reduced affine root systems.

Inversion of the Pieri formula for Macdonald polynomials

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 Ar and Dr Basic 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.

Summation theorems for multidimensional basic hypergeometric series by determinant evaluations

Abstract We derive summation formulas for a specific kind of multidimensional basic hypergeometric series associated to root systems of classical type. We proceed by combining the classical (one-dimensional) summation formulas with certain determinant evaluations. Our theorems include A r extensions of Ramanujans bilateral 1 ψ 1 sum, C r extensions of Baileys very-well-poised 6 ψ 6 summation, and a C r extension of Jacksons very-well-poised 8 φ 7 summation formula. We also derive multidimensional extensions, associated to the classical root systems of type A r , B r , C r , and D r , respectively, of Chus bilateral transformation formula for basic hypergeometric series of Gasper–Karlsson–Minton type. Limiting cases of our various series identities include multidimensional generalizations of many of the most important summation theorems of the classical theory of basic hypergeometric series.

Summations and transformations for multiple basic and elliptic hypergeometric series by determinant evaluations

Abstract Using multiple q -integrals and a determinant evaluation, we establish a multivariable extension of Baileys nonterminating 1009 transformation. From this result, we deduce new multivariable terminating 10 φ 9 transformations, 8 φ 7 summations and other identities. We also use similar methods to derive new multivariable l 1 ψ 1 summations. Some of our results are extended to the case of elliptic hypergeometric series.

Abel-Rothe Type Generalizations of Jacobi's Triple Product Identity

Using a simple classical method we derive bilateral series identities from terminating ones. In particular, we show how to deduce Ramanujans 1ψ1 summation from the q-Pfaff-Saalschutz summation. Further, we apply the same method to our previous q-Abel-Rothe summation to obtain, for the first time, Abel-Rothe type generalizations of Jacobis triple product identity. We also give some results for multiple series.

Some New Applications of Matrix Inversions in Ar

We apply multidimensional matrix inversions to multiple basic hypergeometric summation theorems to derive several multiple (q-)series identities which themselves do not belong to the hierarchy of (basic) hypergeometric series. Among these are A terminating and nonterminating q-Abel and q-Rothe summations. Furthermore, we derive some identities of another type which appear to be new already in the one-dimensional case.

Another proof of Bailey’s 6ψ6 summation

Summary.Adapting a method used by Cauchy, Bailey, Slater, and more recently, the second author, we give a new proof of Bailey’s celebrated 6ψ6 summation formula.

Multiple Hypergeometric Series: Appell Series and Beyond

This survey article provides a small collection of basic material on multiple hypergeometric series of Appell-type and of more general series of related type.

On Warnaar's elliptic matrix inversion and Karlsson-Minton-type elliptic hypergeometric series

Using Krattenthalers operator method, we give a new proof of Warnaars recent elliptic extension of Krattenthalers matrix inversion. Further, using a theta function identity closely related to Warnaars inversion, we derive summation and transformation formulas for elliptic hypergeometric series of Karlsson-Minton type. A special case yields a particular summation that was used by Warnaar to derive quadratic, cubic and quartic transformations for elliptic hypergeometric series. Starting from another theta function identity, we derive yet different summation and transformation formulas for elliptic hypergeometric series of Karlsson-Minton type. These latter identities seem quite unusual and appear to be new already in the trigonometric (i.e., p=0) case.