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Through a finite-dimensional reduction of the difference Lame equation, an elliptic analog of the Kac–Sylvester tridiagonal matrix is found. We solve the corresponding finite discrete Lame equation by constructing an orthogonal basis of eigenvectors for this novel elliptic Kac–Sylvester matrix.

Upon solving a finite discrete reduction of the difference Heun equation, we arrive at an elliptic generalization of the Racah polynomials. We exhibit the three-term recurrence relation and the orthogonality relations for these elliptic Racah polynomials. The well-known $q$-Racah polynomials of Askey and Wilson are recovered as a trigonometric limit., Comment: 18 pages

We construct the orthogonal eigenbasis for a discrete elliptic Ruijsenaars type quantum particle Hamiltonian with hyperoctahedral symmetry. In the trigonometric limit the eigenfunctions in question recover a previously studied $q$-Racah type reduction of the Koornwinder-Macdonald polynomials. When the inter-particle interaction degenerates to that of impenetrable bosons, the orthogonal eigenbasis simplifies in terms of generalized Schur polynomials on the spectrum associated with recently found elliptic Racah polynomials., 23 pages

By means of a truncation condition on the parameters, the elliptic Ruijsenaars difference operators are restricted onto a finite lattice of points encoded by bounded partitions. A corresponding orthogonal basis of joint eigenfunctions is constructed in terms of polynomials on the joint spectrum. In the trigonometric limit, this recovers the diagonalization of the truncated Macdonald difference operators by a finite-dimensional basis of Macdonald polynomials., 14 pages, v2: minor corrections and reference added

We show that the gradient flows associated with a recently found family of Morse functions converge exponentially to the roots of the symmetric continuous Hahn polynomials. By symmetry reduction the rate of the exponential convergence can be improved, which is clarified by comparing with corresponding gradient flows for the roots of the Wilson polynomials.

Starting from a recently found branching rule for the six-parameter family of symmetric Macdonald-Koornwinder polynomials, we arrive by degeneration at corresponding branching formulas for symmetric hypergeometric orthogonal polynomials of Wilson, continuous Hahn, Jacobi, Laguerre, and Hermite type.

We present a second explicit difference equation and—by duality symmetry—a second explicit Pieri formula for the Macdonald polynomials associated with the exceptional root system of type G2, complementing the celebrated difference equation and Pieri formula due to Macdonald. As by-product, we find a complete set of integrals for the corresponding trigonometric Ruijsenaars–Schneider type particle system.

We employ a discrete integral-reflection representation of the double affine Hecke algebra of type $C^\vee C$ at the critical level q=1, to endow the open finite $q$-boson system with integrable boundary interactions at the lattice ends. It is shown that the Bethe Ansatz entails a complete basis of eigenfunctions for the commuting quantum integrals in terms of Macdonald's three-parameter hyperoctahedral Hall-Littlewood polynomials., Comment: 31 pages, LaTeX

We place the hyperbolic quantum Ruijsenaars-Schneider system with an exponential Morse term on a lattice and diagonalize the resulting $n$-particle model by means of multivariate continuous dual $q$-Hahn polynomials that arise as a parameter reduction of the Macdonald-Koornwinder polynomials. This allows to compute the $n$-particle scattering operator, to identify the bispectral dual system, and to confirm the quantum integrability in a Hilbert space set-up., 13 pages, LaTeX

A formula for the large-degree asymptotics of Koornwinder's multivariate Askey–Wilson polynomials is presented. In the special case of a single variable, this asymptotic formula agrees with the known leading asymptotics of the Askey–Wilson polynomials determined by Ismail and Wilson.

We provide explicit formulas for the quantum integrals of a semi-infinite $q$-boson system with boundary interactions. These operators and their commutativity are deduced from the Pieri formulas for a $q\to 0$ Hall-Littlewood type degeneration of the Macdonald-Koornwinder polynomials.

Upon introducing a one-parameter quadratic deformation of the q-boson algebra and a diagonal perturbation at the end point, we arrive at a semi-infinite q-boson system with a two-parameter boundary interaction. The eigenfunctions are shown to be given by Macdonald's hyperoctahedral Hall-Littlewood functions of type BC. It follows that the n-particle spectrum is bounded and absolutely continuous and that the corresponding scattering matrix factorizes as a product of two-particle bulk and one-particle boundary scattering matrices., Comment: 9 pages

Science fiction and Macdonald's polynomials by F. Bergeron and A. M. Garsia On the expansion of elliptic functions and applications by R. Chouikha Generalized hypergeometric functions-Classification of identities and explicit rational approximations by D. V. Chudnovsky and G. V. Chudnovsky Tensor products of $q$-superalgebras and $q$-series identities. I by W. S. Chung, E. G. Kalnins, and W. Miller, Jr. $q$-Racah polynomials for $BC$ type root systems by J. F. van Diejen and J. V. Stokman Intertwining operators of type $B_N$ by C. F. Dunkl Symmetries and continuous $q$-orthogonal polynomials by R. Floreanini, J. LeTourneux, and L. Vinet Addition theorems for spherical polynomials on a family of quantum spheres by P. G. A. Floris On a $q$-analogue of the string equation and a generalization of the classical orthogonal polynomials by F. A. Grunbaum and L. Haine The $q$-Bessel function on a $q$-quadratic grid by M. E. H. Ismail, D. R. Masson, and S. K. Suslov Three statistics on lattice paths by D. Kim and D. Stanton Quantum Grothendieck polynomials by A. N. Kirillov $q$-difference raising operators for Macdonald polynomials and the integrality of transition coefficients by A. N. Kirillov and M. Noumi Great powers of $q$-calculus by B. A. Kupershmidt $q$-special functions: Differential-difference equations, roots of unity, and all that by V. Spiridonov On algebras of creation and annihilation operators by A. Strasburger.

Limiting cases are studied of the Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials. We recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable Wilson, continuous Hahn and Jacobi type polynomials, respectively. For each class of polynomials we provide systems of difference (or differential) equations, recurrence relations, and expressions for the norms of the polynomials in terms of the norm of the constant polynomial., 42 pages, AMSLaTeX 1.1 with amssymb

We prove certain duality properties and present recurrence relations for a four-parameter family of self-dual Koornwinder-Macdonald polynomials. The recurrence relations are used to verify Macdonald's normalization conjectures for these polynomials., 21 pages, AMSLaTeX 1.1 with amssymb