Askey-Wilson type functions with bound states

Abstract

Abstract. The two linearly independent solutions of the three-term recurrence relation of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22], are slightly modified so as to make it transparent that these functions satisfy a beautiful symmetry property. It essentially means that the geometric and the spectral parameters are interchangeable in these functions. We call the resulting functions the Askey-Wilson functions. Then, we show that by adding bound states (with arbitrary weights) at specific points outside of the continuous spectrum of some instances of the Askey-Wilson difference operator, we can generate functions that satisfy a doubly infinite three-term recursion relation and are also eigenfunctions of q-difference operators of arbitrary orders. Our result provides a discrete analogue of the solutions of the purely differential version of the bispectral problem that were discovered in the pioneering work [8] of Duistermaat and Grünbaum.

Bispectral commuting difference operators for multivariable Askey-Wilson polynomials

“…For instance, in the one-dimensional case, for specific values of the free parameters, the Askey-Wilson recurrence operator L n commutes with a difference operator of odd order. Jointly with Haine [12] we explored this fact to connect the above theory to algebraic curves, having specific singularities, or equivalently, to specific soliton solutions of the Toda lattice hierarchy, using the correspondence established by van Moerbeke-Mumford [18,19] and Krichever [16]. In particular, this led to a different explanation of the bispectral property using algebro-geometric considerations [11].…”

“…In particular, this led to a different explanation of the bispectral property using algebro-geometric considerations [11]. Moreover, this approach suggested using techniques from integrable systems (such as the Darboux transformation) to construct extensions of the Askey-Wilson polynomials which satisfy higher-order q-difference equations [12]. In the multivariable case, algebro-geometric methods were used by Chalykh [2], within the context of symmetric functions, to give more elementary proofs of several of Macdonald's conjectures.…”

The spectral matrices of Toda solitons and the fundamental solution of some discrete heat equations

“…(with a time-dependent discrete Schrödinger operator), where D q,t denotes the q-derivative with respect to t. Remarkably, the condition (1.4) precisely defines the so-called Askey-Wilson type solitons [20,23], that provide the rank 1 solutions of a difference -q-difference version of the bispectral problem, that was initiated in [17] in the aim of generalizing the celebrated Askey-Wilson polynomials [2], see also [32].…”

“…In [17,20,23], the following difference -q-difference bispectral problem was studied. To find all doubly infinite (or semi-infinite) Jacobi matrices for which some family of eigenfunctions is also a family of eigenfunctions of a q-difference operator of an arbitrary order, in the spectral variable z.…”

“…The Askey-Wilson orthogonal polynomials [2] provide instances of semi-infinite Jacobi matrices which solve this problem. Rank 1 solutions of the problem, that we called "Askey-Wilson type solitons", were TOME 55 (2005), FASCICULE 6 described in [20] (see also [23], which further contains a description of rank 2 solutions). It is quite remarkable that the conditions (5.12) which lead to a "finite sum formula" for the fundamental solution of (5.2), precisely define the "Askey-Wilson type solitons".…”