Askey-Wilson Type Functions, With Bound States

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.

42 pages, Section 3 moved to the end, minor corrections

References in corpus (5)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Differential equations for generalized Jacobi polynomials
• General methods for constructing bispectral operators
• Bispectral algebras of commuting ordinary differential operators
• Bispectral Darboux Transformations: The Generalized Airy Case

Cited by in corpus (10)

• Bispectrality of multivariable Racah-Wilson polynomials
• Bispectral commuting difference operators for multivariable Askey-Wilson polynomials
• Bispectral extensions of the Askey-Wilson polynomials
• Using D-operators to construct orthogonal polynomials satisfying higher order q-difference equations
• Constructing bispectral orthogonal polynomials from the classical discrete families of Charlier, Meixner and Krawtchouk
• Some Remarks on Very-Well-Poised 8phi7 Series
• On the Krall-type Askey-Wilson Polynomials
• q-special functions, basic hypergeometric series and operators
• Automorphisms of the DAHA of type $\check{C_1}{C}_1$ and non-symmetric Askey-Wilson functions
• Constructing bispectral dual Hahn polynomials