Plamen Iliev

Tau Function Solutions to a q-Deformation of the KP Hierarchy

We construct tau-function solutions to the q-KP hierarchy as deformations of classical tau functions.

The bispectral property of a q-deformation of the Schur polynomials and the q-KdV hierarchy

We show that appropriate q-analogues of the Schur polynomials provide rational solutions of a q-deformation of the Nth KdV hierarchy. This allows us to construct explicit examples of bispectral commutative rings of q-difference operators.

Discrete orthogonal polynomials and difference equations of several variables

The goal of this work is to characterize all second order difference operators of several variables that have discrete orthogonal polynomials as eigenfunctions. Under some mild assumptions, we give a complete solution of the problem.

Askey-Wilson type functions with bound states

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.

A noncommutative version of the bispectral problem

We consider a matrix valued version of the bispectral problem involving a block tridiagonal doubly infinite matrix and a first-order differential operator with matrix coefficients. We give a set of necessary conditions that the coefficients need to satisfy and solve these equations under a variety of conditions. The situations discussed here should make it plain that while the corresponding problem in the scalar case is relatively trivial and devoid of any interest, the noncommutative version of the problem is much richer and subtle. The results here should be useful, for instance, in the study of a noncommutative version of the nonlinear Toda lattice.

Krall-Jacobi commutative algebras of partial differential operators

Abstract We construct a large family of commutative algebras of partial differential operators invariant under rotations. These algebras are isomorphic extensions of the algebras of ordinary differential operators introduced by Grunbaum and Yakimov corresponding to Darboux transformations at one end of the spectrum of the recurrence operator for the Jacobi polynomials. The construction is based on a new proof of their results which leads to a more detailed description of the one-dimensional theory. In particular, our approach establishes a conjecture by Haine concerning the explicit characterization of the Krall–Jacobi algebras of ordinary differential operators.

Krall–Laguerre commutative algebras of ordinary differential operators

In 1999, Grünbaum, Haine and Horozov defined a large family of commutative algebras of ordinary differential operators, which have orthogonal polynomials as eigenfunctions. These polynomials are mutually orthogonal with respect to a Laguerre-type weight distribution, thus providing solutions to Krall’s problem. In the present paper, we give a new proof of their result, which establishes a conjecture, concerning the explicit characterization of the dual commutative algebra of eigenvalues. In particular, for the Koornwinder’s generalization of Laguerre polynomials, our approach yields an explicit set of generators for the whole algebra of differential operators. We also illustrate how more general Sobolev-type orthogonal polynomials fit within this theory.

q-KP hierarchy, bispectrality and Calogero-Moser systems

We show that there is a one-to-one correspondence between the q-tau functions of a q-deformation of the KP hierarchy and the planes in Sate Grassmannian Gr. Using this correspondence, we define a subspace Gr(q)(ad) of Gr, which is a q-deformation of Wilsons adelic Grassmannian Gr(ad). From each plane W is an element of Gr(q)(ad) we construct a bispectral commutative algebra A(W)(q), Of q-difference operators, which extends to the case q not equal 1 all rank one solutions to the bispectral problem. The common eigenfunction Psi(x, z) for the operators from A(W)(q) is a q-wave (Baker-Akhiezer) function for a rational (in x) solution to the q-KP hierarchy. The poles of these solutions are governed by a certain q-deformation of the Calogero-Moser hierarchy

Heat Kernel Expansions on the Integers

In the case of the heat equation ut=uxx+Vu on the real line, there are some remarkable potentials V for which the asymptotic expansion of the fundamental solution becomes a finite sum and gives an exact formula.We show that a similar phenomenon holds when one replaces the real line by the integers. In this case the second derivative is replaced by the second difference operator L0. We show if L denotes the result of applying a finite number of Darboux transformations to L0 then the fundamental solution of ut=Lu is given by a finite sum of terms involving the Bessel function I of imaginary argument.

A Lie-theoretic interpretation of multivariate hypergeometric polynomials

In 1971, Griffiths used a generating function to define polynomials in d variables orthogonal with respect to the multinomial distribution. The polynomials possess a duality between the discrete variables and the degree indices. In 2004, Mizukawa and Tanaka related these polynomials to character algebras and the Gelfand hypergeometric series. Using this approach, they clarified the duality and obtained a new proof of the orthogonality. In the present paper, we interpret these polynomials within the context of the Lie algebra . Our approach yields yet another proof of the orthogonality. It also shows that the polynomials satisfy d independent recurrence relations each involving d2+d+1 terms. This, combined with the duality, establishes their bispectrality. We illustrate our results with several explicit examples.