Tom H. Koornwinder

Two-Variable Analogues of the Classical Orthogonal Polynomials

Publisher Summary This chapter discusses two-variable analogues of the classical orthogonal polynomials. Analogues in severable variables of the Jacobi polynomials are be highly nontrivial generalizations of the one-variable case. The chapter describes a number of distinct classes of orthogonal polynomials in two variables, for which many properties hold that are analogous to properties of Jacobi polynomials. The polynomials belonging to these orthogonal systems are eigenfunctions of two algebraically independent partial differential operators. In the Chebyshev cases, these polynomials can be interpreted as quotients of two eigenfunctions of the Laplacian on a two-dimensional torus or sphere, which satisfy symmetry relations with respect to certain reflections. The classical orthogonal polynomials in one variable are the Jacobi polynomials, the Laguerre, and the Hermite polynomials. The chapter also illustrates examples of two-variable analogues of the Jacobi polynomials.

Jacobi Functions and Analysis on Noncompact Semisimple Lie Groups

A Jacobi function \({\phi _\lambda }^{\left( {\alpha ,\beta } \right)}\left( {\alpha ,\beta ,\lambda \in C,\alpha \ne - 1, - 2,...} \right)\) is defined as the even C∞-function on ℝ which equals 1 at 0 and which satisfies the differential equation

On

\left( {{d^2}/d{t^2} + \left( {\left( {2\alpha + 1} \right)cotht + \left( {2\beta + 1} \right)tht} \right)d/dt + + {\lambda ^2} + {{\left( {\alpha + \beta + 1} \right)}^2}} \right){f_\lambda }^{\left( {\alpha ,\beta } \right)}\left( t \right) = 0.

-analogues of the Fourier and Hankel transforms

(1.1)

Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. III

which generalizes the Mehler-Fok transform, was studied by Titchmarsh [23, w 17], Olevskii [21], Braaksma and Meulenbeld [2], Flensted--Jensen [9], [11, w and w and Flensted--Jensen and Koornwinder [12]. Some papers by Ch6bli [3], [4], [5] deal with a larger class of integral transforms which includes the Jacobi transform. An even more general class was considered by Braaksma and De Shoo [24]. In the present paper short proofs will be given of a Paley--Wiener type theorem and the inversion formula for the Jacobi transform. The L2-theory, i.e. the Plancherel theorem, is then an easy consequence. These results were earlier obtained by Flensted--Jensen [9], [11, w and by Ch6bli [5]. However, to prove the Paley-Wiener theorem these two authors needed the L2-theory, which can be obtained as a corollary of the Weyl S t o n e T i t c h m a r s h K o d a i r a theorem about the spectral decomposition of a singular Sturm--Liouville operator (cf. for instance Dunford and Schwartz [6, Chap. 13, w The proofs presented here exploit the properties of Jacobi functions as hypergeometric functions and no general theorem needs to be invoked. Furthermore, it turns out that the Paley--Wiener theorem, which was proved by Flensted---Jensen [11, w for real c~, fi, ~ > 1, holds for all complex values of ~ and ft. The key formula in this paper is a generalized Mehler formula

The convolution structure for Jacobi function expansions

For H. Extons q-analogue of the Bessel function (going back to W. Hahn in a special case, but different from F. H. Jacksons q-Bessel functions) we derive Hansen-Lommel type orthogonality relations, which, by a symmetry, turn out to be equivalent to orthogonality relations which are q-analogues of the Hankel integral transform pair. These results are implicit, in the context of quantum groups, in a paper by Vaksman and Korogodskii. As a specialization we get q-cosines and q-sines which admit q-analogues of the Fourier-cosine and Fourier-sine transforms

Askey-Wilson polynomials as zonal spherical functions on the SU (2) quantum group

Abstract Let the region S={(x, y)∥μ(x+iy, x−iy)>0} be the interior of Steiners hypocycloid, where μ(z, z )=−z 2 z 2 +4z 3 +4 z 3 −18z z +27 . For each real α>− 5 6 an orthogonal system of polynomials p m,n α (z, z ), m, n⩾0 , can be defined on this region S such that p m,n α (z, z )−z m z n has degree less than m+n and ∫∫ S p α m,n (z, z ) q(z, z ) (μ(z, z )) α dx dy=0 for each polynomial q of degree less than m+n. If z=e i(s+ t √3 ) +e i(−s+ t √3 ) + e − 2it √3 then, in terms of s and t, the functions p m,n − 1 2 and μ 1 2 p m−1,n−1 1 2 are the regular eigenfunctions of the operator ∂ 2 ∂s 2 + ∂ 2 ∂t 2 which remain invariant or change sign, respectively, under the reflections in the edges of a certain equilateral triangle. Two explicit partial differential operators D1α and D2α in z and z of orders two and three, respectively, are obtained such that the polynomials pm,nα are eigenfunctions of D1α and D2α. The operators D1α and D2α commute and are algebraically independent, and they generate the algebra of all differential operators for which the polynomials pm,nα are eigenfunctions. If α=0, 1 2 , 3 2 or 7 2 then the operator D1α expressed in terms of s and t is the radial part of the Laplace-Beltrami operator on certain compact Riemannian symmetric spaces of rank two.

Orthogonal polynomials in connection with quantum groups

The product ϕλ(α,β)(t1)ϕλ(α,β)(t2) of two Jacobi functions is expressed as an integral in terms of ϕλ(α,β)(t3) with explicit non-negative kernel, when α≧β≧−1/2. The resulting convolution structure for Jacobi function expansions is studied. For special values of α and β the results are known from the theory of symmetric spaces.