Duality of q-polynomials, orthogonal on countable sets of points

2 Bogolyubov
de Matem´ aticas, UNAM, CP 62210 Cuernavaca, Morelos, M´ exico Institute for Theoretical Physics, 03143 Kiev, Ukraine
E-mail: natig@matcuer.unam.mx and aklimyk@bitp.kiev.ua
Key words. q -orthogonal polynomials, duality, Jacobi matrix, orthogonality relations AMS subject classification. 33D80, 33D45, 17B37
”We mathematicians are particularly fond of duality theorems; translating mathematical statements from one category to another often gives us new and unexpected insight”, M.Harris, ”Postmodern at an Early Age”, Notices of the American Mathematical Society, Vol.50, No.7, p.792, 2003. 1. INTRODUCTION It is well known that each family pn (x), n = 0, 1, 2, · · · , of orthogonal polynomials of one variable corresponds to the determinate or indeterminate moment problem. If a polynomial family corresponds to the determinate moment problem, then there exists only one positive orthogonality measure µ for these polynomials and they constitute a complete orthogonal set in the Hilbert space L2 (µ). If a family corresponds to the indeterminate moment problem, then there exists infinitely many orthogonality measures µ for these polynomials and these measures are divided into two parts: extremal measures and non-extremal measures. If a measure µ is extremal, then the corresponding set of polynomials constitute a complete orthogonal set in the Hilbert space L2 (µ). If a measure µ is not extremal, then the corresponding family of polynomials is not complete in the Hilbert space L2 (µ) (see [ST]). It is also well known that there exists a close relation of the theory of orthogonal polynomials with the theory of symmetric (self-adjoint) operators, representable by a Jacobi matrix. The point is that with each family of orthogonal polynomials one can associate a closed symmetric (or self-adjoint) operator A, representable by a Jacobi matrix. If the corresponding 1
moment problem is indeterminate, then the operator A is not self-adjoint and it has infinitely many self-adjoint extensions. If the operator A has a physical meaning, then these self-adjoint extensions are especially important. These extensions correspond to extremal orthogonality measures for the same set of polynomials and can be constructed by means of these measures (see, for example, [Ber], Chapter VII, and [Sim]). If the family of polynomials corresponds to the determinate moment problem, then the corresponding operator A is self-adjoint and its spectrum is determined by an orthogonality relation for the polynomials. Moreover, the spectral measure for the operator A is constructed by means of the orthogonality measure for the corresponding polynomials (see [Ber], Chapter VII). In section 2 we briefly review the relations between the theory of orthogonal polynomials, the theory of operators, representable by a Jacobi matrix, and the theory of moment problem. This information is basic for the exposition in further sections. In section 2 we also discuss how one can naturally extend the conventional notion of duality to sets of polynomials, orthogonal on countable sets of points. In order to find orthogonality measures for dual sets of polynomials, we use two symmetric (or self-adjoint) operators, representable (with respect to different bases) by Jacobi matrices. This approach is applied to several sets of dual q -orthogonal polynomials from the Askey scheme. Pairs of operators (A, I ), employed for studying some sets of q -orthogonal polynomials and their duals, belong to the discrete series representations of the quantum algebra Uq (su1,1 ) (see, for example, [AK1] and [AK6]). However, in order to facilitate ease of comprehending to a larger number of readers we have not exploited this deep algebraic fact; that is, we exhibit explicit forms of these operators without using the representation theory of the quantum algebra Uq (su1,1 ). These pairs of operators are, in fact, a generalization of Leonard pairs, introduced by P. Terwilliger [Ter1] (for the definition and references see section 3). When one considers dual sets of q -polynomials, orthogonal on countable sets of points, then one member of these sets corresponds to the determinate moment problem and another to the indeterminate moment problem. One of the two operators (A, I ) (that is, the operator A) for a given dual pair of sets of q -orthogonal polynomials corresponds to a three-term recurrence relation for the set of polynomials, which corresponds to the determinate moment problem. This operator is bounded and self-adjoint; moreover, it has the discrete spectrum. We diagonalize this self-adjoint bounded operator and find its spectrum with the aid of the second operator I , which corresponds to a q -difference equation for the same set of polynomials. An explicit form of all eigenfunctions for the operator A is found for each dual set of polynomials, considered by us. They are expressed in terms of q -polynomials, which belong to the set, associated with the determinate moment problem. Since the spectrum of A is simple, its eigenfunctions form an orthogonal basis in the Hilbert space. One can normalize this basis. This normalization is effected by means of the second operator I from the corresponding pair. As a result of this normalization, two orthonormal bases in the Hilbert space emerge: the canonical (or the initial) basis and the basis of eigenfunctions of the operator A. They are interrelated by a unitary matrix U , whose entries umn are explicitly expressed in terms of polynomials Pm (x), corresponding to the determinate moment problem. Since the matrix U is unitary (and in fact it is real in our case), there are two orthogonality relations for its elements, namely (1.1) umn umn′ = δnn′ . umn um′ n = δmm′ ,