By using the three-term recurrence equation satisfied by a family of orthogonal polynomials, the Christoffel-Darboux-type bilinear generating function and their asymptotic expressions, we obtain quadrature formulas for integral transforms generated by the classical orthogonal polynomials. These integral transforms, related to the so-called Poisson integrals, correspond to a modified Fourier Tra...

For any orthogonal polynomials system on real line we construct an appropriate oscillator algebra such that the polynomials make up the eigenfunctions system of the oscillator hamiltonian. The general scheme is divided into two types: a symmetric scheme and a non-symmetric scheme. The general approach is illustrated by the examples of the classical orthogonal polynomials: Hermite, Jacobi and La...

The aim of this paper is to define and study of the Gegenbauer matrix polynomials of two variables. An explicit representation, a three-term matrix recurrence relations, differential recurrence relations and hypergeometric matrix representation for the Gegenbauer matrix polynomials of two variables are given. The Gegenbauer matrix polynomials are solutions of the matrix differential equations a...

We show that the equilibrium positions of the Ruijsenaars-Schneider-van Diejen systems with the trigonometric potential are given by the zeros of the Askey-Wilson polynomials with five parameters. The corresponding single particle quantum version, which is a typical example of “discrete” quantum mechanical systems with a q-shift type kinetic term, is shape invariant and the eigenfunctions are t...

We show that the formalism of hybrid polynomials, interpolating between Hermite and Laguerre polynomials, is very useful in the study of Motzkin numbers and central trinomial coefficients. These sequences are identified as special values of hybrid polynomials, a fact which we use to derive their generalized forms and new identities satisfied by them.

Dmitrii Karp Institute of Applied Mathematics 7 Radio Street, Vladivostok, 690041, Russia [email protected] Abstra t Let fk be the k-th Fourier oe ient of a fun tion f in terms of the orthonormal Hermite, Laguerre or Ja obi polynomials. We give ne essary and su ient onditions on f for the inequality k |fk|θ < ∞ to hold with θ > 1. As a by-produ t new orthogonality relations for the Hermite and La...