The bivariate big q-Jacobi polynomials are defined by [3] Pn,k(x, y; a, b, c, d; q) := Pn−k(y; a, bcq , dq; q) y(dq/y; q)k Pk (x/y; c, b, d/y; q) (n ≥ 0; k = 0, 1, . . . , n), where q ∈ (0, 1), 0 < aq, bq, cq < 1, d < 0, and Pm(t;α, β, γ; q) are univariate big q-Jacobi polynomials, Pm(t;α, β, γ; q) := 3φ2 ( q−m, αβq, t αq, γq ∣∣∣∣ q; q) (m ≥ 0) (see, e.g., [1, Section 7.3]). We give structure r...

Some families of orthogonal matrix polynomials satisfying second order differential equations with coefficients independent of n have recently been introduced (see [DG1]). An important difference with the scalar classical families of Jacobi, Laguerre and Hermite, is that these matrix families do not satisfy scalar type Rodrigues’ formulas of the type (ΦnW )W, where Φ is a matrix polynomial of d...

The main purpose of this article is to solve the connection problems between (p, q)−Fibonacci polynomials and the two polynomials, namely Chebyshev polynomials of third and fourth kinds which are considered as two nonsymmetric polynomials of the Jacobi polynomials. Moreover, the inversion connection formulae for the latter formulae are given. We show that all the connection coefficients are exp...

We use the "tridiagonal representation approach" to solve time-independent Schr\"odinger equation for bound states of generalized versions trigonometric and hyperbolic P\"oschl-Teller potentials. These new solvable potentials do not belong conventional class exactly problems. The solutions are finite series square integrable functions written in terms Jacobi polynomial.

We will outline in this paper a general procedure for constructing whole systems of orthogonal Laurent polynomials on the real line from systems of orthogonal polynomials. To further explain our intentions, we proceed with some basic definitions and results, all of which appear, or are modifications of those that appear, in the literature. In particular, [1,2,3,4,5,6,7,8,9,10,11,12] were consul...

We introduce here a generalization of the modified Bernstein polynomials for Jacobi weights using the q-Bernstein basis proposed by G.M. Phillips to generalize classical Bernstein Polynomials. The function is evaluated at points which are in geometric progression in ]0, 1[. Numerous properties of the modified Bernstein Polynomials are extended to their q-analogues: simultaneous approximation, p...

We study Cesàro (C, δ) means for two-variable Jacobi polynomials on the parabolic biangle B = {(x1, x2) ∈ R2 : 0 ≤ x1 ≤ x2 ≤ 1}. Using the product formula derived by Koornwinder & Schwartz for this polynomial system, the Cesàro operator can be interpreted as a convolution operator. We then show that the Cesàro (C, δ) means of the orthogonal expansion on the biangle are uniformly bounded if δ > ...

We consider the perturbation of the classical Bessel moment functional by the addition of the linear functional M 0 (x) + M 1 0 (x), where M 0 and M 1 2 IR. We give necessary and suucient conditions in order for this functional to be a quasi-deenite functional. In such a situation we analyze the corresponding sequence of monic orthogonal polynomials B ;M0;M1 n (x). In particular, a hypergeometr...