I. Area

Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: discrete case

Abstract We present a simple approach in order to compute recursively the connection coefficients between two families of classical (discrete) orthogonal polynomials (Charlier, Meixner, Kravchuk, Hahn), i.e., the coefficients Cm(n) in the expression P n (X)= ∑ n m=0 C m (n)Q m (x) , where Pn(x) and Qm(x) belong to the aforementioned class of polynomials. This is SCV2 done by adapting a general and systematic algorithm, recently developed by the authors, to the discrete classical situation. Moreover, extensions of this method allow to give new addition formulae and to estimate Cm(n)-asymptotics in limit relations between some families.

Solving connection and linearization problems within the Askey scheme and its q -analogue via inversion formulas

For the polynomial families {Pn(x)}n belonging to the Askey scheme or to its q-analogue, the hypergeometric represen- tation provides a natural expansion of the form Pn(x )= � n=0 Dm(n)� m(x), where the expanding basism(x) is, in general, a product of Pochhammer symbols or q-shifted factorials. In this paper we solve the corresponding inversion problem, i.e. we compute the coe6cients Im(n) in the expansionn(x )= � n m=0 Im(n)Pm(x), which are then used as a tool for solving any connection and linearization problem within the Askey scheme and its q-analogue. Extensions of this approach for polynomials outside these two schemes are also given. c

Inversion Problems in theq-Hahn Tableau

If { Pn(x;q)}nis a family of polynomials belonging to the q -Hahn tableau then each polynomial of this family can be written as Pn(x;q) =?m=0nDm(n)?m(x) where ?m(x) stands for (x;q)mor xm. In this paper we solve the corresponding inversion problem, i.e. we find the explicit expression for the coefficients Im(n) in the expansion ?n(x) =?m=0nIm(n)Pm(x;q).

Zeros of Gegenbauer and Hermite polynomials and connection coefficients

In this paper, sharp upper limit for the zeros of the ultraspherical polynomials are obtained via a result of Obrechkoff and certain explicit connection coefficients for these polynomials. As a consequence, sharp bounds for the zeros of the Hermite polynomials are obtained.

Results for some inversion problems for classical continuous and discrete orthogonal polynomials

Explicit expressions for the coefficients in the expansion of classical discrete orthogonal polynomials (Charlier, Meixner, Krawtchouck, Hahn, Hahn - Eberlein) into the falling factorial basis are given. The corresponding inversion problems are solved explicitly. This is done by using a general algorithm, recently developed by the authors, which is also applied to this kind of inversion problem but relating the basis and the classical (continuous) orthogonal polynomials of Jacobi, Laguerre, Hermite and Bessel.

On the limit relations between classical continuous and discrete orthogonal polynomials

Abstract Limit relations between classical continuous (Jacobi, Laguerre, Hermite) and discrete (Charlier, Meixner, Kravchuk, Hahn) orthogonal polynomials are well known and can be described by relations of type limλ → ∞ Pn(x;λ) = Qn(x). Deeper information on these limiting processes can be obtained from the expansion P n (x;λ) = ∑ k=0 ∞ R k (x;n) λ k . In this paper a method for the recursive computation of coefficients Rk(x;n) is designed being the main tool the consideration of a closely related connection problem which can be solved, also recurrently, by using an algorithm recently developed by the authors.

Classical orthogonal polynomials: dependence on parameters

Most of the classical orthogonal polynomials (continuous, discrete and their q-analogues) can be considered as functions of several parameters ci. A systematic study of the variation, innitesimal and nite, of these polynomials Pn(x;ci) with respect to the parameters ci is proposed. A method to get recurrence relations for connection coecients linking (@ r =@c r i)Pn(x;ci )t oPn(x;ci) is given and, in some situations, explicit expressions are obtained. This allows us to compute new integrals or sums of classical orthogonal polynomials using the digamma function. A basic theorem on the zeros of (@=@ci)Pn(x;ci) is also proved. c 2000 Elsevier Science B.V. All rights reserved. MSC: 33C25; 42C05; 33B15

Classification of all δ-Coherent Pairs

In a recent work we have proved that if (u 0 u 1) is a δ-coherent pair of linear functionals, then at least one of them must be a classical discrete linear functional under certain conditions. In this paper we present a similar result for Dw -coherent pairs and the description of all δ-coherent pairs. Furthermore, by using a limit process we recover the classification of all coherent pairs of positive definite linear functionals.

Hypergeometric-type differential equations: second kind solutions and related integrals

A Rodrigues-type representation for the second kind solutions of a second-order differential equation of hypergeometric type is given. This representation contains some integrals related with relevant special functions. For these integrals, a general recurrence relation, which only involves the coefficients of the differential equation, is also presented. Finally, an extension of the Rodrigues type representation for the second solution of a second-order difference equation of hypergeometric type is indicated.

Orthogonal polynomials of two discrete variables on the simplex

In this article, a method for constructing families of orthogonal polynomials of two discrete variables on the triangle is presented. Some examples, related with Hahn, Meixner, Kravchuk and Charlier polynomials, are discussed in detail, giving recurrence relation in vector–matrix form for the orthonormal families, partial difference equation, limit relations between them, as well as forward and backward structure relations. Finally, as a limit case of Hahn polynomials of two variables we recover the Jacobi polynomials of two variables on the triangle.