André Ronveaux

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.

Recurrence relations for connection coefficients between two families of orthogonal polynomials

We describe a simple approach in order to build recursively the connection coefficients between two families of orthogonal polynomial solutions of second- and fourth-order differential equations.

On orthogonal polynomials with perturbed recurrence relations

Abstract Orthogonal polynomials may be fully characterized by the following recurrence relation: Pn(x) = (x − βn-1)Pn-1(x)-γn-1Pn-2(x), with P0(x)=1, P1(x) = x - β0 and γn ≠ 0. Here we study how the structure and the spectrum of these polynomials get modified by a local perturbation in the β and γ parameters of a co-recursive (βk → βk + μ), co-dilated (γk → λγk and co-modified (βk → βk + μ; γk → λγk) nature for an arbitrary (but fixed) kth element (1 ⩽ k). Specifically, Stieltjes functions, differential equations and distributions of zeros as well as representations of the new perturbed polynomials in terms of the old unperturbed ones are given. This type of problems is strongly related to the boundary value problems of finite-difference equations and to the quantum mechanical study of physical many-body systems (atoms, molecules, nuclei and solid state systems).

On a class of polynomials orthogonal with respect to a discrete Sobolev inner product

This paper analyzes polynomials orthogonal with respect to the Sobolev inner product @(Lg) = I f(x)g(x)e(x)dx+~-‘f”‘(c)g”‘(c) iF with I E IR+, c E [R, and p(x) is a weight function. We study this family of orthogonal polynomials, as linked to the polynomials orthogonal with respect to Q(X) and we find the recurrence relation verified by such a family. If the weight Q is semiclassical we obtain a second order differential equation for these polynomials. Finally, an illustrative example is shown.

On recurrence relations for Sobolev orthogonal polynomials

This paper discusses recurrence relations for sequences of polynomials which are orthogonal with respect to the Sobolev inner product defined on the set of polynomials

On a system of “classical” polynomials of simultaneous orthogonality

\mathcal{P}

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

by \[ (p,q)w = \sum_{k = 0}^N {\int_\mathbb{R} {p^{(k)} (x)\bar q^{(k)} (x)d\mu _k (x)\quad (p,q \in \mathcal{P})} } \] for some integer

Co-recursive orthogonal polynomials and fourth-order differential equations

N \geq 1

Inversion Problems in theq-Hahn Tableau

, where each

Zeros of Gegenbauer and Hermite polynomials and connection coefficients

\mu _k