We consider a varying discrete Sobolev inner product involving the Laguerre weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and of their zeros. We are interested in Mehler–Heine type formulas because they describe the asymptotic differences between these Sobolev orthogonal polynomials and the classical Laguerre polynomials. Moreover, they give u...

This result was obtained also by N. Bari [1] using very different methods. The factor n in (1.1) cannot be replaced by any function tending to infinity more slowly. Namely, for each n exist polynomials P (t) of degree n such that the left side of (1.1) is ≤ Bn, where B is a constant of the same nature as A. Under only a little restriction on the zeros of P (z), M. A. Malik [4] found the followi...

Three specializations of a set of orthogonal polynomials with “8 different q’s” are given. The polynomials are identified as q-analogues of Laguerre polynomials, and the combinatorial interpretation of the moments give infinitely many new Mahonian statistics on permutations.

The Laguerre derivative and its iterations have been used to define new sets of special functions, showing the possibility generating a kind parallel universe for mathematical entities this kind. In paper, we introduce Laguerre-type Appell polynomials, in particular, Bernoulli Euler case, examine set hypergeometric Laguerre–Bernoulli polynomials. We show their main properties indicate possible ...

Although rational orthogonal bases can be used to model any L2[0,∞[ system, they fail to capture the aperiodic multi-mode behaviour of fractional systems in a limited number of terms. The classical definition of orthogonal Laguerre, Kautz, and GOB functions has been extended for the use of fractional derivatives. An appropriate diagram is thus proposed for simulation. Copyright c ©2005 IFAC

The aim of the present paper is to discuss in some detail the Laguerre geometry (cf. [1], [6]) which arises from the 3-dimensional real algebra L := R(ε), where ε = 0. This algebra generalizes the algebra of real dual numbers D = R(ε), where ε = 0. The Laguerre geometry over D is the geometry on the so-called Blaschke cylinder (Figure 1); the non-degenerate conics on this cylinder are called ch...

We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of weighted sums of Laguerre polynomials with parameter α = −1. We describe how such a series can be computed by finding an appropriate ordinary generating function and applying a certain transformation. We use this technique to find the generating function for the numb...

We decribe various aspects of the Al-Salam-Chihara q-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients. It is remarkable that the corresponding moment sequence appears also in the recent work of Postnikov and Williams on enumeration of totally positive Gr...

Using the well-known fact that the Fourier transform is unitary, we obtain a class of orthogonal polynomials on the unit circle from the Fourier transform of the Laguerre polynomials (with suitable weights attached). Some related extremal problems which arise naturally in this setting are investigated.