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.

Let W (ψ; x, y) be the wideband ambiguity function. It is obtained in this note that y− α+2 2 W (ψ;x, y)(α > −1) is SO(2)-invariant if and only if the Fourier transform of ψ is a Laguerre function.

We present a direct proof of a known result that the Hardy operator Hf(x) = 1 x R x 0 f(t) dt in the space L = L2(0,∞) can be written as H = I − U , where U is a shift operator (Uen = en+1, n ∈ Z) for some orthonormal basis {en}. The basis {en} is constructed by using classical Laguerre polynomials. We also explain connections with the Euler differential equation of the first order y′− 1 x y = ...

We consider the compressive wave for the modified Korteweg–de Vries equation with background constants c > 0 for x→ −∞ and 0 for x→ +∞. We study the asymptotics of solutions in the transition zone 4ct− εt < x < 4ct − βt ln t for ε > 0, σ ∈ (0, 1), β > 0. In this region we have a bulk of nonvanishing oscillations, the number of which grows as εt ln t . Also we show how to obtain Khruslov–Kotlyar...

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.