Analysis of function spaces and special functions are closely related to the representation theory of Lie groups. We explain here the connection between the Laguerre functions, the Laguerre polynomials, and the Meixner-Pollacyck polynomials on the one side, and highest weight representations of Hermitian Lie groups on the other side. The representation theory is used to derive differential equa...

where α + β + 1 > β > 1 −m, σ + 1 > α + β > 0, m is a positive integer, and 0 < h < ∞, 0 ≤ b <∞, and h and b are finite constants. L n [(x + b)h] is a Laguerre polynomial, An are unknown coefficients, and f (x) and g(x) are prescribed functions. Srivastava [5, 6] has solved the following dual series equations: ∞ ∑ n=0 AnL (α) n (x) Γ(α+n+ 1) = f (x), 0 < x < a, (1.3) ∞ ∑ n=0 AnL (σ) n (x) Γ(α+n...

The explicit double sum for the associated Laguerre polynomials is derived combinatorially. The moments are described using certain statistics on permutations and permutation tableaux. Another derivation of the double sum is provided using only the moment generating function.

The support of the orthogonality measure of so-called little q-Laguerre polynomials {ln(.; a|q)}n=0, 0 < q < 1, 0 < a < q−1, is given by Sq = {1, q, q, . . .} ∪ {0}. Based on a method of MÃlotkowski and Szwarc we deduce a parameter set which admits nonnegative linearization. We additionally use this result to prove that little q-Laguerre polynomials constitute a so-called Faber basis in C(Sq).

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 ...

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...