In this paper we give new proofs of some elementary properties of the Hermite and Laguerre orthogonal polynomials. We establish Rodriguestype formulae and other properties of these special functions, using suitable operators defined on the Lie algebra of endomorphisms to the vector space of infinitely many differentiable functions.

We study the combinatorics of a continued fraction formula due to Wall. We also derive the orthogonality of little q-Jacobi polynomials from this formula, as Wall did for little q-Laguerre polynomials.

1. Fourier transform on Rn 1 2. Fourier analysis on the Heisenberg group 2 2.1. Representations of the Heisenberg group 2 2.2. Group Fourier transform 3 2.3. Convolution and twisted convolution 5 3. Hermite and Laguerre functions 6 3.1. Hermite polynomials 6 3.2. Laguerre polynomials 9 3.3. Special Hermite functions 9 4. Group Fourier transform of radial functions on the Heisenberg group 12 Ref...

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 ...

In some recent investigations involving differential operators for generalized Laguerre polynomials, Herman Bavinck (1996) encountered and proved a certain summation formula for the classical Laguerre polynomials. The main object of this sequel to Bavinck’s work is to prove a generalization of this summation formula for a class of hypergeometric polynomials. The demonstration, which is presente...

Multivariate versions of classical orthogonal polynomials such as Jacobi, Hahn, Laguerre, Meixner are reviewed and their connection explored by adopting a probabilistic approach. Hahn and Meixner polynomials are interpreted as posterior mixtures of Jacobi and Laguerre polynomials, respectively. By using known properties of Gamma point processes and related transformations, an infinite-dimension...

A simple structure of the multiple Laguerre polynomial expansions is used to study the Cesàro summability above the critical index for the convolution type Laguerre expansions. The multiple Laguerre polynomial expansion of an `1-radial function f0(|x|) is shown to be an `1-radial function that coincides with the Laguerre polynomial expansion of f0, which allows us to settle the problem of summa...

where S is an n × n positive definite Hermitian matrix, dS is equivalent to the dH in (1) restricted in the subset of positive definite Hermitian matrices, and E+ is a subset of R+. The integral H(E) is in some sense the limiting case of L(E+), since H(E) can be evaluated by Hermitian polynomials, while L(E+) by Laguerre polynomials in the same way [8]. It is a classical result that Hermitian p...

In this paper, an approximate formula of the fractional derivatives (Caputo sense) is derived. The proposed formula is based on the generalized Laguerre polynomials. Special attention is given to study the convergence analysis of the presented formula. The spectral Laguerre collocation method is presented for solving a class of fractional optimal control problems (FOCPs). The properties of Lagu...