Certain dual series equations involving Laguerre polynomials

Dual series equations involving generalized Laguerre polynomials

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...

Differential equations for deformed Laguerre polynomials

The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite size may be expressed in terms of a solution of the fifth Painlevé transcendent. The generating function of a certain discontinuous linear statistic of the Laguerre unitary ensemble can similarly be expressed in terms of a solution of the fifth Painlevé equation. The methodology used to derive ...

On the Irreducibility of a Certain Class of Laguerre Polynomials

Regularization of Certain Divergent Series of Polynomials

We investigate the generalized convergence and sums of series of the form P n≥0 anT P (x), where P ∈ R[x], an ∈ R, ∀n ≥ 0, and T : R[x] → R[x] is a linear operator that commutes with the differentiation d dx : R[x]→ R[x]. CONTENTS 1. The main result 1 2. Some applications 3 References 9 1. THE MAIN RESULT We consider series of the form ∑ n≥0 anT P (x), (†) where P ∈ R[x], and T : R[x]→ R[x] is ...

Differential equations for discrete Laguerre-Sobolev orthogonal polynomials

The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Laguerre–Sobolev bilinear form with mass point at zero. In particular we construct the orthogonal polynomials using certain Casorati determinants. Using this construction, we prove that they are eigenfunctions of a differential operator (which will be explicitly constructed). Moreover,...