An Introductory Overview of Bessel Polynomials, the Generalized Bessel Polynomials and the q-Bessel Polynomials

The Generalized Bessel Matrix Polynomials

Abstract.In this paper, the generalized Bessel matrix polynomials are introduced, starting from the hypergeometric matrix function. Integral form, Rodrigues’s formula and generating matrix function are then developed for the generalized Bessel matrix polynomials. These polynomials appear as finite series solutions of second-order matrix differential equations and orthogonality property for the ...

Linearization coefficients of Bessel polynomials

We prove positivity results about linearization and connection coefficients for Bessel polynomials. The proof is based on a recursion formula and explicit formulas for the coefficients in special cases. The result implies that the distribution of a convex combination of independent Studentt random variables with arbitrary odd degrees of freedom has a density which is a convex combination of cer...

Asymptotics of basic Bessel functions and q-Laguerre polynomials

We establish a large n complete asymptotic expansion for q-Laguerre polynomials and a complete asymptotic expansion for a q-Bessel function of large argument. These expansions are needed in our study of an exactly solvable random transfer matrix model for disordered electronic systems. We also give a new derivation of an asymptotic formula due to Littlewood (1907).

Bessel Polynomials and the Partial Sums of the Exponential Series

Let e k (x) denote the k-th partial sum of the Maclaurin series for the exponential function. Define the (n + 1) × (n + 1) Hankel determinant by setting Hn(x) = det[e i+j (x)] 0≤i,j≤n. We give a closed form evaluation of this determinant in terms of the Bessel polynomials using the method of recently introduced γ-operators.

The Irreducibility of All but Finitely Many Bessel Polynomials