Khalfa Douak

Copyright © 1999 Khalfa Douak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let {Pn}n≥0 be a sequence of 2-orthogonal monic polynomials relative to linear functionals ω0 and ω1 (see Definition 1.1). Now, let {Qn}n≥0 be the sequence of polynomials defined by Qn:=(n+1)−1P′n+1,n≥0. When {Qn}n≥0 is, also, 2-orthogonal, {Pn}n≥0 is called “classical” (in the sense of having the Hahn property). In this case, both {Pn}n≥0 and {Qn}n≥0 satisfy a third-order recurrence relation (see below). Working on the recurrence coefficients, under certain assumptions and well-chosen parameters, a classical family of 2-orthogonal polynomials is presented. Their recurrence coefficients are explicitly determined. A generating function, a third-order differential equation, and a differential-recurrence relation satisfied by these polynomials are obtained. We, also, give integral representations of the two corresponding linear functionals ω0 and ω1 and obtain their weight functions which satisfy a second-order differential equation. From all these properties, we show that the resulting polynomials are an extention of the classical Laguerre's polynomials and establish a connection between the two kinds of polynomials.