International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 1, Pages 29-48
On 2-orthogonal polynomials of Laguerre type
Laboratoire d' Analyse Numérique, Université Pierre et Marie Curie, 4, place Jussieu, Cedex 05, Paris 75252, France
Received 7 July 1997; Revised 10 February 1998
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.
Let be a sequence of 2-orthogonal monic polynomials relative to linear functionals and (see Definition 1.1). Now, let be the sequence of polynomials defined by . When is, also, 2-orthogonal, is called classical (in the sense of having the Hahn property). In this case, both and 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 and 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.