On 2-orthogonal polynomials of Laguerre type

Abstract

Let{Pn}n≥0be a sequence of 2-orthogonal monic polynomials relative to linear functionalsω0andω1(see Definition 1.1). Now, let{Qn}n≥0be the sequence of polynomials defined byQn:=(n+1)−1P′n+1,n≥0. When{Qn}n≥0is, also, 2-orthogonal,{Pn}n≥0is called “classical” (in the sense of having the Hahn property). In this case, both{Pn}n≥0and{Qn}n≥0satisfy 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 functionalsω0andω1and 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.