On 2-orthogonal polynomials of Laguerre type

Let {Pn}n ￠ ‰ ￥0 be a sequence of 2-orthogonal monic polynomials relative to linear functionals ‰0 and ‰1 (see Definition 1.1). Now, let {Qn}n ￠ ‰ ￥0 be the sequence of polynomials defined by Qn:=(n+1) ￠ ’1P ￠ € 2n+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 ‰0 and ‰1 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.