On 2-orthogonal polynomials of Laguerre type

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