On 2‐orthogonal polynomials of Laguerre type

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Let be a sequence of 2‐orthogonal monic polynomials relative to linear functionals ω0 and ω1 (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 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.