%0 Journal Article
%T On 2-orthogonal polynomials of Laguerre type
%A Khalfa Douak
%J International Journal of Mathematics and Mathematical Sciences
%D 1999
%I Hindawi Publishing Corporation
%R 10.1155/s0161171299220297
%X Let {Pn}n ¡é ¡ë £¤0 be a sequence of 2-orthogonal monic polynomials relative to linear functionals    ¡ë0 and    ¡ë1 (see Definition 1.1). Now, let {Qn}n ¡é ¡ë £¤0 be the sequence of polynomials defined by Qn:=(n+1) ¡é   ¡¯1P ¡é ? 2n+1,n ¡é ¡ë £¤0. When {Qn}n ¡é ¡ë £¤0 is, also, 2-orthogonal, {Pn}n ¡é ¡ë £¤0 is called    ¡°classical   ¡± (in the sense of having the Hahn property). In this case, both {Pn}n ¡é ¡ë £¤0 and {Qn}n ¡é ¡ë £¤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    ¡ë0 and    ¡ë1 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.
%K Orthogonal polynomials
%K d-orthogonal polynomials
%K Laguerre polynomials
%K Sheffer polynomials
%K recurrence relations
%K integral representations.
%U http://dx.doi.org/10.1155/S0161171299220297