On 2‐orthogonal polynomials of Laguerre type

Abstract

Abstract. 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 differentialrecurrence 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. They are defined by the generating functionTheir corresponding monic polynomials L (α) n n≥0 are defined byLn , n ≥ 0 and satisfy the second-order recurrence relation 30KHALFA DOUAK as well as the following relations:Recently, within the framework of the d-orthogonality of polynomials or polynomials of simultaneous orthogonality studied in [12,11,16] which does not really have the same orthogonality relations but are considered to be orthogonal relative to positive measures, new kinds of d-orthogonal polynomials have been the subject of various investigations [1,3,5,9,15]. In particular, those having some properties that are analogous to the classical orthogonal polynomials.In this paper, when d = 2, under special conditions and well-chosen parameters, we give a family of 2-orthogonal "classical" polynomials which are a natural extension of the classical Laguerre polynomials. These polynomials have some properties analogous to those satisfied by the classical Laguerre polynomials. Their recurrence coefficients and generating function are explicitly determined, a differential-recurrence relation and a third-order differential equation are obtained. We denote these polynomials by P n (·; α), where α is an arbitrary parameter. They are called the 2-orthogonal polynomials of Laguerre type related to the two linear functionals ω 0 ,ω 1 , where ω 0 satisfies a second-order differential (distributional) equation and ω 1 is given in terms of ω 0 and ω 0 (see equations (4.13), (4.14)). Finally, one of the problems is to determine integral representations of both functionals ω 0 and ω 1 . Indeed, applying the method explained in [8], if we denote by ᐃ 0 (resp., ᐃ 1 ) the weight function representing the functional ω 0 (resp., ω 1 ), we obtain that when α > −1, ᐃ 0 (x) = e −1 (x)I * α (x) on the interval 0 ≤ x < +∞, with (x) = x α e −x being the weight function related t...

Frobenius–Padé approximants for d-orthogonal series: Theory and computational aspects

“…Actually, there is, in the literature (see for example [3,4,5,6,10,11]), many information about orthogonal polynomials of dimension d, especially for the case d = 2. These polynomials can be used most likely orthogonal polynomials.…”

“…We compute FPA using the recursive schemes of the preceding section. We begin by computing 2-FPA from some known 2-OS, namely a 2-OS of Jacobi type [4,11], and a 2-OS of Laguerre type [6]. We observe fast convergence of approximants along the diagonals.…”

On two-orthogonal polynomials related to the Bateman’s $J^{u,v}_{n}$-function

“…In this case, the polynomials B%>P , n ^ 0, will be called the two-parameter family of 2-orthogonal polynomials of Laguerre type. Recall that the polynomials {i :> n(-;^)}n^o , studied in [6], constitutes a one-parameter family of 2-orthogonal polynomials of Laguerre type of other kind. Finally, by differentiating (3.5) and making use of (3.7), we obtain after the shift n -> n -2 B?"…”

“…Recall that the two sequences of 2-OPS of Hermite type [7,8] {Pn}n^o and the oneparameter family of Laguerre type [6] {-Pn(-;aO}n^o are obtained by solving a nonlinear system satisfied by the recurrence coefficients. In such cases, we have obtained the two sets of solutions: <J n = 0,n^0,/9 n = l,n^l and e n = 1, n ^ 1, give the 2-OPS of Hermite type, and 5 n --l,n^0,p n = l,n^l and s n = 1, n ^ 1, give the 2-OPS of Laguerre type with one parameter.…”

“…Recently, in this context, the second author introduced in [6] a one-parameter family, denoted by {Pn(-,«)}n^o 5 of classical 2-orthogonal polynomials of Laguerre type. Some of their interesting properties which are analogous of those satisfied by the classical Laguerre polynomials as well as the connection between the two families of polynomials are given.…”