ON 2-ORTHOGONAL POLYNOMIALS OF LAGUERRE TYPE

Let {Pn}n≥0 be a sequence of 2-orthogonal monic polynomials relative to linear functionalsω0 andω1 (see Definition 1.1). Now, let {Qn}n≥0 be the sequence of polynomials defined by Qn := (n+1)−1P ′ n+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 differentialrecurrence 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.


Introduction.
It is well known (see, e.g., [4]) that the generalized Laguerre polynomials L (α) n n≥0 , for α > −1, are orthogonal with respect to the weight function (x) = x α e −x on the interval 0 ≤ x < +∞, that is, +∞ 0 L (α)  m (x)L (α)  n (x)x α e −x dx = Γ (n + α + 1) n! δ m,n , m,n≥ 0. (1.1) They are defined by the generating function Their corresponding monic polynomials L(α) 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 to the classical Laguerre polynomials, and I * α an entire function defined by , where I α is the modified Bessel function of the first kind.
Let us now recall some results which we need below.Let {P n } n≥0 be a sequence of monic polynomials and {ω n } n≥0 be its dual sequence defined by ω m ,P n = δ m,n , m, n ≥ 0, where , is the duality brackets between ᏼ (the vector space of polynomials with coefficients in C) and its dual ᏼ .Lemma 1.1 [14].For any linear functional u and integer p ≥ 1, the following two statements are equivalent: Definition 1.1.The sequence {P n } n≥0 is said to be d-orthogonal polynomials sequence (d-OPS) with respect to the d-dimensional functional Ω = t (ω 0 ,...,ω d−1 ) if it fulfills [14,17] ω ν ,P m P n = 0, n≥ md for each integer ν with ν = 0, 1,...,d− 1 and m ≥ 0. The linear functionals ω ν are not necessarily positive definite and the d-dimensional functional Ω is not unique.Indeed, according to Lemma 1.1, if {P n } n≥0 is dorthogonal relative to the functional ᐁ = t (u 1 ,...,u d ), we have Thus, this notion of d-orthogonality for polynomials, defined and studied in a different context in [17], appears as a particular case of the general notion of biorthogonality described in [2].A remarkable characterization of the d-orthogonal polynomials is that they satisfy a standard (d + 1)-order recurrence relation, that is, a relation between d + 2 consecutive polynomials [17].Here, we work only with the canonical d-dimensional functional Ω = t (ω 0 ,...,ω d−1 ) and in all the sequel we only consider the case d = 2, that is, {P n } n≥0 is 2-OPS with respect to the linear functionals ω 0 and ω 1 .In this case, the orthogonality relations are ω 0 ,P m P n = 0, n≥ 2m + 1; and ω 1 ,P m P n = 0, n≥ 2m + 2; ω 1 ,P m P 2m+1 = 0, m ≥ 0. (1.10) Then {P n } n≥0 satisfies a third-order recurrence relation [14,17] which we write in the form (1.11) Let us now introduce the sequence of monic polynomials {Q n := (n+1) −1 P n+1 } n≥0 .We denote by { ωn } n≥0 the dual sequence of {Q n } n≥0 .According to the Hahn's property [10], if the sequence {Q n } n≥0 is also 2-orthogonal, then {P n } n≥0 is called a "classical" 2-OPS.In this case, the sequence {Q n } n≥0 , too, satisfies a third-order recurrence relation (1.12) By differentiating (1.11) and using (1.12), we easily obtain with the initial conditions (1.14) Otherwise, when the 2-OPS {P n } n≥0 is "classical", it was obtained in [6] that the recurrence coefficients {β n }, { βn }, {γ ν n }, and {γ ν n }(ν = 0, 1) satisfy the following non-linear system which is valid for n ≥ 1 : (1.15) ) (1.17) To solve this system, we pose In its full generality, the above system has remained unsolved.However, if we impose symmetry [6] or are interested in particular conditions [5,9,15,8], some solutions have been obtained.For instance, under the assumption (δ n = 0,n ≥ 0, then β n = βn = const.),the system was solved in [6] and a class of "classical" 2-orthogonal polynomials were obtained.In this paper, we give another particular solution when δ n = 0,n ≥ 0.

The case (B)
Secondly, from (2.22), for n → 2n and n → 2n + 1 we, respectively, obtain These give easily ) Finally, from (2.23) and taking into account the last results, we, also, obtain which yield , n≥ 0, (2.37) , n≥ 0, (2.40) From now on, we are only interested in the sequence of polynomials obtained in case (A) with δ 0 = 0 (fixed), and, under some conditions, we shall be concerned mainly with the properties of the resulting polynomials.We show that some of these properties are analogous to the Laguerre's polynomials ones.For this, we note first that when θ n = 1 and ρ n = 1, the relations (2.1) and (2.2) become (2.41) This relation plays a fundamental role in the next section, it allows us to derive some properties as differential-recurrence relation, a third-order differential equation, and a generating function of the resulting polynomials.
Proof.From (3.1), we have Differentiating twice, we successively obtain and by differentiation, we obtain (3.18) Using (3.5) to eliminate P n+2 , the last equation becomes which yields by differentiating again and for n → n + 1 (3.21) Using (3.5) again, we obtain (1) Since the values β 0 ,δ 0 ,α 1 , and γ 1 are arbitrary, with δ 0 = 0 and γ 1 = 0, then the polynomials obtained in case (A) constitute, a priori, a four-parameter family.But, in the rest of this paper, we fix δ 0 ,γ 1 and give special importance to the case when δ 0 = −1 and γ 1 = 2.In this case, the differential equation (3.13) becomes In order to determine the family of 2-OPS analogous to the classical Laguerre's one, we first transform the singularity of the above differential equation to the origin.Indeed, it is clear that equation (3.24) has a singularity at the point x 0 = β 0 −α 1 +1.Then by an appropriate change of variable, this singularity may be transformed to the origin, that is, β 0 − α 1 + 1 = 0, i.e., α 1 = β 0 + 1.In this case, we pose β 0 = α + 2 so that α 1 = α + 3. Thus, the differential equation (3.24) takes the form Recall that the classical Laguerre polynomials satisfy the following second-order differential equation: (2) Further, in this case the third-order recurrence relations (1.11) and (1.12) become and (3.28) From these, by taking into account the dependence on the parameter α and putting P n (x) = P n (x; α), n ≥ 0, it may be seen that Q n (x) = P n (x; α + 1), n ≥ 0, i.e.,

A generating function. Let G(x, t) be the generating function of the polynomials
By using the relations (3.30), (3.31) and the definition (3.32), it is easy to verify that G(x, t) satisfies the properties Now, from (3.33) and (3.34), we easily obtain Remark 3.2.(1) The generating function obtained here has the form G(x, t) = A(t)e xH (t) , with A(t) = (1+t) −α−1 e −t , and H(t) = t/(1+t), that is, generating function of type Scheffer.Thus, the polynomials P n (•; α), n = 0, 1,... appear as particular case of the class of 1/2-orthogonal polynomials studied in [1].
(2) On the other hand, if we substitute t by −t in (3.36), we obtain L (α)  n (x)t n .
(3.39)By using the expansion of the two members of this identity as a power series in t, we obtain, by identification, Examples.4. Integral representations.We are, now, interested in this section in the integral representation of the linear functionals ω 0 and ω 1 with respect to which the sequence {P n (•; α)} n≥0 is 2-orthogonal.For this, we use the same technique explained in [8].First, we start with the 2-OPS satisfying the recurrence relation (1.11) with β n ,α n and γ n given in (2.29).Since the sequence {P n } n≥0 is "classical", according to the characterization given in [7], we have the following theorem which is an easy application of [7, Thm.3.1] and which we adapt to our situation: Theorem 4.1 [7].For the 2-OPS {P n } n≥0 satisfying the recurrence relation (1.11) with β n ,α n and γ n given by (2.29), its associated vector functional Ω = t (ω 0 ,ω 1 ) satisfies the following vector distributional equation: where Φ and Ψ are two 2 × 2 polynomials matrices ) Moreover, the vector functional Ω = t ω0 , ω1 associated with the sequence {Q n } n≥0 is given by Ω = Φ(x)Ω.

.24)
A solution of (4.24) satisfying the condition (4.18) is to be found.Thus, by choosing Ꮿ = [0, +∞[, a solution of (4.24) can be taken as V (s) = KI α (s), where I α is the modified Bessel function of the first kind and K is the normalization constant which we determine below.Then a solution of (4.22) is given by

.25)
The function I ν (z), for arbitrary order ν, is defined by (see, e.g., [13]) where J ν (z) is the Bessel function of order ν.It is evident from the expansion given in the right-hand side of (4.26) that I ν (z) is an analytic function of the complex variable z and, for z > 0 and ν ≥ 0, it is a positive function which increases monotonically as z → +∞.The asymptotic behavior of this function as z → +∞ is given by