ON 2-ORTHOGONAL POLYNOMIALS OF LAGUERRE TYPE

Transcription

1 Internat. J. Math. & Math. Sci. Vol. 22, No S Electronic Publishing House ON 2-ORTHOGONAL POLYNOMIALS OF LAGUERRE TYPE KHALFA DOUAK Received 7 July 1997 and in revised form 1 February 1998 Abstract. Let {P n } n be a sequence of 2-orthogonal monic polynomials relative to linear functionals ω and ω 1 see Definition 1.1. Now, let {Q n } n be the sequence of polynomials defined by Q n := n+1 1 P n+1, n. When {Q n} n is, also, 2-orthogonal, {P n } n is called classical in the sense of having the Hahn property. In this case, both {P n } n and {Q n } n 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 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. Keywords and phrases. Orthogonal polynomials, d-orthogonal polynomials, Laguerre polynomials, Sheffer polynomials, recurrence relations, integral representations Mathematics Subject Classification. 33C45, 42C5. 1. Introduction. It is well known see, e.g., [4] that the generalized Laguerre polynomials { L α } n n, for α> 1, are orthogonal with respect to the weight function ϱx = x α e x on the interval x<+, that is, + L α m xlα n xxα e x dx = Γ n+α+1 δ m,n, m,n. 1.1 n! They are defined by the generating function 1 t α 1 e xt/1 t = n= L α n xtn. 1.2 {ˆL α n Their corresponding monic polynomials }n are defined by ˆL α n n and satisfy the second-order recurrence relation = 1 n n! L α n, ˆL α n+2 x = α α x 2n+α+3 ˆL n+1 x n+1n+α+1ˆl n x, n, ˆL α x = 1, 1.3 ˆL α 1 x = x α+1,

2 3 KHALFA DOUAK as well as the following relations: d α α+1 ˆL n+1 x = n+1ˆl n x, dx n, 1.4 ˆL α n+1 x = ˆL α+1 α+1 n+1 x+n+1ˆl n x, n, 1.5 xˆl α+1 n x = ˆL α α n+1 x+n+α+1ˆl n x, n. 1.6 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,ω 1, where ω satisfies a second-order differential distributional equation and ω 1 is given in terms of ω and ω see equations 4.13, Finally, one of the problems is to determine integral representations of both functionals ω and ω 1. Indeed, applying the method explained in [8], if we denote by resp., 1 the weight function representing the functional ω resp., ω 1, we obtain that when α> 1, x = e 1 ϱxiα x on the interval x<+, with ϱx = x α e x being the weight function related to the classical Laguerre polynomials, and Iα an entire function defined by I α x = x α/2 I α 2 x, 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 be a sequence of monic polynomials and {ω n } n be its dual sequence defined by ω m,p n =δ m,n, m,n, where, is the duality brackets between the vector space of polynomials with coefficients in C and its dual. Lemma 1.1 [14]. For any linear functional u and integer p 1, the following two statements are equivalent: i u,p p 1 ; u,p n =, n p; ii λ ν C, ν p 1, λ p 1 such that u = p 1 ν= λ νu ν. Definition 1.1. The sequence {P n } n is said to be d-orthogonal polynomials sequence d-ops with respect to the d-dimensional functional Ω = t ω,...,ω d 1 if it fulfills [14, 17] ων,p m P n =, n md+ν +1, ων,p m P md+ν, 1.7 for each integer ν with ν =,1,...,d 1 and m.

3 ON 2-ORTHOGONAL POLYNOMIALS OF LAGUERRE TYPE 31 Remark 1.1. a When d = 1, we meet again the ordinary orthogonality. b 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 is d- orthogonal relative to the functional = t u 1,...,u d, we have κ 1 u κ = λ κ ν ω ν, λ κ κ 1, 1 κ d ν= ν+1 ω ν = τ ν k u k, τν+1 ν, ν d k=1 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,...,ω d 1 and in all the sequel we only consider the case d = 2, that is, {P n } n is 2-OPS with respect to the linear functionals ω and ω 1. In this case, the orthogonality relations are ω,p m P n =, n 2m+1; 1.9 ω,p m P 2m, m and ω 1,P m P n =, n 2m+2; ω 1,P m P 2m+1, m. 1.1 Then {P n } n satisfies a third-order recurrence relation [14, 17] which we write in the form P n+3 x = x β n+2 P n+2 x α n+2 P n+1 x γ n+1 P n x, n, P x = 1, P 1 x = x β, P 2 x = x β 1 P 1 x α 1, 1.11 γ n+1, n regularity conditions. Let us now introduce the sequence of monic polynomials {Q n := n+1 1 P n+1 } n.we denote by { ω n } n the dual sequence of {Q n } n. According to the Hahn s property [1], if the sequence {Q n } n is also 2-orthogonal, then {P n } n is called a classical 2- OPS. In this case, the sequence {Q n } n, too, satisfies a third-order recurrence relation Q n+3 x = x β n+2 Qn+2 x α n+2 Q n+1 x γ n+1 Q n x, n, Q x = 1, Q 1 x = x β, Q 2 x = x β 1 Q1 x α 1, 1.12 γ n+1, n>. By differentiating 1.11 and using 1.12, we easily obtain P n+3 x = Q n+3 x+n+3 β n+3 β n+2 Qn+2 x + n+2α n+3 n+3 α n+2 Qn+1 x + n+1γ n+2 n+3 γ n+1 Qn x, n, 1.13

4 32 KHALFA DOUAK with the initial conditions P x = Q x = 1, P 1 x = Q 1 x+β 1 β, P 2 x = Q 2 x+2 β 2 β 1 Q1 x+α 2 α Otherwise, when the 2-OPS {P n } n is classical, it was obtained in [6] that the recurrence coefficients {β n }, { β n },{γn ν}, and { γν n }ν =,1 satisfy the following non-linear system which is valid for n 1: n+2 β n n β n 1 = n+1β n+1 n 1β n ; β = β +β 1, 1.16 n+3 α n+1 n+1 α n = n+1α n+2 n 1α n+1 +n+1β n+1 β n 2 ; α 1 = α 2 +α 1 +β 1 β 2, 1.18 n+4 γ n+1 n+2 γ n = n+1γ n+2 n 1γ n+1 +n+1α n+2 βn+2 +β n+1 2 β n 1.19 n+2 α n+1 2βn+2 β n+1 β n ; 4 γ 1 = γ 2 +γ 1 +α 2 β 2 +β 1 2 β 2 α 1 2β 2 β 1 β, 1.2 nα n+1 α n+2 +n+2 α n α n+1 2n+1 α n α n+2 = n+2 γ n 2βn+2 β n+1 β n nγ n+1 βn+2 +β n 2 β n 1 ; nα n+1 γ n+2 +nγ n+1 α n+3 = γ n 2n+2αn+3 n+3 α n+2 + α n 2n+1γn+2 n+3 γ n+1, 1.22 nγ n+1 γ n+3 = γ n 2n+2γn+3 n+4 γ n To solve this system, we pose β n = β n+1 +δ n, n, 1.24 n α n = α n+1 n+1 ρ n, n 1 ρ n, 1.25 n γ n = γ n+1 n+2 θ n, n 1 θ n 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 =,n,then β n = β n = 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,n.

5 ON 2-ORTHOGONAL POLYNOMIALS OF LAGUERRE TYPE Another particular solution of the system First, under the transformations , the relation 1.13 becomes P n+3 x = Q n+3 x n+3δ n+2 Q n+2 x +n+2α n+3 1 ρ n+2 Q n+1 x +n+1γ n+2 1 θ n+1 Q n x, n, 2.1 with P x = Q x = 1, P 1 x = Q 1 x δ, P 2 x = Q 2 x 2δ 1 Q 1 x+α 2 1 ρ Likewise, by using , the system is, also, transformed to β n+1 β n = nδ n 1 n+2δ n, n 1, 2.3 β 1 β = 2δ, 2.4 [ n+3ρn+1 n+2 ] α n+2 n+2 [ nρ n n 1 ] α n+1 n+1 = δ2 n, n 1, 2.5 3ρ 1 2 α 2 2 α 1 = δ 2, 2.6 [ n+4θn+1 n+3 ] γ n+2 n+3 [ nθ n n 1 ] γ n+1 n+1 { } = α n+2 βn+2 β n+1 2δ n ρn+1 βn+2 β n+1 δ n+1 δ n, n 1, 2.7 4θ 1 3 γ 2 3 γ { 1 = α 2 β2 β 1 2δ ρ 1 β 2 β 1 δ 1 δ }, n 1, 2.8 2ρ n ρ n+1 ρ n 1α n+1 α n+2 { } = γ n+1 βn+2 β n δ n+1 δ n 1 θn β n+2 β n 2δ n 1, n 1, 2.9 [ α n+1 γ n+2 2ρn ρ n θ n+1 1 ] [ +α n+3 γ n+1 2θn θ n ρ n+2 1 ] =, n 1, 2.1 θ n θ n = 2, n As the starting point for the solution of the above system, we begin with equation 2.11 which is Riccati equation and for which θ n = 1,n 1 is a trivial solution. In that case, the above system becomes β n+1 β n = nδ n 1 n+2δ n, n 1; 2.12 β 1 β = 2δ, 2.13 [ n+3ρn+1 n+2 ] α n+2 n+2 [ nρ n n 1 ] α n+1 n+1 = δ2 n, n 1; ρ 1 2 α 2 2 α 1 = δ 2, 2.15

6 34 KHALFA DOUAK γ n+2 n+3 γ n+1 n+1 { } = α n+2 βn+2 β n+1 2δ n ρn+1 βn+2 β n+1 δ n+1 δ n, n 1; 2.16 γ 2 3 γ { 1 = α 2 β2 β 1 2δ ρ 1 β 2 β 1 δ 1 δ }, 2.17 [ 2ρn ρ n+1 ρ n 1 ] α n+1 α n+2 = γ n+1 δn+1 δ n 1, n 1, 2.18 α n+1 γ n+2 [ρ n 1] α n+3 γ n+1 [ρ n+2 1] =, n It may be seen that equation 2.19 is verified if ρ n = 1,n 1. Thus, in view of this remark, the two following cases may be distinguished: ρ n = 1, n 1 and ρ n 1, n In this paper, we consider only the first case, that is, ρ n = 1,n 1 the simplest case. Hence, the system becomes α n+2 β n+1 β n = nδ n 1 n+2δ n, n,δ 1 =, 2.21 n+2 α n+1 n+1 = δ2 n, n, 2.22 γ n+2 n+3 γ n+1 n+1 = α n+2δ n+1 δ n, n, 2.23 γ n+2 δ n+2 δ n =, n Since γ n+1, shifting n n + 1, equation 2.24 leads to δ n+2 δ n =, n, whose solutions are given in the following δ 2n = δ, n, δ 2n+1 = δ 1, n, 2.25 with δ = β β 1 /2 and δ 1 = β + β 1 2β 2 /6. It may be seen that δ 1 = δ if and only if β 2 = 2β 1 β. Thus, two cases arise. A δ 1 = δ, B δ 1 δ The case A δ 1 = δ. In this case, the system leads to α n+2 From these, we easily obtain β n+1 β n = 2δ, n, 2.26 n+2 α n+1 n+1 = δ2, n, 2.27 γ n+2 n+3 γ n+1 =, n+1 n β n = β 2δ n, n, α n+1 = n+1 δ 2 n+α 1, n, 2.29 γ n+1 = 1 2 γ 1n+2n+1, n,

7 ON 2-ORTHOGONAL POLYNOMIALS OF LAGUERRE TYPE 35 and β n = β δ 2n+1, n, α n+1 = n+1δ 2 n+1+α 1, n, 2.3 γ n+1 = 1 2 γ 1n+2n+1, n. Note that if δ =, we obtain β n = β n, α n = α n and γ n = γ n. Then P n = Q n,n =,1,2,... and {P n } n is, at the same time, 2-orthogonal and Appell sequence [5, 6]. In this case, the polynomials obtained are the analogous of the classical orthogonal polynomials of Hermite The case B δ 1 δ. First, from 2.21, we obtain, for n 2n+1 and n 2n,β 2n = β δ +3δ 1 n,n and β 2n+1 = β 1 δ 1 +3δ n, n, respectively. Secondly, from 2.22, for n 2n and n 2n+1 we, respectively, obtain These give easily α 2n+2 2n+2 α 2n+1 2n+1 = δ2, n, 2.31 α 2n+3 2n+3 α 2n+2 2n+2 = δ2 1, n, 2.32 α 2n+1 = 2n+1 δ 2 +δ2 1 n+α 1, n, 2.33 α 2n+2 = 2n+2 δ 2 +δ2 1 n+α1 +δ 2, n Finally, from 2.23 and taking into account the last results, we, also, obtain γ 2n+2 2n+3 γ 2n+1 2n+1 = δ 1 δ α 2n+2, n, 2.35 γ 2n+3 2n+4 γ 2n+2 2n+2 = δ δ 1 α 2n+3, n, 2.36 which yield γ 2n+1 = 2n+22n+1 δ δ 1 δ 2 1 n+ γ 1 2 γ 2n+2 = 2n+32n+2 Two subcases arise. B 1 δ =., n, 2.37 δ 1 δ δ 2 n+1+ γ δ 1 δ α 1, n β 2n = β 3δ 1 n, n ; β 2n+1 = β 1 δ 1 n, n, α 2n+1 = 2n+1δ 2 1 n+α 1, n ; α 2n+2 = 2n+2δ 2 1 n+α 1, n, γ 2n+1 = 2n+22n+1 δ 3 1 n+ γ 1, n, 2 γ 2n+2 = 2n+32n+2 δ 1 α 1 + γ 1, n

8 36 KHALFA DOUAK B 2 δ 1 =. β 2n = β δ n, n ; β 2n+1 = β 1 3δ n, n, α 2n+1 = 2n+1δ 2 n+α 1, n ; α 2n+2 = 2n+2δ 2 n+1+α 1, n, 2.4 γ 2n+1 = γ 1 2 γ 2n+2 = 2n+32n+2 2n+22n+1, n ; δ 3 n+1 δ α 1 + γ 1 2, n. From now on, we are only interested in the sequence of polynomials obtained in case A with δ 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 P n+1 x = Q n+1 x n+1δ n Q n x, n 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. 3. Some properties of the sequence obtained in case A 3.1. A differential-recurrence relation Proposition 3.1. The sequence {P n } n obtained in case A satisfies the following differential-recurrence relation: φxq n x = P n+1 x a n P n x b n P n 1 x, n P 1 =, 3.1 where Proof. First, from 2.41, we have φx = x β + α 1 + γ 1 δ 2δ 2, 3.2 a n = δ n+ α 1 + γ 1 δ 2δ 2, n, 3.3 b n = γ 1 n, n δ P n+2 x = n+2p n+1x+δ n+2p n+1 x, n 3.5 which we write for the subscripts n,n+1, and n+3 as follows: P n x = np n 1x+δ np n 1 x, n, 3.6 P n+1 x = n+1p nx+δ n+1p n x, n, 3.7 P n+3 x = n+3p n+2x+δ n+3p n+2 x, n. 3.8

9 ON 2-ORTHOGONAL POLYNOMIALS OF LAGUERRE TYPE 37 Substituting for P n+2 from 3.5 into 3.8, we get P n+3 x = n+3p n+2x+δ n+3n+2p n+1 x +δ 2 n+3n+2p n+1 x, n. 3.9 Now, according to 2.29, the recurrence relation 1.11 gives, for n, P n+3 x = x β +2δ n+2 P n+2 x n+2 δ 2 n+1+α 1 Pn+1 x γ 1 2 n+2n+1p nx, 3.1 which gives by differentiation P n+2 x = P n+3 x x β +2δ n+2 P n+2 x +n+2 δ 2 n+1+α 1 P n+1 x γ 1 2 n+2n+1p n x, n. Substituting for P n+3, P n+2, and P n from 3.9, 3.5, and 3.7, respectively, into 3.11 yields δ x β +α 1 + γ 1 P n+1 2δ x = P n+2 x+ x β +δ n+1 P n+1 x+ γ 1 n+1p n x δ From this, using the recurrence relation 3.1 by shifting n n 1, we easily obtain the differential relation A third-order differential equation Proposition 3.2. The polynomials P n, n =,1,... satisfy the following third-order differential equation: δ 2 φxy + 2δ x β α 1 +δ 2 Y + x β δ n 1 Y ny =, n, 3.13 where Y = P n x. Proof. From 3.1, we have φxp n+1 x = n+1p n+1x n+1a n P n x n+1b n P n 1 x, n Differentiating twice, we successively obtain φxp n+1 x = np n+1 x n+1a np n x n+1b np n 1 x, n 3.15 and φxp n+1 x = n 1P n+1 x n+1a np n x n+1b np n 1 x, n. 3.16

10 38 KHALFA DOUAK Now, from the recurrence relation 3.1, by shifting n n 1, we get γ 1 2 n+1np n 1x = x β +2n+1δ Pn+1 x n+1 nδ 2 +α 1 Pn x P n+2 x, 3.17 and by differentiation, we obtain γ 1 2 n+1np n 1 x = P n+1x+ x β +2δ n+1 P n+1 x n+1 δ 2 n+α P n x P n+2 x. Using 3.5 to eliminate P n+2, the last equation becomes γ 1 2 n+1np n 1 x = n+1p n+1x+ x β +δ n P n+1 x n+1 δ 2 n+α 1 P n x, 3.19 which yields by differentiating again and for n n+1 γ 1 n+1np n 1 2 x = np n+1 x+ x β +δ n P n+1 x n+1 δ 2 n+α P n x, γ 1 n+2n+1p n 2 x = n+1p n+2 x+ x β +δ n+1 P n+2 x n+2 δ 2 n+1+α P n+1 x. Using 3.5 again, we obtain γ 1 n+1p n 2 x = δ x β α 1 P n+1 x+x β P n+1 x n+1p n+1x Then 3.2 becomes γ 1 n+1np n 1 2 x = 2 δ 2 γ n+α 1 n+1pn+1 x 1 2 δ 2 γ n+α 1 x β +n P n+1 x 1 + x β +δ n 2 δ 2 γ n+α 1 δ x β α 1 P n+1 x Now, by using 3.22 and 3.23 to eliminate P n and P n 1 in 3.16 and shifting n n 1, the differential equation 3.13 follows easily. Remark Since the values β,δ,α 1,and γ 1 are arbitrary, with δ and γ 1, then the polynomials obtained in case A constitute, a priori, a four-parameter

11 ON 2-ORTHOGONAL POLYNOMIALS OF LAGUERRE TYPE 39 family. But, in the rest of this paper, we fix δ,γ 1 and give special importance to the case when δ = 1 and γ 1 = 2. In this case, the differential equation 3.13 becomes x β +α 1 1Y + 2x +2β α 1 +1Y +x β 1+nY ny =, n 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 = β α Then by an appropriate change of variable, this singularity may be transformed to the origin, that is, β α 1 +1 =, i.e., α 1 = β +1. In this case, we pose β = α+2 so that α 1 = α+3. Thus, the differential equation 3.24 takes the form xy + 2x +α+2y +x α 3+nY ny =, n Recall that the classical Laguerre polynomials satisfy the following second-order differential equation: xy + x +α+1y +ny =, n Further, in this case the third-order recurrence relations 1.11 and 1.12 become P n+3 x = x 2n+3+α P n+2 x n+2n+α+4p n+1 x n+2n+1p n x, n, P x = 1, P 1 x = x α+2, P 2 x = x 2 2α+3x +α+4α+1+1 and Q n+3 x = x 2n+3+α+1 Q n+2 x n+2n+α+5q n+1 x n+2n+1q n x, n, Q x = 1, Q 1 x = x α+3, Q 2 x = x 2 2α+4x +α+5α From these, by taking into account the dependence on the parameter α and putting P n x = P n x;α,n, it may be seen that Q n x = P n x;α+1,n, i.e., d dx P n+1x;α = n+1p n x;α+1, n Further, the relations 2.41 and 3.1, also, become P n+1 x;α = P n+1 x;α+1+n+1p n x;α+1, n, 3.3 xp n x;α+1 = P n+1 x;α+n+α+2p n x;α+np n 1 x;α, n The relations are analogous to the relations satisfied by classical Laguerre polynomials {L α n } n. Therefore, the polynomials P n ;α, n =,1,... are called the 2-orthogonal polynomials of Laguerre type.

12 4 KHALFA DOUAK 3.3. A generating function. Let Gx,t be the generating function of the polynomials P n x = P n x;α,n, say Gx,t = n= P n x;α tn n! By using the relations 3.3, 3.31 and the definition 3.32, it is easy to verify that Gx,t satisfies the properties 1+t x Gx,t = tgx,t, 3.33 x x Gx,t = t1+t t Gx,t+tt+α+2Gx,t Now, from 3.33 and 3.34, we easily obtain where x = / x and t = / t. 1+t 2 t Gx,t = [ x 1+tt +α+2 ] Gx,t, 3.35 Proposition 3.3. The generating function Gx,t has the form Gx,t = e t 1+t α 1 e xt/1+t Proof. From 3.33, we have x Gx,t = t Gx,t. By integrating both sides of 1+t this equality with respect to x, fromtox, we obtain Gx,t = Ate xt/1+t, 3.37 with At = G,t. Now, to calculate At, substituting 3.37 in 3.34, we get 1 + ta t = t + α + 2At. This yields At = C1 + t α 1 e t, where the constant C is given by C = A = 1. Thus, At = 1+t α 1 e t. Remark The generating function obtained here has the form Gx,t = Ate xht, with At = 1+t α 1 e t, and Ht = t/1+t, that is, generating function of type Scheffer. Thus, the polynomials P n ;α,n =,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 e t 1 t α 1 e xt/1 t = 1 n P n x;α tn n! = P n x;αt n, 3.38 n= where P n := 1 n /n! P n,n=,1,.... The polynomials P n ;α,n =,1,... are the non monic polynomials corresponding to the family of polynomials P n ;α,n =,1,.... They are analogous to the Laguerre polynomials L α n,n =,1,... defined by the generating function 1.2. From the expression of the two generating functions of the polynomials L α n and P n ;α, we can write P n x;αt n = e t n= n= n= L α n xtn. 3.39

13 ON 2-ORTHOGONAL POLYNOMIALS OF LAGUERRE TYPE 41 By using the expansion of the two members of this identity as a power series in t, we obtain, by identification, Examples. n 1 P n x;α = n k! Lα k x, n =,1,..., 3.4 k= n n P n x;α = 1 n+k ˆL α k x, n =,1, k k= P x;α = ˆL α x = 1, P 1 x;α = ˆL α 1 x ˆL α x = x α+2, P 2 x;α = ˆL α α 2 x 2ˆL 1 x+ ˆL α x = x2 2α+3x +α+4α+1+1, P 3 x;α = ˆL α α α 3 x 3ˆL 2 x+3ˆl 1 x ˆL α x = x 3 3α+4x 2 +3α 2 +7α+11x α 3 +9α 2 +23α+16, The Languerre polynomials { L α } n n are, also, defined by the Rodrigues s formula ˆL α n x = 1n x α e x dn dx n xα+n e x, n=,1, Then from 3.41 we can also write P n x;α = 1 n x α e x n k= n d k x α+k e x, n=,1, k dx k 4. Integral representations. We are, now, interested in this section in the integral representation of the linear functionals ω and ω 1 with respect to which the sequence {P n ;α} n 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 Since the sequence {P n } n 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 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,ω 1 satisfies the following vector distributional equation: where Φ and Ψ are two 2 2 polynomials matrices DΦ +Ψ =, 4.1 ϕ Φx = x ϕ 1 x 1 ϕ1 x ϕ1 1 x, Ψx =, 4.2 ψx ζ

14 42 KHALFA DOUAK with ϕ x = 1, ϕ1 x = δ,ϕ1 x = 2γ 1 1 δ x β, ϕ1 1 x = 1+2γ 1 1 δ α 1, ψx = 2γ1 1 x β and ζ = 2γ1 1 α 1. Moreover, the vector functional Ω = t ω, ω 1 associated with the sequence {Qn } n is given by Ω = ΦxΩ. Recall that, for a linear functional ω, the derivative Dω = ω and the left-multiplication of ω by a polynomial h are defined by Dω,f = ω,f, and hω, f = ω,hf,f. The main result here is Theorem 4.2. The two linear functionals ω,ω 1, associated with the 2-classical sequence {P n ;α} n, for α> 1, respectively, have the following integral representations: + ω,f = xf xdx, ω 1,f = + 1 xf xdx, with the weight functions and 1 being given by x = e 1 ϱxi α x; f ; f ; 1 x = e 1 ϱx [ xi α+1 x I α x], where ϱx = x α e x and Iα = + k=x k /k! Γ k+α+1. Proof of Theorem 4.1. For the proof of 4.1 and the explicit formulas for the elements of the matrix Ψ, see [7]. To calculate the expressions of the polynomials ϕ i j, we proceed as follows: Applying the functional ω resp., ω 1 to 2.41, since { ω n } n is the dual sequence of {Q n } n we, respectively, obtain ω,p =1, ω,p 1 = δ, 4.5 ω,p n =; n 2, and ω 1,P =, ω 1,P 1 =1, ω 1,P 2 = 2δ, ω 1,P n =; n 3, 4.6 so that, by Lemma 1.1, we have ω = λ ω + λ 1 ω 1 and ω 1 = η ω + η 1 ω 1 + η 2 ω 2, with λ = 1,λ 1 = δ,η =,η 1 = 1,and η 2 = 2δ. Now, from [14, théorème 2.1], the functional ω 2 can be written in terms of ω and ω 1 as ω 2 = ax + bω + cω 1. To determine the coefficients a,b, and c, we apply ω 2 to the polynomials P,P 1, and P 2. Using the fact that {ω n } n is the dual sequence of {P n } n, we easily obtain = ω 2,P =aβ +b, = ω 2,P 1 =aα 1 +c, = ω 2,P 2 =aγ 1,

15 ON 2-ORTHOGONAL POLYNOMIALS OF LAGUERRE TYPE 43 so that a = γ1 1,b = γ 1 1 β and c = γ1 1 α 1. Whence, ω = ω δ ω 1 and ω 1 = 2γ1 1 δ x β ω +1+2γ1 1 δ α 1 ω 1. Consequently, ω 1 δ ω = ω 1 2γ1 1 δ x β 1+2γ1 1 δ, 4.8 α 1 ω 1 and then 1 δ Φx = 2γ1 1 δ x β 1+2γ1 1 δ. 4.9 α 1 Proof of Theorem 4.2. In all the sequel, the values δ = 1, β = α + 2, α 1 = α+3 and γ 1 = 2 are retained. Then, from Theorem 4.1, we obtain Φx =, Ψx =. 4.1 x α+2 α+2 x α+2 α+3 Then, from 4.1, we obtain ω +ω 1 +ω 1 =, 4.11 x α 2ω α+2ω 1 +x α 1ω α+3ω 1 = From these, we easily obtain and xω +2x α+1ω +x αω = 4.13 ω 1 = xω +x α 1ω To find integral representations of both the functionals ω and ω 1, we use equations 4.13 and The problem consists in determining weight functions and 1 depending in such that ω,f = fx xdx, f, 4.15 ω 1,f = fx 1 xdx, f We suppose that function is regular as far as it is necessary. From 4.13, by integration by parts, we obtain { } x x+2x α+1 x+x α x dx ] + [x xf x {2x α x+x x}fx If the following condition holds: = [ } ] x xf x {2x α x+x x fx =, 4.18

16 44 KHALFA DOUAK we obtain { } x x+2x α+1 x+x α x dx =, 4.19 which leads to x x+2x α+1 x+x α x = λgx, 4.2 where λ is an arbitrary constant and g is a function representing the null functional over the path gxx n dx =, n Here, we consider only the representation in terms of integrals evaluated along an interval of the real axis. Setting λ =, the differential equation 4.2 becomes x x+2x α+1 x+x α x = Under the transformation x = e x Us with s = 2 x, the last becomes su s 2α 1U s sus = Now, by setting Us= s α Vs in 4.23, we obtain that V satisfies the following Bessel differential equation of imaginary argument: s 2 V s+sv s α 2 +s 2 Vs= A solution of 4.24 satisfying the condition 4.18 is to be found. Thus, by choosing = [,+ [, a solution of 4.24 can be taken as Vs= 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 x = 2 α Kx α/2 e x I α 2 x The function I ν z, for arbitrary order ν, is defined by see, e.g., [13] I ν z = i ν J ν iz = + k= z/2 ν+2k, k! Γ k+ν +1 z < +, arg z <π, 4.26 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> and ν, it is a positive function which increases monotonically as z +. The asymptotic behavior of this function as z + is given by I ν z ez 2πz, z For small z, we have the asymptotic formula I ν z z ν, 2 ν Γ ν +1 z 4.28

17 ON 2-ORTHOGONAL POLYNOMIALS OF LAGUERRE TYPE 45 and, therefore, I ν = ifν>, I = 1. Otherwise, I ν satisfies some simple recurrence relations 2I ν z = I ν 1z+I ν+1 z, 2νz 1 I ν z = I ν 1 z I ν+1 z Now, we are ready to determine the constant K and verify that the condition 4.18 holds. The normalization condition ω,1 = + xdx = leads to + 2 α K x α/2 e x I α 2 xdx = We first evaluate the integral + = x α/2 e x I α 2 + xdx = 2 t α+1 e t2 I α 2tdt But, from 4.26, we have Then I α 2t = + k= t α+2k k! Γ k+α = 2 t α+1 t α+2k e t2 dt k! Γ k+α+1 k= = 2 e t2 t 2α+2k+1 dt. k! Γ k+α+1 k= 4.34 The exchange of integration and summation is justified by an absolute convergence argument. Next, under the transformation u = t 2, we easily obtain that = + k= 1 k! Γ k+α+1 + e u u α+k du = e Whence, 4.31 gives K = 2 α e 1 and the weight function is given by Let us define the function I α by Thus, can be written as x = x α/2 e x 1 I α 2 x I α x = x α/2 I α 2 x = + k= x k k! Γ k+α x = x α e x 1 Iα x = e 1 ϱxiα x, 4.38

18 46 KHALFA DOUAK where, ϱx = x α e x is the weight function with respect to which the classical Laguerre polynomials are orthogonal. It is clear that the weight function is positive definite. Finally, we must verify that the condition 4.18 holds [ } ] + x xf x {2x α x+x x fx = Indeed, from 4.38, we have x x = x α+1 e x 1 Iα x = e 1 xiα xϱx. 4.4 On the other hand, by differentiating, we obtain } x = e 1{ ϱxiα x+ϱ xiα x = e 1{ ϱxiα+1 x+ αx 1 1 } ϱxiα x = e 1 ϱxiα+1 x+ αx 1 1 x Because I α x = I α+1 x and ϱ x = αx 1 1 ϱx. Then, and x = e 1 xi α+1 xϱx+α x x x α x+x x = x x+e 1 xϱxi α+1 x = e 1 xϱx [ I α+1 x+i α x] From 4.4 and 4.43, it is easy to see that at the origin when α> 1, the condition 4.18 is verified. Otherwise, from the asymptotic behavior 4.27 of I α,asx +, we obtain I α x 4π 1/2 x 2α+1/4 e 2 x and I α+1 x 4π 1/2 x 2α+3/4 e 2 x Then, xϱxi α x 4π 1/2 x 2α+3/4 e x+2 x 4.45 and xϱxi α+1 x 4π 1/2 x 2α+1/4 e x+2 x Whence, x x, as x + and 2x α x + x x, as x +. Then the condition 4.18 holds. Now, from 4.14, we have ω 1,f = xω,f + x α 1ω,f = ω,xfx + ω,x α 1f x = ω,xf x + ω,x α 1f x, f. 4.47

19 ON 2-ORTHOGONAL POLYNOMIALS OF LAGUERRE TYPE 47 By integrating by parts, we obtain ω,xf x + = xf x xdx = xf x x ] xf x xdx. Since xf x x ] + = and by writing ω 1,f = + fx 1 xdx, we easily obtain that the weight function 1 is given by ] 1 x = x x+x α 1 x = e 1 ϱx [xi α+1 x I α x Note that if we take into account the dependence on the parameter α and setting x = x;α and 1 x = 1 x;α, it may be seen that 1 x;α = x;α+1 x;α. 4.5 References [1] A. Boukhemis and P. Maroni, Une caractérisation des polynômes strictement 1/porthogonaux de type Scheffer. Étude du cas p = 2. [A characterization of strictly 1/p-orthogonal polynomials of Scheffer type. The case p = 2], J. Approx. Theory , no. 1, French. MR 89h:3313. Zbl [2] C. Brezinski, Biorthogonality and its applications to numerical analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol. 156, Marcel Dekker, Inc., New York, MR 92m:411. Zbl [3] C. Brezinski and M. Redivo-Zaglia, On the zeros of various kinds of orthogonal polynomials, Ann. Numer. Math , no. 1-4, 67 78, The heritage of P. L. Chebyshev: a Festschrift in honor of the 7th birthday of T. J. Rivlin. CMP Zbl [4] T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Application, vol. 13, Gordon and Breach Science Publishers, New York, MR Zbl [5] K. Douak, The relation of the d-orthogonal polynomials to the Appell polynomials, J. Comput. Appl. Math , no. 2, MR 97k:337. Zbl [6] K. Douak and P. Maroni, Les polynômes orthogonaux classiques de dimension deux. [Two-dimensional classical orthogonal polynomials], Analysis , no. 1-2, French. MR 93d:334. Zbl [7], Une caractérisation des polynômes d-orthogonaux classiques. [A characterization of classical d-orthogonal polynomials], J. Approx. Theory , no. 2, French. MR 96j:4211. Zbl [8], On d-orthogonal Tchebychev polynomials. I, Appl. Numer. Math , no. 1, CMP Zbl [9], On d-orthogonal Tchebychev polynomials. II, Methods Appl. Anal , no. 4, [1] W. Hahn, Über die Jacobischen Polynome und zwei verwandte Polynomklassen, Math. Zeit , Zbl [11] V. Kaliaguine, On a class of polynomials defined by two orthogonality relations, Math. Sbornik , , English translated in Math. USSR-Sb [12] V. Kaliaguine and A. Ronveaux, On a system of classical polynomials of simultaneous orthogonality, J. Comput. Appl. Math , no. 2, MR 97f:4239. Zbl

20 48 KHALFA DOUAK [13] N. N. Lebedev, Special functions and their applications, Prentice Hall, Inc., Englewood Cliffs, N.J., 1965, Revised English edition.translated and edited by Richard A. Silverman. MR 3#4988. Zbl [14] P. Maroni, L orthogonalité et les récurrences de polynômes d ordre supérieur à deux. [Orthogonality and recurrences of polynomials of order greater than two], Ann. Fac. Sci. Toulouse Math , no. 1, French. MR 97g:4219. Zbl [15], Two-dimensional orthogonal polynomials, their associated sets and the corecursive sets, Numer. Algorithms , no. 1-4, , Extrapolation and rational approximation Puerto de la Cruz, MR 94e:4229. Zbl [16] V. N. Sorokin, A generalization of a classical orthogonal polynomials and the convergence of simultaneous Padé approximants, J. Soviet Math , [17] J. van Iseghem, Vector orthogonal relations. Vector QD-algorithm, J. Comput. Appl. Math , no. 1, MR 89c:6525. Zbl Douak: Laboratoire d Analyse Numérique, Université Pierre et Marie Curie, 4, place Jussieu, 75252, Paris, Cedex 5, France address: douak@ann.jussieu.fr