View metadata, citation and similar papers at core.ac.uk brought to you byCORE provided by Elsevier - Publisher Connector JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS ELSEVIER Journal of Computational and Applied Mathematics 70 (1996) 279-295 The relation of the d-orthogonal polynomials to the Appell polynomials Khalfa Douak Universitb Pierre et Marie Curie, Laboratoire d'Analyse Numbrique, Tour 55-65, 4 place Jussieu, 75252 Paris Cedex 05, France Received 10 November 1994; revised 28 June 1995 Abstract We are dealing with the concept of d-dimensional orthogonal (abbreviated d-orthogonal) polynomials, that is to say polynomials verifying one standard recurrence relation of order d + 1. Among the d-orthogonal polynomials one singles out the natural generalizations of certain classical orthogonal polynomials. In particular, we are concerned, in the present paper, with the solution of the following problem (P): Find all polynomial sequences which are at the same time Appell polynomials and d-orthogonal. The resulting polynomials are a natural extension of the Hermite polynomials. A sequence of these polynomials is obtained. All the elements of its (d + 1)-order recurrence are explicitly determined. A generating function, a (d + 1)-order differential equation satisfied by each polynomial and a characterization of this sequence through a vectorial functional equation are also given. Among such polynomials one singles out the d-symmetrical ones (Definition 1.7) which are the d-orthogonal polynomials analogous to the Hermite classical ones. When d = 1 (ordinary orthogonality), we meet again the classical orthogonal polynomials of Hermite. Keywords: Appell polynomials; Hermite polynomials; Orthogonal polynomials; Generating functions; Differential equations; Recurrence relations AMS classification: 42C99; 42C05; 33C45 O. Introduction The orthogonal polynomials in general and the classical orthogonal polynomials in particular have been the object of extensive works. They are connected with numerous problems of applied mathematics, theoretical physics, chemistry, approximation theory and several other mathematical branches. In particular, their applications are being widely used in theories as Pad~ approximants, continued fractions, spectral study of SchriSdinger discrete operators, polynomial solutions of second-order differential equations and others. The notions of d-dimensional orthogonality for polynomials [14], vectorial orthogonality as defined and studied in [19] or simultaneous orthogonality in [7] are obviously generalizations of 0377-0427/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0377-0427(95)00211-1 280 K. Douak/Journal of Computational and Applied Mathematics 70 (1996) 279-295 the notion of ordinary orthogonality for polynomials. Such polynomials are characterized by the fact that they satisfy an order d + 1 recurrence relationship, that is a relation between d + 2 consecutive polynomials [19]. All these new notions of d-orthogonality for polynomials and, equivalently, 1/d-orthogonality [5] appear as particular cases of the general notion of bior- thogonality studied in [6]. Recently they have been the subject of numerous investigations and applications. In particular, they are connected with the study of vector Pad6 approximants [19, 20], simultaneous Pad6 approximants [-7] and other problems such as vectorial continued fractions, polynomials solutions of the higher-order differential equations, spectral study of multidiagonal nonsymmetric operators [4]. Otherwise, according to a point of view based only upon the d-orthogonality conditions, especially the (d + 1)-order recurrence relation, some studies which look for generalizations of the orthogonal polynomials properties in general and the classical orthogonal polynomial ones in particular, are made in [-5, 8-11, 15]. In this paper we present the results in this direction. It is well known that the Hermite polynomials (up to a linear change of variable) form the only sequence of polynomials that are simultaneously orthogonal and Appell polynomials. This charac- terization of the Hermite polynomials was first given by Angelesco [2], and later by other authors (see, e.g., [18] and for additional references [-1]). Let {H,}, > o be the sequence of Hermite polynomials and {H,}, ~ o the corresponding monic polynomials, i.e.,/~,(x) = 2-"H,(x), n >~ 0. The previous property is translated by the fact that the sequence {/4,}, ~ o verifies: (a) the three-term recurrence relation H,+2(x)=xH,+,(x)-½(n+l)/~,(x), n>~0; /4o(X) = 1, /41(x) = x and (b) the Appell character /~'+l(x) = (n + 1)/~,(x), n >/0. Our main objective in this paper is the investigation of sequences {P,}. ~ o of monic polynomials which are simultaneously Appell polynomials and d-orthogonal. Section 1 is devoted to some preliminaries and notations necessary for the sequel. In particular, the exposition of the definition and characterizations of the d-orthogonal polynomials. In Section 2, we state and solve the problem (P). The proof is based only on the (d + 1)-order recurrence where an explicit expression for each of its elements is obtained. In Section 3, we give some characteriza- tions of these polynomials. First, by using the (d + 1)-order recurrence and the Appell property, we obtain a generating function and a (d + 1)-order differential equation of the polynomials P.(x), n/> 0. Next, by introducing the d-dimensional functional with respect to which the sequence {P,}, ~ o is d-orthogonal, we give a characterization of this novel sequence through a vectorial functional equation. Lastly, we express the derivative of the product of two consecutive poly- nomials in the form of a differential relation which generalizes McCarthy's characterization of a Hermite classical polynomials. K. Douak/Journal of Computational and Applied Mathematics 70 (1996) 279-295 281 I. Preliminaries and notations Let ~ be the vector space of polynomials with coefficients in C and let ~' be its dual. We denote by (u, f ) the effect of u ~ 2' onf~ 2. In particular, we denote by (u), =(u, x"), n >/O, the moments of the functional u. Since a linear functional is uniquely determined by its action on a basis, u is uniquely determined by the sequence of constants (u),. By a polynomial set (PS), we mean a sequence of monic polynomials {P,},>~ 0 in which deg P,(x) = n for all n, say, P,(x) = x" + .... n >~ O. Let {P,}, >~ o be a polynomial set; there exists a sequence of linear functionals {u,}, >~ o, such that (Urn, P,) = 6,.,,, m, n >70. (1.1) The sequence {u,}, ~> o is called the dual sequence of {P,}, >~ o; it is unique [14]. Lemma 1.1 (Maroni [14]). For each u ~ ~' and p >1 1 integer, the followin9 two propositions are equivalent: (a) <u, Pp-I> ¢ O; (u, Pn> = O, n/> p; (b) 32~eC, O~<v~<p-l, 2p_l #O, suchthat p-1 u = • 2~Uv. (1.2) v=O We now consider a polynomial set {P,},~> o with its derivative sequence {Q.}. ~ o defined by Q,(x) = [1/(n + 1)] P,] +1 (x), n >/0. We denote by {v,}, ~ o the dual sequence of {Q.}, ~> o. Proposition 1.2 (Maroni [14]). We have Dv,=-(n+l)u,+l, n>~0, (1.3) where Du, the derivative of a linear functional u, is defined by (Du, f ) := -- (u, f' ), V u ~ 2' and Vf~ 2. 1.1. The regular orthogonality Let us recall the definition of regular orthogonality. The linear functional u will be called regular if we can associate with it a sequence of polynomials {P,}, ~ o such that (u,P,P,,)=r,6,,m, n,m>>,O, r,v~O, n>~O. Then for all n the degree of P,(x) is exactly n and we can always suppose each P,(x) to be monic. Therefore, the PS {P,}. ~>o is unique. In this case, it is called the orthogonal polynomial set (OPS) relative to u. Necessarily, u = 2Uo, 2 ¢ 0. It is an old result that such OPSs are characterized by the fact that they verify a three-term recurrence relation, namely: Pn+2(x) = (x -- fl,+l)P.+l(x) - 7.+lP,(x), n >~ 0 (monic form); (1.4) Po(x) = 1, Pl(x) = x -- flo; (1.5) 282 K. Douak/Journal of Computational and Applied Mathematics 70 (1996) 279-295 with 7.+a #0, n~>0. Definition 1.3. The OPS {P.}. ~ o will be called classical if {Q,}. ~>o is also an OPS [13]. This gives (1.4), (1.5) and Qn+2(x) = (x - o~n+a)Qn+x(X) - ~.+IQn(x), n >i 0; (l.6) Qo(x) = 1, Qx(x) = x - ~o; (1.7) with 6n+1#0, n >>.O. Theorem 1.4 (Geronimus [12]). The sequence {P,}, ~ o, orthog °nal with respect to u, is classical if and only if there exist two polynomials dp and ~b with deg ~b = t ~< 2, deg ~k = 1, such that @u + D(dpu) = 0 (d? monic), writing ~(x) = atx + ao, we must have alCN* if t = 2. 1.2. The d-orthogonality Let us consider d linear functionals F 1, ..., F n (d >f 1). Definition 1.5. A polynomial set {P,}, ~>o is called a d-dimensional orthogonal polynomial set or simply a d-orthogonal polynomial set (d-OPS) with respect to F = (F 1, ..., Fd) x if it fulfils [19]: (F~,Pmp,) =O, n>~md +~, m>~O, (1.8) (F~,P,,P,,,d+~_I) 4=0, m>~O, (1.9) for each integer e with 1 ~< c~ ~< d. Remark 1.6. The inequalities (1.9) are the regular conditions equivalent to the ones given in [14, p. 110] or [20, p. 142]. They are a natural generalization of those given in the regular orthogonality case. In this case, the d-dimensional functional F is called regular. It is not unique. Indeed, from Lemma 1.1, we have ~-1 v+l F ~ ~ 2~u~, 2~-l#0, l<~e<<.d ~ u~= ~v,F, '/Sv+ 1 #0, O<<.v<<.d 1.