www.elsevier.com/locate/cam Expansions in series of varying Laguerre polynomials and some applications to molecular potentials J. SÃanchez-Ruiza;b;∗,P.LÃopez-ArtÃesc, J.S. Dehesab;d aDepartamento de Matematicas,à Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganes,à Madrid, Spain bInstituto Carlos I de FÃsica Teoricaà y Computacional, Universidad de Granada, 18071 Granada, Spain cDepartamento de EstadÃstica y Matematicaà Aplicada, Universidad de AlmerÃa, 04120 AlmerÃa, Spain dDepartamento de FÃsica Moderna, Universidad de Granada, 18071 Granada, Spain Received 10 January 2002 Abstract The expansion of a large class of functions in series of linearly varying Laguerre polynomials, i.e., Laguerre polynomials whose parameters are linear functions of the degree, is found by means of the hypergeometric functions approach. This expansion formula is then used to obtain the Brown–Carlitz generating function (which gives a characterization of the exponential function) and the connection formula for these polynomials. Finally, these results are employed to connect the bound states of the quantum–mechanical potentials of Morse and Poschl–Teller,which are frequently used to describe molecular systems. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Varying orthogonal polynomials; Laguerre polynomials; Connection problems; Generalized hypergeometric functions; Morse potential; Poschl–Tellerpotential 1. Introduction Let {!n(x)} be a sequence of weights on ∈ R, and pn;m(x) denote the mth polynomial orthogonal with respect to !n(x), pn;i(x)pn;j(x)!n(x)dx = hn;ii;j: ∗ Corresponding author. Departamento de MatemÃaticas,Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 LeganÃes,Madrid, Spain. Tel.: +34-91-624-9470; fax: +34-91-624-9430. E-mail addresses: [email protected] (J. SÃanchez-Ruiz), [email protected] (P. LÃopez-ArtÃes), [email protected] (J.S. Dehesa). 0377-0427/03/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved. PII: S0377-0427(02)00615-5 1 In recent years, much attention has been paid to the study of the asymptotic behaviour and the distribution of zeros of the sequence of polynomials {pn;n(x)}, which are called orthogonal polyno- mials with varying weights [3,4,6,7,9,12,18,29]. These polynomials are interesting not only because of their own mathematical properties, but also due to their numerous applications in matrix the- ory, numerical analysis, and classical and quantum physics. Indeed, (a) they have been used to prove universality for various spectral statistical quantities arising in random matrix models [9], (b) they play a relevant role in the theory of convergent sequences of interpolatory integration rules [3], as well as in the study of Hermite–PadÃe approximants [7], (c) they have been shown to represent spectral functions of turbulence elds [28], and (d) they are often encountered in the wavefunctions of some quantum–mechanical potentials modelling a wide variety of physical systems [2]. Here we deal with the expansion problem for varying orthogonal polynomials, which, up to our knowledge, has not yet been considered in the literature. We conne ourselves to the expansion problem for Laguerre orthogonal polynomials with linearly varying weights; i.e., the problem of nding the coecients in the expansion ∞ (an+) f(x)= cnLn (x); (1) n=0 (an+) where f(x) is an arbitrary function and the varying Laguerre polynomials Ln (x) are an+ −x (an+) orthogonal with respect to !n(x)=x e . The polynomials Ln (x) with a = −2, which are closely related to the Bessel polynomials [13,14], appear in the wavefunctions of the Morse potential (see Section 4) and have also found application in problems of classical physics [28]. (an+) The polynomials Ln (x) with a = −1 were considered as early as 1952–53 by Toscano [30] and Shively [27]. The latter author pointed out that these polynomials have a generating func- tion of exponential type, so that they are of Sheer A-type zero (see, e.g., [22, Chapter 13]), and this property was later shown to hold for arbitrary complex values of a [5,8]. More recently, the asymptotic behaviour of the polynomials orthogonal with respect to the more general weights n !n(x)=x exp(−ÿnx), where n = n +o(n), ÿn = n +o(n)(; ÿ ¿ 0) has been intensively studied [4,6,18,29]. To solve the expansion problem (1), we shall use the hypergeometric functions method, which has been already employed for the corresponding problem involving classical orthogonal polyno- mials with nonvarying weights [10,11,15,17,23,25]. This is briey outlined in Section 2, where the well-known connection formula for the standard Laguerre polynomials is obtained. In Section 3, a theorem of Verma [31] is used to determine expansions of hypergeometric functions (an+) in series of the varying Laguerre polynomials {Ln (x)}. As a by-product, the poorly known exponential generating function of Brown and Carlitz [5,8] is obtained; moreover, we solve the connection problem between Laguerre polynomials whose parameters are linear functions of the degree. Problems of this kind are very often encountered in the study of the relationships be- tween the wavefunctions of some quantum–mechanical potentials employed for describing phys- ical and chemical properties in atomic and molecular systems. As an illustration, in Section 4 we give explicit relationships among the wavefunctions of the Morse and Poschl–Teller potentials. Finally, in Section 5, some concluding remarks and a few open problems are given. 2 2. The hypergeometric functions approach to series expansions for classical orthogonal polynomials The expansion of a general function f(x) in series of a given system of classical orthogonal polynomials {pn(x)} with nonvarying weights is a long-standing and well-known problem in the theory of special functions. If the polynomial sequence {pn(x)} is orthogonal on an interval ∈ R with respect to the weight function !(x), pn(x)pm(x)!(x)dx = hnn;m; and f(x) admits an expansion of the form ∞ f(x)= cnpn(x); n=0 every Fourier coecient cn can be computed as 1 cn = f(x)pn(x)!(x)dx; hn provided that the integral in the right-hand side does exist and the resulting series satises the appropriate convergence conditions (see, e.g., the detailed discussion of the Jacobi case in [17, Chapter 8]). Other characterizations of the polynomials {pn(x)}, such as their generating functions or recurrence relations, can also be used to compute the expansion coecients. When f(x) and pn(x) can both be expressed in terms of the generalized hypergeometric function, closed analytical expressions for the expansion coecients can often be found by taking advantage of known theorems from the theory of generalized hypergeometric functions. For instance, the following theorem of Fields and Wimp [11] (see also [10, 17, Vol. II, p. 7] for more general results of the same kind), ∞ [a ]; [c ] − n p r [ap]n()n( z) p+rFq+s zx = [bq] n! [bq]; [ds] n=0 n [n + ap];n+ −n; [cr] × p+1Fq z r+1Fs+1 x ; (2) [n + bq] ; [ds] when used together with the hypergeometric representation of Laguerre polynomials, ( +1) −n () n Ln (x)= 1F1 x ; (3) n! +1 leads to the general expansion formula [11] ∞ [ap] [a ] (−z)n [n + ap];n+ +1 p n () pFq zx = p+1Fq z Ln (x): (4) [bq] [bq] n=0 n [n + bq] 3 As rst pointed out by Lewanowicz [15], Eq. (4) includes as a particular case the well-known connection formula for Laguerre polynomials (see, e.g., the proofs in [1, Section 7.1, 22, p. 209, 24]), m (ÿ − ) − L(ÿ)(x)= m n L()(x): (5) m (m − n)! n n=0 We recall that the denition of the generalized hypergeometric function is ∞ [a ] k p [ap]k x pFq x = ; [bq] k! [bq] k=0 k where [ap] and [bq] denote, respectively, sets {a1;a2;:::;ap} and {b1;b2;:::bq} of complex param- eters such that −bj ∈ N0, and we use the contracted notation p q [ap]k = (ai)k ; [bq]k = (bj)k ; i=1 j=1 where (z)n = (z + n)=(z) is Pochhammer’s symbol. When the series in the right-hand side does not terminate after a nite number of terms, it converges provided that either p 6 q,orp = q +1 and |x| ¡ 1. In Eq. (2), the parameters ,[cr], [ÿu], [ds] and [t] are assumed to be independent of n. This formal power series identity holds whenever the series in the left- and right-hand sides are both either terminating or convergent, a remark that also applies to the expansions given in Section 3 below. 3. Expansions in series of varying Laguerre polynomials 3.1. Expansion of general functions We seek for the coecients in expansion (1), where we assume that the function f(x) can be represented in terms of a hypergeometric series. Let us bring here that according to Eq. (3) the varying Laguerre polynomials (an+)( ) have the hypergeometric representation Ln x (an + +1) −n (an+) n Ln (x)= 1F1 x : (6) n! an + +1 To solve this problem we take advantage of the following expansion formula, rst derived by Verma [31]: ∞ [a ]; [b ] − − n p s (h n +1)n−1[bs]n[eu]n( z) p+sFq zx = h n![cq] [cq] n=0 n [n + bs]; [n + eu];h+ n(1 − ) × s+u+1Fq z [n + cq] −1 −n; [ap]; 1+h(1 − ) × F x : (7) p+2 u+2 −1 h − n +1; [eu];h(1 − ) 4 −1 −1 −1 Taking p =1,[ap]=h(1 − ) , u =1,[eu]=1+h(1 − ) , q = r +1,[cq]={[cr];h(1 − ) }, (7) simplies to ∞ [b ] − − n s (h n +1)n[bs]n( z) sFr zx = n![cr] [cr] n=0 n −1 [n + bs];n+1+h(1 − ) ;h+ n(1 − ) × F z s+2 r+1 −1 n + h(1 − ) ; [n + cr] −n ×1F1 x : h − n +1 Equivalently, writing = −a, h = and using (6), we have a general expansion formula in series of varying Laguerre polynomials, ∞ [b ] − n s [bs]n( z) sFr zx = [cr] [cr] n=0 n −1 [n + bs];n+1+(1 + a) ;+ n(1 + a) × F z L(an+)(x); (8) s+2 r+1 −1 n n + (1 + a) ; [n + cr] which in the particular case a = 0 reduces to the expansion formula (4) in series of standard (i.e., nonvarying) Laguerre polynomials.