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1 Cover picture: Campanile della Cattedrale,Sec. XII, Melfi (PZ), Italia (Cathedral tower, XII century, Melfi (PZ), Italy)

2 1 Proceedings of the Melfi School on Advanced Topics in Mathematics and Physics PROCEEDINGS OF THE WORKSHOP ADVANCED SPECIAL FUNCTIONS AND APPLICATIONS MELFI (PZ), ITALY, 9 12 MAY 1999 Edited by DECIO COCOLICCHIO Università della Basilicata, Potenza, Italy GIUSEPPE DATTOLI ENEA Frascati, Frascati (RM), Italy HARI M. SRIVASTAVA University of Victoria, Victoria, British Columbia, Canada Aracne Editrice Roma

3 Copyright MM ARACNE EDITRICE S.r.l Roma, via R. Garofalo, 133 tel fax ISBN X I diritti di traduzione, di memorizzazione elettronica, di riproduzione e di adattamento anche parziale, con qualsiasi mezzo, sono riservati per tutti i Paesi. I edizione: marzo 2000

4 Scientific Committe Chairmen G. DATTOLI ENEA Frascati, Frascati (RM), Italy H.M. SRIVASTAVA University of Victoria, Victoria, Canada Scientific Secretary C. CESARANO Universität Ulm, Ulm, Germany ENEA Frascati, Frascati (RM), Italy D. COCOLICCHIO Università della Basilicata, Potenza, Italy INFN Milano, Milano, Italy G.M. MATROIANNI Università della Basilicata, Potenza, Italy D. SACCHETTI Università di Roma La Sapienza, Roma, Italy A. RENIERI ENEA Frascati, Frascati (RM), Italy F. MAINARDI Università di Bologna, Bologna, Italy A. TORRE ENEA Frascati, Frascati (RM), Italy Organizing Committe Chairman G. DATTOLI ENEA Frascati, Frascati (RM), Italy Secretary of the workshop G. BARTOLOMEI ENEA Frascati, Frascati (RM), Italy D. OCCORSIO Università della Basilicata, Potenza, Italy M. QUATTROMINI ENEA Frascati, Frascati (RM), Italy M. VIGGIANO Università della Basilicata, Potenza, Italy

5 List of Participants G. BRICOGNE MRC Lab. of Molecular Biology, Cambridge, England C. CESARANO Universität Ulm, Ulm, Germany ENEA Frascati, Frascati (RM), Italy D. COCOLICCHIO Università della Basilicata, Potenza, Italy INFN Milano, Milano, Italy G. DATTOLI ENEA Frascati, Frascati (RM), Italy M.C. DE BONIS Università della Basilicata, Potenza, Italy G. KORCHMAROS Università della Basilicata, Potenza, Italy C. LAURITA Università della Basilicata, Potenza, Italy F. MAINARDI Università di Bologna, Bologna, Italy G.M. MASTROIANNI Università della Basilicata, Potenza, Italy D. OCCORSIO Università della Basilicata, Potenza, Italy W.A. PACIOREK Global Phasing Ltd., Cambridge, England M. QUATTROMINI ENEA Frascati, Frascati (RM), Italy D. SACCHETTI Università di Roma La Sapienza, Roma, Italy J. SANCHEZ RUIZ Universidad Carlos III de Madrid, Spain A. SONNINO Università della Basilicata, Potenza, Italy H.M. SRIVASTAVA University of Victoria, Victoria, Canada A. TORRE ENEA Frascati, Frascati (RM), Italy M. VIGGIANO Università della Basilicata, Potenza, Italy A. ZARZO Universidad de Granada, Granada, Spain G. BARTOLOMEI ENEA Frascati, Frascati (RM), Italy M. PIEROTTI ENEA Frascati, Frascati (RM), Italy Major N.G. PAGLIUCA Comune di Melfi (PZ), Italy L. BRANCHINI Comune di Melfi (PZ), Italy S. CALABRESE Comune di Melfi (PZ), Italy T. LASALA Comune di Melfi (PZ), Italy E. NAVAZIO Public Library, Melfi (PZ), Italy E.A. NAVAZIO Comune di Melfi (PZ), Italy G. NUCCI Roma News, Napoli, Italy

6 Table of Contents Introduction H.M. SRIVASTAVA A unified theory of mixed generating functions and other related results A. TORRE Theory and applications of generalized Bessel function I A. TORRE Theory and applications of generalized Bessel function II C. CESARANO Theory of Bessel functions with many indices J. SÁNCHEZ RUIZ, P.L. ARTÉS, A. MARTÍNEZ FINKELSHTEIN, J. DEHESA Linearization problems of hypergeometric polynomials in quantum physics P. MARTÍNEZ, R.J. YÁÑEZ, A. ZARZO A study of relativistic Hermite polynomials from the differential equation that they satisfy G. DATTOLI Hermite Bessel and Laguerre Bessel functions: a by product of the monomiality principle F. MAINARDI, R. GORENFLO Fractional calculus: special functions and applications D. SACCHETTI On the functions with absolutely monotonic inverse in zero M.C. DE BONIS, M.G. RUSSO Computation of the Cauchy Principal Value Integrals on the real line

7 C. LAURITA, D. OCCORSIO Numerical solution of the generalized airfoil equation M. QUATTROMINI Unitary, non perturbative integration schemes for differential equations of the Raman Nath type D. COCOLICCHIO Special functions and quantum field theory of the kaon oscillations D. COCOLICCHIO, M. VIGGIANO The squeeze expansion and the dissipative effects in coupled oscillations W.A. PACIOREK Generalized Bessel functions in incommensurate structure analysis G. BRICOGNE Crystallographic moment generating and likelihood functions G. KORCHMAROS, A. SONNINO Coding theory and algebraic geometry

8 Introduction The Special Functions have played and are playing a crucial role in pure and applied Mathematics, in Physics, Engineering and other fields of research involving Mathematics as an operative tool. The history of special functions is so strongly entangled with that of the development of Mathematics itself and of the problems, very often physical problems, that suggested the introduction of specific families of functions going beyond those of elementary nature (trigonometric, logarithmic, exponential...), that it is quite difficult to give a simple picture of their genesis and development. It is also difficult to frame in an univocal way the concept of Special Function itself. Just to make an attempt, we can associate Special Functions with the solutions of particular families of ordinary differential equations with non constant coefficients. During the end of the last century, Sophus Lie pondering on the deep reasons underlying the solution by quadrature of differential equations was led to the notion of group symmetry. This concept inspired the work of Cartan, who was the first to point out that Special Functions can be framed within the context of the Lie theory. This point of view culminated in the work of Wigner who regarded the Special Functions as matrix elements of irreducible representations of Lie groups. This method is certainly interesting and powerful, it provides an elegant and coherent framework for the apparently chaotic wealth of recurrence relations, addition theorems and so on, characterizing the theory of Special Functions. However such a point of view, we will define maximal, is sometimes too general to be useful, in addition it leaves out from the realm of Special Functions classes of functions like the Mathieu or the Elliptic Functions, whose genuine non linear nature can hardly be reconciled with a Lie group approach. On the other side there is a also a minimal point of view, which reduces the special functions to a mere problem of computation. Albeit too pragmatic, this approach has its own importance. Very often, in spite of the 11

9 12 Introduction synthetic properties which can be derived from a general theory, one is faced with the difficulty of computing the function with very high accuracy. During the last few decades new families of special functions have been suggested by problems in various branches of Physics, including Nuclear Physics, Quantum Optics, Synchrotron Radiation, Crystallography and so on. Some of these functions appeared sporadically at the beginning of this century and then have been forgotten. This is indeed the case of multi-variable and multi- index Bessel functions, touched on by Appell and Humbert and then rediscovered by non mathematicians within the context of scattering problems going beyond the dipole approximation. The reason why this class of functions raised initially a brief flurry of interest and then forgotten is perhaps due to the fact that they did not find any specific application and because they were difficult to compute. The recent advances in some branches of Theoretical Physics and in computing power provided a more suitable environment, which stimulated further studies and the consequent framing within a coherent context involving generalized classes of Hermite, Laguerre, Gegenbauer...polynomials. The scope of the workshop on advanced Special Functions and Application is that of giving as much as possible an idea of the state of the art, by trying to define what is meant by advanced, how the relevant theory can be developed and what are the fields in which these new families of functions can be employed. The fields in which non standard forms of special functions have applied or are ideally suited tools, is getting wider and wider and combine with advanced methods linked to integration techniques involving those of operational nature which demand for new classes of functions and the generalization of concepts linked to those associated with monomiality and isospectrality. The workshop has not been limited to new families of special functions but also to new results, involving conventional families, of noticeable importance in pure mathematics, approximation theory and quantum mechanics. The workshop, a joint effort of ENEA and of University of Basilicata at Potenza, took place in Melfi, one of the most fascinating town in southern Italy, for the central role it played in historical events, for its importance in cultural life and for its strategic position as a place of industrial development. The success of the initiative has also been due to the enthusiastic support of the town council who provided not only the financial help, but the necessary encouragements. We owe therefore our gratitude to the major of the town On.le Nicola G. Pagliuca, to Prof. Luigi Branchini responsible of the education department and to dr. Tania Lasala responsible of the culture department.

10 Introduction 13 Finally it is a pleasure to thank Miss Giulia Bartolomei and Mister Maurizio Pierotti for taking care of all the bureaucratic and financial problems connected with the realization of the workshop. Giuseppe Dattoli

11 A UNIFIED THEORY OF MIXED GENERATING FUNCTIONS AND OTHER RELATED RESULTS H.M. Srivastava Department of Mathematics and Statistics University of Victoria Victoria, British Columbia V8W 3P4, Canada E Mail: Abstract Various families of linear, bilinear, bilateral, as well as mixed multilateral generating functions are known to play an important rôle in the investigation of several potentially useful properties and characteristics of the sequences which they generate. With this objective in view, in this two part series of lectures, we present some recent developments in the theory of generating functions for the classical orthogonal polynomials including, for example, the Jacobi polynomials (which indeed contain, as their special cases, the Gegenbauer or ultraspherical polynomials, the Legendre or spherical polynomials, and the Chebyshev polynomials of the first and second kinds), the Laguerre polynomials, and the Hermite polynomials. We also consider a family of multivariable polynomial expansions which provide unifications and generalizations of some remarkable product formulas of Harry Bateman ( ) for the Jacobi polynomials. Relevant historical remarks and observations, and connections with other works on applications of some of the results considered in these lectures, are also presented rather briefly. 1. Basic Definitions and Preliminary Results The family of the classical orthogonal polynomials forms a special type of the orthogonal system of functions {φ n (x)} n=0 for which the inner product (φ m,φ n ):= b a φ m (x) φ n (x) dµ(x) =λ n δ m,n (1.1) (m, n N 0 := N {0}; N := {1, 2, 3, }), 15

12 16 H.M. Srivastava where δ m,n is the Kronecker delta, (a, b) is a finite, one sided infinite, or two sided infinite interval on the real axis, and dµ(x) is a distribution along that interval. Here λ n = φ n 2 := (φ n,φ n ) (n N 0 ) (1.2) and µ(x) is a non decreasing function; if µ(x) is absolutely continuous, we may set µ (x) =w(x), and then refer to w(x) as the weight function of the orthogonal system {φ n (x)} n=0. This family is led by the Jacobi polynomials P n (α,β) (x) which indeed are the most general of the aforementioned three classes of orthogonal polynomials. These polynomials may be defined explicitly by P (α,β) n (x) = or, equivalently, by P (α,β) n (x) = n k=0 ( α + n n k ( ) α + n n )( β + n k 2F 1 )( ) x 1 k ( ) x +1 n k (1.3) 2 2 n, α + β + n +1; α +1; 1 x, (1.4) 2 where 2 F 1 is the Gaussian hypergeometric function which corresponds to a special case p 1=q = 1 of the generalized hypergeometric function p F q (with p numerator and q denominator parameters) defined by pf q (α 1,,α p ; β 1,,β q ; z) = p F q α 1,,α p ; z β 1,,β q ; (1.5) (α 1 ) n (α p ) n z n := (β 1 ) n (β q ) n n! (p, q N 0 ; p q +1; p<q+ 1 and z < ; p = q + 1 and z U:= {z : z C and z < 1}; p = q +1, z U, and R(ω) > 0), where, for convenience, (λ) n := Γ(λ + n)/γ(λ), in terms of Gamma functions, and q p ω := β j α j, (1.6) j=1 provided (of course) that no zeros appear in the denominator of (1.5). Clearly, since ( 1) n N! (n =0, 1,,N) ( N) n = (N n)! (1.7) 0 (n = N +1,N +2,N +3, ), n=0 j=1

13 A unified theory of mixed generating functions and other related results 17 the series in (1.5) would terminate when one (or more) of the numerator parameters α 1,,α p is zero or a negative integer, and then the question of convergence of the series will not arise. Thus, if one of α 1,,α p is a nonpositive integer N, and there are no zeros in the denominator of (1.5), the function p F q (z) would reduce to what may be called a hypergeometric polynomial of degree N in z. For such a hypergeometric polynomial, it is not difficult to show from the definition (1.5) that p+1f q n, α 1,,α p ; z = (α 1) n (α p ) n ( z) n β 1,,β q ; (β 1 ) n (β q ) n q+1 F p n, 1 β 1 n,, 1 β q n; 1 α 1 n,, 1 α p n; ( 1) p+q (n N 0 ), z (1.8) which can be applied to rewrite the hypergeometric representation (1.4) in the form: ( )( ) P n (α,β) α + β +2n x 1 n (x) = n 2 (1.9) n, α n; 2 2F 1. α β 2n; 1 x When min{r(α), R(β)} > 1, these polynomials are orthogonal with respect to the Beta distribution on [-1,1]: 1 1 (1 x) α (1 + x) β P (α,β) m (x) P n (α,β) (x) dx (1.10) = 2α+β+1 Γ(α + n + 1) Γ(β + n +1) n! (α + β +2n + 1) Γ(α + β + n +1) δ m,n (m, n N 0 ; min{r(α), R(β)} > 1). Various other members of the family, which are special cases of the Jacobi polynomials, include the Gegenbauer (or ultraspherical) polynomials Cn(x), ν where ( ) C α+ 1 2 α + n 1 ( ) 2α + n n (x) = P (α,α) n n n (x) = n k=0 (α ) k(α ) n k k! (n k)! e i(n 2k)θ (x = cos θ), the relatively more familiar Legendre (or spherical) polynomials: (1.11) P n (x) =P (0,0) n (x) =C 1 2 n (x), (1.12)

14 18 H.M. Srivastava and the Chebyshev polynomials (of the first and second kinds): T n (x) = U n (x) = 1 2 ( n 1 2 n ( n n +1 ) 1 P ( 1 2, 1 2 ) n (x) = 1 2 nc0 n(x), ) 1 P ( 1 2, 1 2 ) n (x) =Cn(x), 1 (1.13) where, by definition, { } Cn(x) 0 = lim λ 1 C λ λ 0 n(x). (1.14) Two other important members of the family of the classical orthogonal polynomials are the Hermite polynomials: H n (x) = [n/2] k=0 ( 1) k n! k! (n 2k)! (2x)n 2k =(2x) n 2F n, 1 2 n ; ; 1 x 2 (1.15) and the Laguerre polynomials: L (α) n (x) = n k=0 ( ) α + n ( x) k n k k! = ( ) α + n n 1F 1 n; α +1; x. (1.16) Indeed, since and H n (x) =( 1) n 2 n/2 n! { lim α { L (α) n (x) = lim β α n/2 L (α) n P (α,β) n ( α + x )} 2α (1.17) ( 1 2x )}, (1.18) β many of the properties of the Hermite and Laguerre polynomials can be deduced from those involving the classical Jacobi polynomials. Another interesting class of orthogonal polynomials is provided by the generalized Bessel polynomials: n ( )( ) ( ) n α + n + k 2 x k y n (x, α, β) = k! k k β k=0 = 2 F 0 n, α + n 1; ; x, β (1.19)

15 A unified theory of mixed generating functions and other related results 19 which were studied systematically by Krall and Frink [37] (and, subsequently, by Grosswald [32]). In view of the relationships: { ( n! y n (x, α, β) = lim P n (λ 1,α λ 1) 1+ 2λx )} (1.20) λ (λ) n β and ( y n (x, α, β) =n! x β ) n L (1 α 2n) n ( β x ), (1.21) the Bessel polynomials are also recoverable from the classical Jacobi and Laguerre polynomials. The so called classical Jacobi, Laguerre, and Hermite polynomials, and many of their aforementioned relatives, are often characterized by one or the other of a number of interesting and useful properties which they are known to have in common. The subject of orthogonal polynomials is treated, in a lucid and systematic manner, by Szegö [71]. Other useful references on this subject include Erdélyi et al. [26, Chapter 10], Sansone [51], Rainville [47], Abramowitz and Stegun [1, Chapter 22], Magnus et al. [39, Chapter 5], Askey [4], Luke [38, Chapter 11], Chihara [24], Brezinski et al. [11], Nikiforov et al. ([42] and [43]), and Nevai [41]. In this two part series of lectures, we aim at presenting some recent developments in the theory of generating functions and polynomial expansions associated with some of the aforementioned polynomials as well as with various other systems in one and more variables. 2. A Class of Mixed Generating Functions For the classical Laguerre polynomials defined by (1.16), the linear generating functions: and n=0 ( L (α) n (x) t n =(1 t) α 1 exp xt ) 1 t n=0 ( t < 1) (2.1) L (α n) n (x) t n =(1+t) α exp( xt) ( t < 1) (2.2) are well known in the mathematical literature (cf., e.g., Erdélyi et al. [26], Rainville [47], and McBride [40]). An interesting unification (and generalization) of a number of generating functions for the classical Laguerre polynomials, including (for example) (2.1) and (2.2), was given by Carlitz [16] in the form: L (α+λn) n (x) t n (1 + v)α+1 = exp( xv), (2.3) 1 λv n=0

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