RENDICONTI LINCEI MATEMATICA E APPLICAZIONI

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1 ATTI ACCADEMIA NAZIONALE LINCEI CLASSE SCIENZE FISICHE MATEMATICHE NATURALI RENDICONTI LINCEI MATEMATICA E APPLICAZIONI Giuseppe Dattoli Derivatio of the Hille-Hardy type formulae ad operatioal methods Atti della Accademia Nazioale dei Licei. Classe di Scieze Fisiche, Matematiche e Naturali. Redicoti Licei. Matematica e Applicazioi, Serie 9, Vol. 14 (2003),.2, p Accademia Nazioale dei Licei < L utilizzo e la stampa di questo documeto digitale è cosetito liberamete per motivi di ricerca e studio. No è cosetito l utilizzo dello stesso per motivi commerciali. Tutte le copie di questo documeto devoo riportare questo avvertimeto. Articolo digitalizzato el quadro del programma bdim (Biblioteca Digitale Italiaa di Matematica) SIMAI & UMI

2 Atti della Accademia Nazioale dei Licei. Classe di Scieze Fisiche, Matematiche e Naturali. Redicoti Licei. Matematica e Applicazioi, Accademia Nazioale dei Licei, 2003.

3 Red. Mat. Acc. Licei s. 9, v. 14:85-90 (2003) Matematica. Derivatio of the Hille-Hardy type formulae ad operatioal methods. Nota di GIUSEPPE DATTOLI, presetata (*) dal Socio C. De Cocii. ABSTRACT. The Hille-Hardy formula is a biliear geeratig fuctio, ivolvig products of Laguerre polyomials. We use the poit of view, developed i previous publicatios, to propose a operatioal method which allows a fairly direct derivatio of this kid of formulae. KEY WORDS: Operatioal methods; Hermite polyomials; Laguerre polyomials; Biliear geeratig fuctios. RIASSUNTO. Derivazioe delle formule di Hille-Hardy e metodi operazioali. La formula di Hille- Hardy è ua fuzioe geeratrice bilieare relativa a prodotti di poliomi di Hermite. I questo lavoro si utilizza il puto di vista sviluppato i precedeti pubblicazioi, per proporre ua derivazioe diretta di tale tipo di formula. 1. INTRODUCTION The Hermite ad Laguerre polyomials ca be respectively derived from the operatioal rules [1] (1) [/2] H (x, y) 4!! r40 y r x 22r r!(22r)! 4exp gy 2 x 2 h x (21) L (x, y) 4!! r y 2r x hg h r r40 (r!) 2 (2r)! 4expg2y x x (2x). x! Both polyomial forms have bee writte by meas of two variables x, y eve though they ca always be reduced to the ordiary oe variable form, amely (2) H (x, y) 4i y /2 H g x 2iky h L (x, y) 4y L g x y h, where H (x) ad L (x) deotes Hermite ad Laguerre polyomials i the caoical forms [2]. The use of a extra variable is however of oticeable importace. The H (x, y) ad L (x, y) ca be, ideed, viewed as the solutios of the followig type of partial differetial equatios (3) y L (x, y) 42 x x x L (x, y) L (x, 0) 4 (2x)! ad (4) y H (x, y) 4 2 x H (x, y) 2 H (x, 0) 4x. It is evidet that the operatioal rules give i eq. (1) are a cosequece of eqs. (*) Nella seduta dell 11 aprile 2003.

4 86 G. DATTOLI (3, 4). Havig clarified these poits, let us ote that the expoetial operator (5) S 4expgy h 2 x 2 allows the formal solutio of the heat equatio (6) y F(x, y) 4 2 F(x, y) 2 x F(x, 0) 4g(x). Accordigly the operatioal solutio (7) F(x, y) 4 S g(x) ca be uderstood i terms of a Gauss type Trasform [3]. I the case i which g(x) 4exp (2x 2 ), the followig relatio (8) S exp (2x 2 ) 4 1 k114y exp g2 h x 2, 114y kow as Glaisher operatioal rule, holds [3, 4]. Eq. (8) has bee a uique tool to derive biliear geeratig fuctios ivolvig Hermite polyomials (see e.g. [4, 5]). I this paper we will take advatage from the properties of the operator (9) T 4expg2y h x x x to derive biliear geeratig fuctios of the Hille-Hardy type [4]. The operator 2. OPERATIONAL CALCULUS AND LAGUERRE DERIVATIVES (10) LD x42 x x x plays a cetral role i the theory of Laguerre polyomials ad is kow as Laguerre derivative [1]. The T operator, defied i the previous sectio, is therefore the expoetial of the Laguerre derivative. Oe of its must importat properties is the followig operatioal idetity [1] (11) T exp (2ax) ya exp g2 ax 12ya h which ca also be viewed as a differet form of the Glaisher rule. The fuctio C 0 (x) 4! (2x) r (12) r40 (r!) 2 is a eigefuctio of the Laguerre derivative [1], so that (13) T C 0 (gx) 4exp (ygx) C 0 (gx).

5 DERIVATION OF THE HILLE-HARDY TYPE FORMULAE AND OPERATIONAL METHODS 87 I passig we ote that C 0 (x) is the 0 th order of the so called Tricomi fuctios (2x) C (x) 4! r (14) r40 r!(1r)! 4x 2/2 J (2kx), where J (x) is the cylidrical Bessel fuctio of first kid. By combiig eqs. (12) ad (13), we fid (15) T [ exp (2bx) C 0 (gx)]4 1 12yb exp g2 h bx2yg C 12yb 0g gx h NybNE1, (12yb) 2 or, what is the same, (16) T [exp (2bx) J 0 (2kgx)]4 1 12yb exp g2 bx2yg 12yb h J 0u 2kgx (12yb) v NybNE1. I the followig sectio we will see how the previous results ca be exploited to treat bilateral geeratig fuctios ivolvig Laguerre polyomials. 3. OPERATIONAL CALCULUS AND LAGUERRE POLYNOMIALS We will itroduce the problem of studyig geeratig fuctios of Laguerre polyomials by usig a fairly direct example. Let us ideed cosider the case G(z, wnt) 4! t (17) 40! L (z, w), the use of eqs. (1), (12) ad (13), allows the followig coclusios G(z, wnt) 4 T! (2twz) (18) 4 T C 40 (!) 2 0 (zt) 4exp (wt) C 0 (zt). We cosider the geeratig fuctio (19) G(x, y; z, wnt) 4! t L (x, y) L (z, w) 40 which ca be recast as (20) G(x, y; z, wnt) 4 T [ exp (2wtx) C 0 (2ztx)]. Therefore, accordig to eq. (15), we fid (NytNE1) (21) G(x, y; z, wnt) 4 1 (wx1yz) t expg2 h C 12ywt 12wyt 0g h 2 zxt (12wyt) 2 Assumig wt4b, 2zt4g, the above relatio reduces to (15), which is essetially, for y4w41, the Hille-Hardy formula. The associated Laguerre polyomials ca be defied by meas of a extesio of eq. (2), which yields (22) L (a) (x, y) 4expy2ygx 2 x 2 1 (a11) x hzk (21) x! l..

6 88 G. DATTOLI By deotig the expoetial operator o the r.h.s. of eq. (22) by T a we ca write the idetity (23) ad T a exp (2xt) 4 1 (12yt) exp g2 h xt NytNE1. a11 12yt (24) T a C a (gx) 4exp (gy) C a (gx). The geeralizatio of eq. (21) is therefore achieved by otig that (25) T a[exp (2bx) C a (gx)]4 1 (12by) exp g2 h bx2gy C a11 12by ag gx h (12by) 2 The use of the above formula allows to recover the complete Hille-Hardy formula. By cosiderig, ideed, the geeratig fuctio (26) G a (x, y; z, wnt) 4! 40! t G(1a11) L (a) (x, y) L (a) (z, w) we fid (NwytNE1) (27) G a (x, y; z, wnt) 4 T (2xt) a! 40 G(1a11) L (a) (z, w) 4 4T a exp (2wxt) C a (2zxt)4 1 (wx1yz) t expg2 h C a11 (12ywt) 12wyt ag h 2 zxt (12wyt) 2 The last term of the above equality is just the Hille-Hardy formula CONCLUDING REMARKS I the previous sectios ad i [5] we have show that operatioal methods may provide us with a uifyig tool for the derivatio of bilateral geeratig fuctios. Just to complete the sceario, we ote that the use of the first eq. (1), alog with the, already metioed, idetificatio of the actio of the operator S i terms of the Gauss trasform, amely S f(x) (x2j) expu 2 (28) 2 v f(j) dj 2kpy 4y 2 allows the proof of the idetity (29)!40 t! H (x, y) L (z, w) 4exp (w 2 t 2 y1wxt)u H C 0g2ztg x 2 1wyt h, z 2 yt 2hv,

7 DERIVATION OF THE HILLE-HARDY TYPE FORMULAE AND OPERATIONAL METHODS 89 where HC (x, y) 4! (21) r H r (x, y) (30) r40 r!(1r)! is the Tricomi Bessel fuctio [1] of the th order. It is evidet that the method we have discussed is fairly powerful ad that it may be exteded to get further results i a direct way. The reaso of iterest for this type of formalism is that distributios of the type P(x, y; a) b e 2 x 12b e 2 y 12b C 0g 2b xy (31) h (12b) 2 appear i problems associated with the statistical properties of chaotic light ad they have bee recetly exploited to study the statistical aspects of the radiatio Spikig i high gai Free Electro Lasers [6]. The relevat mathematical properties have bee recetly discussed i [7], where it has bee show that multidimesioal extesios of (31) are possible, but they require the itroductio of more geeral forms of Laguerre ad Tricomi fuctios, respectively defied as (md2) (32) (21) L (x, ynm) 4!! r y 2r x r r40 (2r)!(r!) m (2x) C ]1, R, m21 ((xnm) 4! r r40 (r!)( 1 1r)!R ( m21 1r)!. The above families of polyomials ad fuctios satisfy iterestig operatioal relatios, which ca be usefully exploited to geeralize the Hille-Hardy formula to the more geeral family of polyomials L (x, ynm) as it will be show i a forthcomig ivestigatio. ACKNOWLEDGEMENTS The author expresses his sicere appreciatio to Prof. P.E. Ricci for clarifyig discussios ad for suggestios improvig the presetatio of the paper. REFERENCES [1] G. DATTOLI, Hermite-Bessel ad Laguerre Bessel Fuctios, a by-product of the moomiality priciple. I: D. COCOLICCHIO - G. DATTOLI - H.M. SRIVASTAVA (eds.), Proceedigs of the Workshop o Special Fuctios ad Applicatios i Mathematics ad Physics (Melfi, 9-12 May 1999). Arace Editrice, Roma [2] L.C. ANDREWS, Special Fuctios for Applied Mathematicias ad Egieers. Mac Milla, New York [3] G. DATTOLI, Geeralized Polyomials, Operatioal Idetities ad their Applicatios. J. Comput. Appl. Math., 118, 2000,

8 90 G. DATTOLI [4] H.M. SRIVASTAVA - H.L. MANOCHA, A Treatise o Geeratig Fuctios. J. Wiley, New York For earlier derivatios by Miller ad Lebedeff, see e.g. A. ERDÉLYI et al., Batema Mauscript project, Vol. 2. [5] G. DATTOLI, Bilateral Geeratig Fuctios ad Operatioal Methods. J. Math. Aal. ad Appl., to appear. [6] S. KRINSKY - R.L. GLUCKSTERN, Aalysis of statistical correlatios ad itesity spikig i the self-amplified spotaeous-emissio free-electro laser. Phys. Rev. ST Accel. Beams, 6(5), 2003, [7] G. DATTOLI - P.E. RICCI, Multi-idex Polyomials ad Applicatios to Statistical Problems. To appear. Perveuta il 18 ottobre 2002, i forma defiitiva l 8 aprile ENEA, Uità Tecico Scietifica Tecologie Fisiche Avazate Cetro Ricerche Frascati C.P FRASCATI RM dattolihfrascati.eea.it