Research Article Sheffer and Non-Sheffer Polynomial Families

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1 Iteratioal Mathematics ad Mathematical Scieces Volume 22, Article ID , 8 pages doi:.55/22/ Research Article Sheffer ad No-Sheffer Polyomial Families G. Dattoli, B. Germao, 2 M. R. Martielli, 2 adp.e.ricci 3 Gruppo Fisica Teorica e Matematica Applicata, Uità Tecico Scietifica Tecologie Fisiche Avazate, ENEA-Cetro Ricerche Frascati, C.P. 65, Via Erico Fermi 45, 44 Frascati, Rome, Italy 2 Dipartimeto di Metodi e Modelli Matematici per le Scieze Applicate, Sapieza Uiversità di Roma, Via A. Scarpa 4, 6 Roma, Italy 3 Iteratioal Telematic Uiversity Uiettuo, Corso Vittorio Emauele II, Rome, Italy Correspodece should be addressed to P. E. Ricci, riccip@uiroma.it Received 2 March 22; Accepted 25 May 22 Academic Editor: Taekyu Kim Copyright q 22 G. Dattoli et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. By usig the itegral trasform method, we itroduce some o-sheffer polyomial sets. Furthermore, we show how to compute the coectio coefficiets for particular expressios of Appell polyomials.. Itroductio The Sheffer polyomials are defied through the geeratig fuctio as follows: t! s x A t e xb t,. where A t ad B t are two aalytic fuctios of the parameter t. The above families of polyomials have bee show to be quasimoomials uder the actio of the operators: P B x, M xb x A x A x,.2

2 2 Iteratioal Mathematics ad Mathematical Scieces ideed, beig D x the ordiary derivative, we easily prove that Ps x s x, Ms x s x..3 The polyomials, σ,α x, specified by the geeratig fuctio: t! σ,α x A t xb t α.4 do ot belog to the Sheffer family. We ca, however, take advatage from the idetity: a α e ξa ξ α dξ.5 to prove that they are liked to the s x by the itegral trasform: σ,α x e ξ s ξx ξ α dξ..6 Before discussig the problem i geeral terms, we cosider the case i which A t e yt2, B t t..7 The use of the previous procedure yields η,α e ξ H ( ξx, y ) ξ α dξ..8 Beig the Hermite-Kampé defériet polyomials defied by /2 ( ) H x, y! x 2r y r 2r!r!,.9 we fid the explicit expressio: η,α! /2 Γ 2r α x 2r y r.. 2r!r!

3 Iteratioal Mathematics ad Mathematical Scieces 3 By otig that the moomiality operators associated with the H x, y are simply give by 2 as P x, M x 2y x,. we obtai the followig recurreces: x η,α αη,α ), y η,α η 2,α ),.2 η,α αxη,α 2 yη,α ). We ca further exted the previous defiitio by addig a extra variable such that η,α β! β α e βξ H ( ξx,y ) ξ α dξ /2 Γ 2r α ( x/β ) 2r y r, 2r! r!.3 thus fidig that y 2 β η,α( x, y β 2 x η,α β )..4 Accordig to the previous example, we have itroduced aew family of polyomials, with otrivial properties, ad applicatios just startig from the correspodig Sheffer family ad by exploitig the wealth of properties that such a family possesses. I the secod part of this paper we show how to costruct particular coectio coefficiets relevat to several Sheffer polyomial sets, icludig multivariable Hermite, Legedre, ad Laguerre polyomials, ad, i geeral, Appell-type polyomial sets. 2. Further No-Sheffer Polyomial Sets It is quite straightforward to see that, iasmuch B t t, thesheffer polyomials are essetially Appell polyomials 3, sice it always happes that the moomial operator P coicides with the ordiary derivative ad therefore the associated polyomials satisfy the recurrece reported i the first of equatios.2. I the more geeral case, the situatio is more iterestig. For example, if A t, B t e t, 2.

4 4 Iteratioal Mathematics ad Mathematical Scieces the moomiality operators, associated with the relevat Sheffer form, amely, the Bell polyomials be x,are P l x, M x x, be x S 2, k x k, k 2.2 with S 2, k beig the Stirlig umbers of secod kid. The correspodig β,α x polyomials are therefore β,α x S 2, k Γ k α x k, k 2.3 ad a recurrece satisfied by the above family is give by β,α x xαβ,α x x x β,α x. 2.4 The Lagrage polyomials are characterized by the geeratig fuctio strictly speakig the Lagrage polyomials, as curretly defied i the literature, are g,α,β x, y /! λ α,β x, y i 4 : t! λ α,β ( ) x, y xt α[ yt ]. β 2.5 They belog therefore to the family.4 with A t [ yt ] β, B t t. 2.6 It is, therefore, evidet that the techique we have proposed allows a geeral tool to frame the theory of polyomial sets i a straightforward cotext, which permits a atural uderstadig of their properties. 3. Special Coectio Coefficiets for Hermite, Gould-Hopper, ad Laguerre Polyomials I the previous sectio, we have discussed differet families of polyomials which ca be framed withi a commo thread. The poit to be clarified is whether they ca be exploited to obtai coveiet expasios.

5 Iteratioal Mathematics ad Mathematical Scieces 5 We itroduce the topics we will discuss i the secod part of this paper, by cosiderig the followig expasio, ivolvig the two variable Hermite polyomials H x, y : ( ) ( ) ( ) H px, qy h m, p, q, y Hm x, y. 3. The problem to be solved is that of derivig the coefficiets h m, p, q, y of this expasio. The use of the idetities: e y 2 x x ( ) H x, y, ( ) ( ) e z 2 x H x, y H x, y z 3.2 allows to recast 3. i the followig form: ( H (px, q p 2) ) y ( ) h m, p, q, y x m. 3.3 Let x e iϑ so that we fid from 3.3 : ( ) 2π ( h m, p, q, y e imϑ H (pe iϑ, q p 2) ) y dϑ 2π! pm[( q p 2) y ] m /2, m! m /2! 3.4 where m eve. The whole procedure is based o the possibility of coectig differet forms of Hermite polyomials usig a expoetial differetial operator. Accordig to the above result, we ca cosider ow the Gould-Hopper polyomials i 2, 5, 6 : e y d x x H d ( ) x, y, H d /d ( ) x dr y r x, y! dr! r!. 3.5 I this case, for the coectio coefficiets h d m, p, q, y, such that H d ( px, qy h d m ( d ( ), p, q, y H x, y, 3.6 m

6 6 Iteratioal Mathematics ad Mathematical Scieces we obtai the expressio h d m ( ) p m[( q p d) y ] m /d, p, q, y!, 3.7 m! m /d! where m dk, k iteger. I the followig, we will discuss a geeral procedure, based o ideas aalogous to that employed so far, allowig the derivatio of the coectio coefficiets of the type 3. ivolvig differet forms of polyomials. The two variable Laguerre polyomials are defied as ( ) r x r y r L x, y! r! 2 r!, L e y L D x [ x r r! ], 3.8 e z L D x L L z ), L D x : x x x, where L D x deotes the Laguerre derivative. We ca, therefore, use the same procedure outlied for the Hermite polyomials to derive the coefficiets of the expasio: ( ) ( ) ( ) L px, qy l m, p, q, y Lm x, y, 3.9 by otig that, o accout of 3.8, the followig idetity holds e y L D x L ( px, qy L ( px, ( q p ) y ( ) [ x m ] l m, p, q, y, 3. m! which yields ( )( ) m [( ) ] m ( ) p q p y l m, p, q, y. 3. m m! I the cocludig sectio, we will discuss alterative derivatios of 3.. The previous idetities have bee proposed as a example aimed at provig the reliability of the proposed method, which is further developed i the last sectio.

7 Iteratioal Mathematics ad Mathematical Scieces 7 4. The Case of Legedre ad Sheffer Polyomials The Legedre polyomials ca be writte i terms of Hermite polyomials accordig to the idetity from 7 : P x! π e s s /2 H 2sx, s ds, 4. ad therefore they are essetially σ polyomials of the type.6. O accout of 3. ad of the fact that the ordiary Hermite polyomials are liked to their two variable couterpart by H 2x, H x, 4.2 ca also be writte as p r P x! π p r H r x, e s s /2 h r, s, s, ds, 4.3 the coefficiets h r, s, s, beig defied by 3.4. The above expasio holds for geeral forms of Appell polyomials: a ( ) ( px C m, p a m x, a m x A x x m. 4.4 I geeral, the above family of polyomials ca be defied through the formal series: a x α r r! x r. 4.5 Alog with the polyomials, we ca defie their couterpart as a x A x x. 4.6 Therefore, we get A x a ( ) ( ) px C m, p x m. 4.7

8 8 Iteratioal Mathematics ad Mathematical Scieces By itroducig the â operator, defied by [ ] â x a x, 4.8 ad by otig that A x a [ ] x a â x α r r! a r x, 4.9 we obtai A x a ( ) ( [ ]) px a p â x, 4. ad lastly ( ) 2π C m, p 2π ( [ e imϑ a p â ( e iϑ)]) dϑ. 4. I a forthcomig more exteded paper, we will recosider the topics developed i this paper by studyig ew expasios of fuctios i terms of Sheffer ad o-sheffer polyomial families. Refereces P. Blasiak, G. Dattoli, A. Horzela, ad K. A. Peso, Represetatios of moomiality priciple with Sheffer-type polyomials ad boso ormal orderig, Physics Letters. A, vol. 352, o. -2, pp. 7 2, G. Dattoli, Hermite-Bessel ad Laguerre-Bessel fuctios: a by-product of the moomiality priciple, i Advaced Special Fuctios ad Applicatios, vol. of Proceedigs of the Melfi School o Advaced Topics i Mathematics ad Physics, May 999, pp , Arace Editrice, Rome, Italy, 2. 3 L. C. Adrews, Special Fuctios of Mathematics for Egieers, McGraw-Hill, New York, NY, USA, 2d editio, H. M. Srivastava ad H. L. Maocha, A Treatise o Geeratig Fuctios, Wiley, New York, NY, USA, P. Appell ad J. Kampé defériet, Foctios Hypergéométriques et Hypersphériques. Polyômes d Hermite, Gauthier-Villars, Paris, Frace, H. W. Gould ad A. T. Hopper, Operatioal formulas coected with two geeralizatios of Hermite polyomials, Duke Mathematical Joural, vol. 29, pp. 5 63, G. Dattoli, B. Germao, M. R. Martielli, ad P. E. Ricci, A ovel theory of Legedre polyomials, Mathematical ad Computer Modellig, vol. 54, o. -2, pp. 8 87, 2.

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