CertainSequencesanditsIntegralRepresentationsinTermsofLaguerrePolynomials

Transcription

1 Global Joural of Sciece Frotier Research: F Mathematics ad Decisio Scieces Volume 5 Issue 5 Versio. Year 5 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic. USA Olie ISSN: & Prit ISSN: Certai Sequeces ad its Itegral Represetatios i Terms of Laguerre Polyomials By Baghdadi Aloui Abstract- I this paper, we itroduce a coectio formula betwee the moomial basis ad the shifted Laguerre basis. As a applicatio, some itegral represetatios i terms of Laguerre polyomials for certai sequeces are obtaied. Keywords: Laguerre polyomials, special fuctios, itegral formulas. GJSFR-F Classificatio : MSC : Primary 33C45; Secodary 4C5 CertaiSequecesaditsItegralRepresetatiosiTermsofLaguerrePolyomials Strictly as per the compliace ad regulatios of : 5. Baghdadi Aloui. This is a research/review paper, distributed uder the terms of the Creative Commos Attributio- Nocommercial 3. Uported Licese permittig all o commercial use, distributio, ad reproductio i ay medium, provided the origial wor is properly cited.

2 Notes Certai Sequeces ad its Itegral Represetatios i Terms of Laguerre Polyomials Baghdadi Aloui Abstract- I this paper, we itroduce a coectio formula betwee the moomial basis ad the shifted Laguerre basis. As a applicatio, some itegral represetatios i terms of Laguerre polyomials for certai sequeces are obtaied. Keywords: Laguerre polyomials, special fuctios, itegral formulas. V V Year 5 Global Joural of Sciece Frotier Research Volume XV Issue ersio I I. Itroductio ad Mai Results By usig some special fuctios ad some particular itegrals, we recall some itegral represetatios for certai iteger or real sequeces. a Some special fuctios The Gamma fuctio is defied by the defiite itegral Γz x z e x dx, Rez >. We ca see directly, that Γ, ad usig itegratio by parts, that Γz + zγz. Notice that, for z N \ {}, the followig formulas hold! Γ +! π Γ +! The Beta fuctio is give i terms of the itegral Bs, t x e x dx, x e x x dx. x s t dx, Res, Ret >. Author: Faculté des Scieces de Gabès, Départemet de Mathématiques, Cité Erriadh, 67 Gabès, Tuisie. Baghdadi.Aloui@fsg.ru.t F 5 Global Jourals Ic. US

3 Certai Sequeces ad its Itegral Represetatios i Terms of Laguerre Polyomials Global Joural of Sciece Frotier Research VolumeYear 5 XV Issue V V ersio I F which is symmetric i s ad t, i.e., Bs, t Bt, s. Notice that, after a chage of variable x, we get +y x s Bs, t dx. + x s+t This fuctio also admits the followig represetatio i terms of the Gamma fuctio [3] Bs, t ΓsΓt Γs + t. I particular, if s ad t are o-zero itegers, the we have!p! + p +! B +, p + x p dx, 3 x dx,, p. 4 + x +p+ The moic Hermite polyomials H x are orthogoal i the iterval, + with respect to the weight fuctio e x ad fulfil the followig orthogoality relatio [] π e x H xh m x dx!δ,m,, m, m where δ,m is the Kroecer delta. The caoical momets, H, of the Hermite polyomials have the represetatio [3] +! b Some other itegrals The Wallis itegral is give by + Γ + H I π By a simple itegratio by parts, we ca obtai I x e x dx,. 5 si x dx,.!π +!, I +! +!,. By the chage of variable t si x, this gives the followig formulas!π +!! +! Now, let cosider the followig itegral x x + dx,, 6 dx,. 7 Ref 3. P. Maroi, Foctios Eulériees, Polyômes Orthogoaux Classiques. Techiques de l'igéieur, Traité Gééralités Scieces Fodametales A 54 Paris, Global Jourals Ic. US

4 Certai Sequeces ad its Itegral Represetatios i Terms of Laguerre Polyomials Notes It is easy to see that T + + T +,. We get by iteratio the two followig formulas + T + + T,. T + T The, by the chage of variable t ta x, we get l + π 4 + π 4 ta x dx, T,. + + x + dx,, 8 + x x dx,, 9 + x V V Year 5 3Global Joural of Sciece Frotier Research Volume XV Issue ersio I with the covetio. We also cosider the followig itegral R x + x dx,. It is easy to see that R + R +,, ad hece the followig formula l + x + x dx,. F Fially, we cosider the itegral B! e x dx,. For, itegratio by parts yields B B, ad we obtai the formula! e! e x dx,.! This gives, after a chage of variable t, the followig relatio! x e x dx,. e! I this paper, we itroduce the followig coectio formula, betwee the moomial {x } ad the shifted Laguerre polyomials, 5 Global Jourals Ic. US

5 Certai Sequeces ad its Itegral Represetatios i Terms of Laguerre Polyomials 4Global Joural of Sciece Frotier Research VolumeYear 5 XV Issue V V ersio I F x + m! t m e t L m tx + dt,, m N \ {}. As a applicatio of our formula, we give the itegral represetatios i terms of Laguerre polyomials for the sequeces give by the equatios. II. Itegral Represetatios i Terms of Laguerre Polyomials Let {L m } be the moic Laguerre polyomial sequece, with parameter m N \ {}, [4] + m! L m x ν ν ν + m! xν,. ν For ay c C ad ay polyomial p, let itroduce i the space of polyomials the liear isomorphism S c, called itertwiig operator, ad give by []: S c px p tx c + c dt. The operator S c ca be characterized taig ito accout its liearity as well as the fact By 3 ad, it is easy to prove that S c x c +!x c,. 3 S x m L m x + m!x m x,. Hece, we ca obtai the followig result. Theorem. For every iteger m N \ {}, the followig formula holds x + m! t m e t L m tx + dt,. 4 Now, as a applicatio of the above formula, we ca express the sequeces give by the equatios by itegral represetatios i terms of Laguerre polyomials. Ideed, substitutig expressio 4 ito, we ca state the followig theorem. Theorem. For every itegers m, ad, p, the followig formulas hold!m! π 4! + m! + m!p! + m! + p +!!p! + m! + p +! t m t m e x+t L m tx + e x+t L m tx + x p t m e t L m tx + t m e t Lm + x +p+ tx + Ref. H. Hochstadt, The Fuctios of Mathematical Physics. Dover Publicatios Ic. New Yor, Global Jourals Ic. US

6 Certai Sequeces ad its Itegral Represetatios i Terms of Laguerre Polyomials +! + m! + Γ +! + m!π! + t m e x +t L m tx + t m e t Lm tx + Notes! + m +! +! t m e t Lm + tx + + m +! l + π + m! m! l +! + m! e + +! with the covetio. Corollary.3 cases t m e t + x Lm + tx + t m e t + x Lm tx + t m e t + x Lm tx + t m e t+x L m tx + For m ad p we have, for every iteger, the followig special V V Year 5 5Global Joural of Sciece Frotier Research Volume XV Issue ersio I F! +!! + π!! +! +! + Γ + +!π + te x+t L tx + t e x+t L tx + x L tx + L + x + tx + te x +t L tx + L tx + 5 Global Jourals Ic. US

7 Certai Sequeces ad its Itegral Represetatios i Terms of Laguerre Polyomials 6Global Joural of Sciece Frotier Research VolumeYear 5 XV Issue V V ersio I F +!! + + l + Remar. Note that, if we tae eve i the fifth formula, we obtai π +! 4 + +! l +! +! e + +! with the covetio.! +!! L + tx + III. Acowledgemets Sicere thas are due to the referee for his/her careful readig of the mauscript ad for his/her valuable commets. Refereces Référeces Referecias. H. Hochstadt, The Fuctios of Mathematical Physics. Dover Publicatios Ic. New Yor, 97.. N. N. Lebedev, Special Fuctios ad their Applicatios. Revised Eglish Editio, Dover Publicatios, New Yor, P. Maroi, Foctios Eulériees, Polyômes Orthogoaux Classiques. Techiques de l'igéieur, Traité Gééralités Scieces Fodametales A 54 Paris, G. Szeg ö, Orthogoal Polyomials, Amer. Math. Soc. Colloq. Publ., Vol. 3, Amer. Math. Soc., Providece, RI, x L + tx + + x L tx + + x L tx + +x L tx + te x +t L tx +. Notes 5 Global Jourals Ic. US