A note on Generalized Hermite polynomials

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1 INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS AND INFORMATICS Volue 8 14 A oe o Geeralized Herie poloials Cleee Cesarao Absrac B sarig fro he geeral heor of he oevariable Herie poloials we will iroduce a bidiesioal geeralizaio of he ha is useful o obai a differe approach wih he haroic oscillaor fucios. We will see soe ieresig properies of his class of Herie poloials ad we also discuss he relaed applicaios o he paricular parial differeial equaios. Kewords Herie Poloials Geeraig Fucios Orhogoal Poloials Special Fucios Haroic oscillaor. I. INTRODUCTION N previous papers [1] we have discussed differe kids I of he Kapé de Ferié Herie poloials ad soe exaples of heir applicaios [34]. The approach we have used o iroduce he differe failies of hese ipora orhogoal poloials has bee exreel varied. We wa ow iroduce he ordiar oe-variable Herie poloials ad he relaed geeralizaio of wo-variable b usig he foralis ad he echiques of he expoeial operaors. If we cosider a fucio f( x ) which is aalic i a eighborhood of he origi i ca be expaded i Talor series ad i paricular we ca wrie: λ f x+ f x (1) ( ) ( λ) ( )! where λ is a coiuous paraeer. The so-called shif or raslaio operaor ca be defied as: λ λ d λ d e f x f x f x f x d dx ( ) ( ) ( ) ( ) ( + λ)! dx! dx () liiig ourselves o real doai ad b assuig ha λ is a real uber ad f( x ) is also aalic i x + λ wihou a oher resricio. The acio of he expoeial operaor o a aalic fucio f( x ) produces a shif of he variable x b λ. The wo-variable Herie poloials ca be defied b usig he relaio saed i (). Afer oig ha: D ( ) e f( x) f( x ) f ( x)! we have: + (3) D f( x) x iplies e x ( x+ ) (4) D f( x) ax iplies e f( x) a ( x+ ) (5) The above procedure ca be easil geeralized o expoeial operaors coaiig higher derivaives. I fac b cosiderig he secod derivaive we ge: D e f x f x (6) ( ) ( ) ( )! ad b oig ha:! D x x ( )! we have: D! (7) e x x. (8)! ( )! We ca ow defie he wo-variable Herie poloials H () ( x ) of Kapé de Férie for [56] b he followig forula: r r () x H ( x )!. (9) r!( r)! r I is ipora o oe ha assuig: f( x) a x (1) he geeral idei reads: C. Cesarao ad Dario Assae are wih he Facul of Egieerig Ieraioal Teleaic Uiversi UNINETTUNO Roe Ial (phoe: ; e-ail: c.cesarao@uieuouiversi.e). D () ( ) ( ) e f x ah x (11) ISSN:

2 INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS AND INFORMATICS Volue 8 14 ad he we ca iediael obai: H (1) ( x ) ( x+ ) (1) which ca also be recas i he for: D (1) e f( x) ah ( x ). (13) I he followig we will idicae he wo-variable Herie poloials of Kapé de Férie for b usig he sbol H ( ) x isead ha H () ( x. ) The wo-variable Herie poloials H ( x ) are liked o he ordiar Herie poloials b he followig relaios [5]: 1 H x He( x) where: r (14) r r ( 1) x He ( x)! (15) r r!( r)! x x H ( x ) e H ( x) e x. () The geeraig fucio [7] of he above wo-variable Herie poloials ca be sae i a was we have for isace afer oig ha he solved he followig differeialdifferece equaio: d Y ( z ) ay ( z ) + b ( 1) Y ( z ) dz Y () δ where a ad b are real ubers ha b exploiig he geeraig fucio ehod seig: (1) Gz (;) Y () z ()! wih coiuous variable we ca rewrie he equaio (1) i he for: d G (;) z ( a + b ) G (;) z dz. (3) G( ) 1 ad: ( ) H x 1 H ( x) (16) where: r r r ( 1) ( x) H ( x)!. (17) r!( r)! Fiall i is also ipora o oe ha he Herie poloials H( x ) saisf he relaio: H ( x) x. (18) Fro he above relaios we ca deduce ha he geeralized Herie poloials saisf he followig parial differeial equaio: H( x ) H( ) x. (19) x This resul help us o derive a ipora operaioal rule for he wo-variable Herie poloials. I fac b cosiderig he differeial equaio (19) as liear ordiar i he variable ad b readig he equaio (18) we ca iediael sae he followig relaio: ha is a liear ordiar differeial equaio ad he is soluio reads: G( x; ) exp( x + ) (4) where we have pued az x ad bz. Fiall b exploiig he r.h.s of he previous relaio we fid he geeraig fucio of he geeralized Herie poloials H( x ) ha is: exp( x ) H ( x )! +. (5) II. HERMITE POLYNOMIALS WITH TWO INDICES AND TWO VARIABLES I he previous secio we have iroduced he oe-variable oe-idex Herie poloials He ( x ) as a paricular case of he poloials H ( x. ) I is possible o use hese poloials o iroduce a differe class of Herie poloials wih wo idices ad wo variables which are a vecorial exesio of he poloials He ( x ); his eas ha fro a idex acs o a oe-diesioal variable we will have a couple of idices acig o a wo-diesioal variable. Le he posiive quadraic for: ISSN:

3 INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS AND INFORMATICS Volue 8 14 q( x ) ax + bx + c ac > ac b > where a b c are real ubers. The associaed arix reads: (6) a b M (7) b c ad sice we have pued ac b > M > ha is a iverible arix. Le ow a vecor: x z i he space ( ) z Mz ( ) ( ) q z i iediael follows ha: a b x q z x ax + bx + c. b c Le: x z ad h u wo vecors of he space u ( u ) <. such ha: (8) We will called wo-idex wo-variable Herie poloials [6] ad we will idicae wih he sbol He ( x ) he poloials defied b he followig geeraig fucio: _ ( ) 1 q z z M z. (3) We defie he adjoi poloials of he wo-idex wovariable Herie poloials he poloials expressed fro he followig relaio: v M k k M k r s e G ( x ) (31)!! where he vecors: ξ r v k (3) η s ad such ha: v Mz k Mh. (33) I is eas o oe ha fro he secod of he relaios coaied i he above equaio i iediael follows ha: r s. ( r s ) <. B usig he relaios i he equaio (33) he expressio of he geeraig fucio (eq. (31)) defiig he adjoi Herie poloials of wo-idex ad wo-variable could be recas i he followig for: 1 1 z k k M k r s e G ( x ). (34)!! I he followig secio(s) we will derive a uber of ideiie regardig he wo-idex wo-variable Herie poloials ad heir adjoi. 1 h Mz h Mh e He ( x ). (9)!! (he superscrip deoes raspose). These poloials are exploied i a fields of pure ad applied aheaics [89]. The are ver useful i descripio of he quau reae [1] of coupled haroic oscillaor. B usig he defiiio of he quadraic for (eq. (6)) we ca iroduce he adjoi class of he wo-idex wo-variable Herie poloials He ( x ). Sice we have saed ha he associaed arix M a he quadraic for is iverible (eq. (6)(7)) we ca defie he quadraic for adjoi b seig: III. GENERALIZED IDENTITIES INVOLVING TWO-INDEX TWO- VARIABLE HERMITE POLYNOMIALS Before o proceed i is ecessar o reid soe basic rules of he vecorial differeial calculus. B he posiios saed i he previous secio we have: a b x a b 1 z Mz ( 1 ) ( x ) x + (35) b c b c a b 1 z Mz 1 ( ) x (36) b c ISSN:

4 INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS AND INFORMATICS Volue 8 14 z Mz x for 3. (37) Siilar relaios could be saed b derivig respec o. The Herie poloials of he pe He ( x ) [56] saisf he followig recurrece relaios: B usig he sae procedure ad he relaed foralis i is possible o sae ohers ipora relaios ivolvig he woidex wo-variable Herie poloials. B oiig he proof sice i copleel siilar o he above procedure we ca sae he relaios: He ( x ) ahe 1 ( x ) bhe 1( x ) (44) x He+ 1 ( x ) ( ax + b) He ( x ) ahe 1 ( x ) + (38) bhe ( x ) 1 He ( x ) bhe 1 ( x ) che 1( x ). (45) He + 1 ( x ) ( bx + c) He ( x ) bhe 1 ( x ) + (39) che ( x ) 1 where a b c are real ubers defied i he relaios (6). To prove he firs of he above recurrece relaios i is eough o oe ha b derivig wih respec o i he equaio (9) [7] we have: 1 z Mh h Mh He ( x )!! 1 He ( x )!! (4) ad b exploiig he l.h.s usig he resuls saed i he equaios (35-37) we ge: 1 1 z Mh h Mh He ( x ) ( x ) M +!! 1 1 ( 1 ) M + ( ) M He ( x ) u!! (41) ha is: ( ax + b a bu) He ( x )!! 1 He ( x ).!! Afer aipulaig he l.h.s of he above equaio ad b equaig he like -power we iediael obai he relaio (38). B followig he sae procedure bu b derivig wih respec o u i he equaio (9) we have: 1 ( bx + c) (b cu) He ( x )!! 1 He ( x )!! (4) (43) The four relaios explicaed i he equaios (38-39) ad (44-45) ca be cobied o defie he shif operaors relaed o he Herie poloials of he pe He ( x ). For isace bi puig he relaio (38) i he (44) we ge: He ( x ) ( ax + b) He ( x ) He+ 1 ( x ) (46) x ad he: ( ax + b) He ( x ) He+ 1 ( x ) x. (47) B cobiig he oher relaios usig he sae procedure of he above we fiall obai: ( bx + c) He ( x ) He + 1( x ) (48) 1 b a He ( x ) He 1( x ) (49) x 1 c b He ( x ) He 1 ( x ). (5) x I is ow aural iroduce he shif operaors b puig: E + ( ax + b) (51) x E + ( bx + c) (5) 1 E c b (53) x 1 E b a. (54) x ad he he recurrece relaio (39) follows. ISSN:

5 INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS AND INFORMATICS Volue 8 14 If we idicae wih he sbol f he geeric Herie poloial fucio i is possible o read he acio of he shif operaors i a ore copac wa; ha is: E± f f± 1 (55) E ± f. f ± 1 The operaors defied above are discree operaor i he sese ha he deped b he idices ad ha as we said are posiive ieger. IV. DIFFERENTIAL EQUATIONS RELATED TO THE POLYNOMIALS He ( x ) The relaios saed i previous secios ad i paricular he acio of he shif operaors defied i he equaios (51-54) allow us o sae a ipora resul for he wo-idex wo-variable Herie poloials discussed i his paper [6]. The poloials He ( x ) saisfied he followig parial differeial equaio: 1 M + z He ( x ) ( + ) He ( x ) (56) z z z where we have idicaed he parial derivaive vecor wih: x z. Fro he relaios (55) i is evide ha he followig equaios hold: B suig up hese las expressios we have: acx b c + b + x x x (61) b x + ac a He ( x ) ( + ) He ( x ) x ad oce we rearrage he ers i he l.h.s. we ca wrie: 1 1 c + b a x + x x + x He ( x ) ( + ) He ( x ). B oig ha: 1 1 c b M (6) ad (63) b a z x we iediael ge: 1 1 c + b a M (64) x x z z ad: x + z. (65) x z B subsiuig hese las ideiies i he equaio (6) we obai: + E E He ( x ) He ( x ) (57) 1 M + z He ( x ) ( + ) He ( x ) (66) z z z + E E He ( x ) He ( x ). (58) B exploiig hese relaios we easil ge: 1 c b ( ax b) He ( x ) ( 1) x + x + He ( x ) c + b a + x + x x x He ( x ) ( + ) He ( x ). (59) (6) which is exacl he parial differeial equaio defied i he equaio (56). B usig a aalogous procedure i is also possible o prove ha he adjoi wo-idex wo-variable Herie poloials ael G ( x ) saisf a siilar parial differeial equaio. I a forhcoig paper we will discuss ore aspecs relaed o his fail of Herie poloials ad we will see i paricular soe applicaios o he haroic oscillaor fucios [113] b usig he proper of bi-orhogoali saisfied b he Herie fucios derived fro he Herie poloials reaed i he prese paper. Moreover hese poloials are a ver useful ool o ivesigae a probles coeced wih he heor of special poloials as Chebshev ad Geebauer [311] ad he field of special fucios o derive releva operaioal resuls [114]. ISSN:

6 INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS AND INFORMATICS Volue 8 14 REFERENCES [1] Cesarao C. Mooiali Priciple ad relaed operaioal echiques for Orhogoal Poloials ad Special Fucios I. J. of Mah. Models ad Mehods i Appl. Sci. o appear 13. [] Daoli G. Cesarao C. O a ew fail of Herie poloials associaed o parabolic clider fucios applied Maheaics ad Copuaio 141 (1) pp [3] Cesarao C. Ideiies ad geeraig fucios o Chebshev poloials Georgia Maheaical Joural 19 (3) pp [4] Daoli G. Lorezua Cesarao C. Beresei poloials ad operaioal ehods Joural of Copuaioal Aalsis ad Applicaios 8 (4) pp [5] Gould H.W. Hopper A.T. Operaioal forulas coeced wih wo geeralizaios of Herie poloials Duke Mah. J. 9 pp [6] Appell P. Kapé de Férie J. Focios hpergéoériques e hpersphériques. Polioes d Herie Gauhier-Villars Paris 196. [7] Srivasava H.M. Maocha H.L. A reaise o geeraig fucios Wile New York [8] Daoli G. Lorezua S. Ricci P.E. Cesarao C. O a fail of hbrid poloials Iegral Trasfors ad Special Fucios 15 (6) pp [9] Daoli G. Ricci P.E. Cesarao C. Vazquez L. Special poloials ad fraioal calculus Maheaical ad Copuer Modellig 37 (7-8) pp (3). [1] Ki Y.S. Noz M.E. Phase-space picure of quau echaics World Scieific Sigapore [11] Daoli G. Lorezua S. Ricci P.E. Cesarao C. O a fail of hbrid poloials Iegral Trasfors ad Special Fucios 15 (6) pp [1] Cesarao C. Gerao B. Ricci P.E. Laguerre-pe Bessel fucios Iegral Trasfors ad Special Fucios 16 (4) pp [13] Koch H. Taaru D. Lp Eigefucios bouds for he Herie operaor Duke Mah. J. 18 () pp [14] Brei G. Cesarao C. Ricci P.E. Laguerre-pe expoeials ad geeralized Appell poloials Copuers ad Maheaics wih Applicaios 48 (5-6) pp Cleee Cesarao is assisa professor of Maheaical Aalsis a Facul of Egieerig- Ieraioal Teleaic Uiversi UNINETTUNO- Roe ITALY. He is coordiaor of didacic plaig of he Facul ad he also is coordiaor of research aciviies of he Uiversi. Cleee Cesarao is Hoorar Research Associaes of he Ausralia Isiue of High Eergeic Maerials His research acivi focuses o he area of Special Fucios Nuerical Aalsis ad Differeial Equaios. He has work i a ieraioal isiuios as ENEA (Ial) Ul Uiversi (Gera) Copluese Uiversi (Spai) ad Uiversi of Roe La Sapieza (Ial). He has bee visiig researcher a Research Isiue for Sbolic Copuaio (RISC) Johaes Kepler Uiversi of Liz (Ausria). He is Ediorial Board eber of he Research Bullei of he Ausralia Isiue of High Eergeic Maerials he Global Joural of Pure ad Applied Maheaics (GJPAM) he Global Joural of Applied Maheaics ad Maheaical Scieces (GJ-AMMS) he Pacific-Asia Joural of Maheaics he Ieraioal Joural of Maheaical Scieces (IJMS) he Advaces i Theoreical ad Applied Maheaics (ATAM) he Ieraioal Joural of Maheaics ad Copuig Applicaios (IJMCA). Cleee Cesarao has published wo books ad over ha fif papers o ieraioal jourals i he field of Special Fucios Orhogoal Poloials ad Differeial Equaios. ISSN:

A note on Generalized Bessel Functions

A note on Generalized Bessel Functions A oe o Geeralized Bessel Fucios Clemee Cesarao ad Dario Assae Absrac I his paper we will discuss he rea iolig arious forms of geeralized Bessel fucios of wo-ariable, i paricular by ouliig he likig bewee