Integral representations and new generating functions of Chebyshev polynomials

Transcription

1 Inegral represenaions an new generaing funcions of Chebyshev polynomials Clemene Cesarano Faculy of Engineering, Inernaional Telemaic Universiy UNINETTUNO Corso Viorio Emanuele II, Roma, Ialy ABSTRACT In his paper we use he wo-variable Hermie polynomials an heir operaional rules o erive inegral represenaions of Chebyshev polynomials. The conceps an he formalism of he Hermie polynomials H nx, y) are a powerful ool o obain mos of he properies of he Chebyshev polynomials. By using hese resuls, we also show how i is possible o inrouce relevan generalizaions of hese classes of polynomials an we erive for hem new ieniies an inegral represenaions. In paricular we sae new generaing funcions for he firs an secon kin Chebyshev polynomials. Keywors: Chebyshev polynomials, Inegral represenaions, Generaing funcions. 2 Mahemaics Subjec Classificaion: 33C45, 33D45. 1 Inroucion The Hermie polynomials [1] can be inrouce by using he concep an he formalism of he generaing funcion an relae operaional rules. In he following we recall he main efiniions an properies. Definiion The wo-variable Hermie Polynomials H 2) m x, y) of Kampé e Férie form [2], [3] are efine by he following formula: [ m 2 ] H m 2) m! x, y) = n!m 2n)! yn x m 2n 1.1) 1

2 In he following we will inicae he wo-variable Hermie polynomials of Kampé e Férie form by using he symbol H m x, y) insea han H 2) m x, y). The wo-variable Hermie polynomials H m x, y) are linke o he orinary Hermie polynomials by he following relaions: H m x, 1 ) = He m x) 2 where: an: where: [ m 2 ] 1) r x n 2r He m x) = m! r!n 2r)!2 r r= H m 2x, 1) = H m x) [ m 2 ] H m x) = m! r= 1) r 2x) n 2r r!n 2r)! an i is also imporan o noe ha he Hermie polynomials H m x, y) saisfy he relaion: H m x, ) = x m. 1.2) Proposiion 1 The polynomials H m x, y) solve he following parial ifferenial equaion: 2 x 2 H mx, y) = y H mx, y) 1.3) Proof By eriving, separaely wih respec o x an o y, in he 1.1), we obain: x H mx, y) = mh m 1 x, y) y H mx, y) = H m 2 x, y). From he firs of he above relaion, by eriving again wih respec o x an by noing he secon relaion, we en up wih he 1.3). The Proposiion 1 help us o erive an imporan operaional rule for he Hermie polynomials H m x, y). In fac, by consiering he ifferenial equaion 2

3 1.3) as linear orinary in he variable y an by remaning he 1.2) we can immeiaely sae he following relaion: H m x, y) = e y 2 x 2 x m 1.4) The generaing funcion of he above Hermie polynomials can be sae in many ways, we have in fac Proposiion 2 The polynomials H m x, y) saisfy he following ifferenial ifference equaion: z Y nz) = a ny n 1 z) + b nn 1)Y n 2 z) 1.5) Y n ) = δ n, where a an b are real numbers. Proof By using he generaing funcion meho, by puing: G z; ) = n n! Y nz) wih coninuous variable,we can rewrie he 1.5) in he form: z G z; ) = a + b 2) G z; ) G ; ) = 1 ha is a linear orinary ifferenial equaion an hen is soluion reas: G z; ) = exp x + y 2) where we have pue az = x an bz = y. Finally, by exploiing he r.h.s of he previous relaion we fin he hesis an also he relaion linking he Hermie polynomials an heir generaing funcion: exp x + y 2) = n n! H nx, y). 1.6) The use of operaional ieniies, may significanly simplify he suy of Hermie generaing funcions an he iscovery of new relaions, harly achievable by convenional means. By remaning ha he following ieniy e x 2 2x) n = 2x ) n 1) 1.7) x 3

4 is linke o he sanar Burchnall ieniy, we can immeiaely sae he following relaion. Proposiion 3 The operaional efiniion of he polynomials H n x) reas e x 2 2x) n = H n x) 1.8) Proof By exploiing he r.h.s of he 1.7), we immeiaely obain he Burchnall ieniy 2x ) n = n! x n 1) s 1 n s)!s! H n sx) s x s 1.9) s= afer using he ecoupling Weyl ieniy, since he commuaor of he operaors of l.h.s. is no zero. The erivaive operaor of he 1.9) gives a no rivial conribuion only in he case s = an hen we can conclue wih: 2x ) n 1) = H n x), x which prove he saemen. The Burchnall ieniy can be also invere o give anoher imporan relaion for he Hermie polynomials H n x). We fin in fac Proposiion 4 The polynomials H n x) saisfy he following operaional ieniy: H n x + 1 ) = 2 x n s= n n s s )2x) s x s 1.1) Proof By muliplying he l.h.s. of he above relaion by n n! an hen summing up, we obain n n! H n x ) x = e 2x+ 1 2) x) 2. By using he Weyl ieniy, he r.h.s. of he above equaion reas e 2x+ 1 2) x) 2 = e 2x e x an from which he 1.1) immeiaely follows, afer expaning he r.h.s an by equaing he like -powers. The previous resuls can be use o erive some aiion an muliplicaion relaions for he Hermie polynomials. 4

5 Proposiion 5 The polynomials H n x) saisfy he following ieniy n, m N: H n+m x) = minn,m) s= 2) s n s ) ) m s!h n s x)h m s x). 1.11) s Proof By using he Proposiion 3, we can wrie H n+m x) = 2x ) n 2x ) m = 2x ) n H m x) x x x an by exploiing he r.h.s. of he above relaion, we fin H n+m x) = n n 1) )H s n s x) s s x s H mx). s= Afer noing ha he following operaional ieniy hols s x s H mx) = 2s m! m s)! H m sx) we obain immeiaely our saemen. From he above proposiion we can immeiaely erive as a paricular case, he following ieniy H 2n x) = 1) n 2 n n!) 2 n s= 1) s [H s x)] 2 2 s s!) 2 n s)!. 1.12) The use of he ieniy 1.1), sae in Proposiion 4, can be exploie o obain he inverse of relaion conaine in he 1.12). We have inee Proposiion 6 Given he Hermie polynomial H n x), he square [H n x)] 2 can be wrien as Proof We can wrie H n x)h n x) = [H n x)] 2 = 2 n n!) 2 [H n x)] 2 = e 1 4 [ 2 x 2 n s= H n x + 1 ) H n x + 1 )]. 2 x 2 x H 2n x) 2 s s!) 2 n s)!. 1.13) By using he relaion 1.1), we fin, afer manipulaing he r.h.s. [ ] [H n x)] 2 = e 1 n 2 4 x 2 2 n n!) 2 2x) 2n 2 s s!) 2 n s)! s= an hen, from he Burchnall ieniy 1.7), he hesis. 5

6 A generalizaion of he ieniies sae for he one variable Hermie polynomials can be easily one for he polynomials H n x, y). We have in fac Proposiion 7 The following ieniy hols x + 2y ) n 1) = x n 2y) s n )H n x, y) s 1). 1.14) s xs s= Proof By muliplying he l.h.s. of he above equaion by n n! an hen summing up, we fin n x + 2y ) n = e x+2y x) 1). n! x By noing ha he commuaor of he wo operaors of he r.h.s. is [ x, 2y ] = 2 2 y x we obain n x + 2y ) n = e x+y2 e 2y x 1). 1.15) n! x Afer expaning an manipulaing he r.h.s. of he previous relaion an by equaing he like powers we fin immeiaely he 1.14). By using he Proposiion 6 an he efiniion of polynomials H n x, y), we can erive a generalizaion of he Burchnall-ype ieniy: e y 2 x 2 x n = x + 2y ) n, 1.16) x an he relae inverse: H n x 2y ) x, y = n 2y) s n n s s )x s x s. 1.17) s= We can also generalize he muliplicaion rules obaine for he Hermie polynomials H n x), sae in Proposiion 5. Proposiion 8 Given he Kampé e Férie Hermie polynomials H n x, y). We have minn,m) H n+m x, y) = m!n! 2y) s H n s x, y)h m s x, y). 1.18) n s)!m s)!s! s= 6

7 Proof By using he relaions sae in he 1.14) an 1.16), we can wrie: H n+m x, y) = x + 2y ) n H m x, y), x an hen By noing ha we obain H n+m x, y) = n 2y) s n )H n x, y) s s x s H mx, y). 1.19) s= s m! x s xm = m 2s)! xm 2s s x s H m! mx, y) = m s)! H m sx, y),. Afer subsiuing he above relaion in he 1.19) an rearranging he erms we immeiaely obain he hesis. From he previous resuls, i also immeiaely follows minn,m) H n x, y)h m x, y) = n!m! 2y) s H n+m 2s x, y) n s)!m s)!s!. 1.2) s= The previous ieniy an he equaion 1.18) can be easily use o erive he paricular case for n = m. We have in fac H 2n x, y) = 2 n n!) 2 n s= [H n x, y)] 2 = 2y) n n!) 2 n [H s x, y)] 2 s)! 2 n s)!2 s, 1.21) s= 1) s H 2s x, y) n s)!s!) 2 2 s. 1.22) Before concluing his secion we wan prove wo oher imporan relaions saisfie by he Hermie polynomials H n x, y). Proposiion 9 The Hermie polynomials H n x, y) solve he following ifferenial equaion 2y 2 x 2 H nx, y) + x x H nx, y) = nh n x, y) 1.23) Proof By using he resuls erive from he Proposiion 7, we can easily wrie ha: x + 2y ) H n x, y) = H n+1 x, y) x 7

8 an from he recurrence relaions x H nx, y) = nh n 1 x, y) we have x + 2y ) ) H n x, y) = nh n x, y), x x which is he hesis. From his saemen can be also erive an imporan recurrence relaion. In fac, by noing ha an hen we can conclue wih H n+1 x, y) = xh n x, y) + 2y x H nx, y) 1.24) H n+1 x, y) = xh n x, y) + 2nyH n 1 x, y). 1.25) 2 Inegral represenaions of Chebyshev polynomials In his secion we will inrouce new represenaions of Chebyshev polynomials [4], [5], by using he Hermie polynomials an he meho of he generaing funcion. Since he secon kin Chebyshev polynomials U n x) reas U n x) = sin[n + 1) arccosx)] 1 x 2, 2.1) by exploiing he righ han sie of he above relaion, we can immeiaely ge he following explici form U n x) = [ n 2 ] k= 1) k n k)!2x) n 2k. 2.2) k!n 2k)! Proposiion 1 The secon kin Chebyshev polynomials saisfy he following inegral represenaion [6]: U n x) = 1 n! e n H n 2x, 1 ). 2.3) 8

9 Proof By noing ha: we can wrie: n! = n k)! = e n e n k. 2.4) From he explici form of he Chebyshev polynomials U n x), given in he 2.1), an by recalling he sanar form of he wo-variable Hermie polynomials: we can immeiaely wrie: an hen he hesis. U n x) = [ n 2 ] y k x n 2k H n x, y) = n! k!n 2k)! k= e n [ n 2 ] k= 1) k k 2x) n 2k k!n 2k)! By following he same proceure we can also obain an analogous inegral represenaion for he Chebyshev polynomials of firs kin T n x), since heir explici form is given by T n x) = n 2 [ n 2 ] k= 1) k n k 1)!2x) n 2k. 2.5) k!n 2k)! By using he same relaions wrien in he previous proposiion, we have: T n x) = 1 2n 1)! e n 1 H n 2x, 1 ). 2.6) In he previous Secion we have sae some useful operaional resuls regaring he wo-variable Hermie polynomials; in paricular we have erive heir funamenal recurrence relaions. These relaions can be use o sae imporan resuls linking he Chebyshev polynomials of he firs an secon kin. Theorem 1 The Chebyshev polynomials T n x) an U n x) saisfy he following recurrence relaions x U nx) = nw n 1 x) 2.7) U n+1 x) = xw n x) n n + 1 W n 1x) 9

10 an T n+1 x) = xu n x) U n 1 x) 2.8) where W n x) = 2 n + 1)! e n+1 H n 2x, 1 ). Proof The recurrence relaions for he sanar Hermie polynomials H n x, y) sae in he firs Secion, can be cosume in he form [ 2x) + 1 ) ] H n 2x, 1 ) = H n+1 2x, 1 ) x 1 2 x H n 2x, 1 ) = nh n 1 2x, 1 ). 2.9) From he inegral represenaions sae in he relaions 2.3) an 2.6), relevan o he Chebyshev polynomials of he firs an secon kin, an by using he secon of he ieniies wrien above, we obain an x U nx) = 2n n! x T n nx) = n 1)! e n H n 1 2x, 1 ) 2.1) e n 1 H n 1 2x, 1 ). 2.11) I is easy o noe ha he above relaion gives a link beween he polynomials T n x) an U n x); in fac, since U n 1 x) = we immeiaely obain 1 n 1)! e n 1 H n 1 2x, 1 ) x T nx) = nu n 1 x). 2.12) By applying he muliplicaion operaor o he secon kin Chebyshev polynomials, sae in he firs of he ieniies 2.9), we can wrie ha is: U n+1 x) = 2 = x n + 1)! 1 n + 1)! e n+1 H n 2x, 1 e [2x) n ) ] H n 2x, 1 ) x U n+1 x) = 2.13) ) n 2 + e n H n 1 2x, 1 n + 1 n! ). 1

11 The secon member of he r.h.s. of he above relaion sugges us o inrouce he following polynomials W n x) = 2 n + 1)! e n+1 H n 2x, 1 ) 2.14) recognize as belonging o he families of he Chebyshev polynomials. Thus, from he relaion 2.1), we have: x U nx) = nw n 1 x) 2.15) an, from he ieniy 2.13), we ge: U n+1 x) = xw n x) n n + 1 W n 1x). 2.16) Finally, by using he muliplicaion operaor for he firs kin Chebyshev polynomials, we can wrie T n+1 x) = 1 2n! e n [2x) + 1 ) ] H n 2x, 1 ) 2.17) x an hen, afer exploiing he r.h.s. of he above relaion, we can fin which compleely prove he heorem. T n+1 x) = xu n x) U n 1 x) 2.18) 3 Generaing funcions By using he inegral represenaions an he relae recurrence relaions, sae in he previous Secion, for he Chebyshev polynomials of he firs an secon kin, i is possible o erive a sligh ifferen relaions linking hese polynomials an heir generaing funcions [4] [6]. We noe inee, for he Chebyshev polynomials U n x), ha by muliplying boh sies of equaion 2.3) by ξ n, ξ < 1 an by summing up over n, i follows ha: ξ n U n x) = + e ξ) n H n 2x, 1 ). 3.1) n! By recalling he generaing funcion of he polynomials H n x, y) sae in he relaion 1.6) an by inegraing over, we en up wih: ξ n U n x) = 1 1 2ξx + ξ ) 11

12 We can now sae he relae generaing for he firs kin Chebyshev polynomials T n x) an for he polynomials W n x), by using he resuls prove in he previous heorem. Corollary 1 Le x, ξ R, such ha x < 1, ξ < 1; he generaing funcions of he polynomials T n x) an W n x) rea: an ξ n T n+1 x) = n + 1)n + 2ξ n W n+1 x) = x ξ 1 2ξx + ξ 2 3.3) 8x ξ) 1 2ξx + ξ 2 ) ) Proof By muliplying boh sies of he relaion 2.8) by ξ n an by summing up over n, we obain: ha is which gives he 3.3). ξ n T n+1 x) = x ξ n U n x) ξ n T n+1 x) = ξ n U n 1 x) x 1 2ξx + ξ 2 ξ 1 2ξx + ξ 2 In he same way, by muliplying boh sies of he secon relaion sae in he 2.7) by ξ n an by summing up over n, we ge an hen he hesis. ξ n U n+1 x) = x ξ n W n x) n n + 1 ξn W n 1 x) These resuls allows us o noe ha he use of inegral represenaions relaing Chebyshev an Hermie polynomials is a fairly imporan ool of analysis allowing he erivaion of a wealh of relaions beween firs an secon kin Chebyshev polynomials an he Chebyshev-like polynomials W n x). References [1] H.W. Srivasava, H.L. Manocha, A reaise on generaing funcions, Wiley, New York,

13 [2] P. Appell J. Kampé e Férie, Foncions Hypergéomériques e Hypersphériques. Polynômes Hermie, Gauhier-Villars, Paris, [3] H.W. Goul, A.T. Hopper, Operaional formulas connece wih wo eneralizaions of Hermie polynomials, Duke Mah. J., 29, 1962, [4] P. Davis, Inerpolaion an Approximaion, Dover, New York, [5] L. C. Anrews, Special Funcions for Engineers an Applie Mahemaics, MacMillan, New York, [6] G. Daoli, C. Cesarano an D. Sacchei, A noe on Chebyshev polynomials, Ann. Univ. Ferrara, 7 21), no. 47,