Generalized Chebyshev polynomials

Transcription

1 Generalized Chebyshev polynomials Clemene Cesarano Faculy of Engineering, Inernaional Telemaic Universiy UNINETTUNO Corso Viorio Emanuele II, Roma, Ialy ABSTRACT We generalize he firs and second kind Chebyshev polynomials by using he conceps and he operaional formalism of he Hermie polynomials of he Kampé de Férie ype. We will see how i is possible o derive inegral represenaions for hese generalized Chebyshev polynomials. Finally we will use hese resuls o sae several relaions for Gegenbauer polynomials. Keywords: Two-variable Chebyshev and Gegenbauer polynomials, Generaing funcions, Inegral represenaions. 2 Mahemaics Subjec Classificaion: 33C45, 33D5. Inroducion I is well known ha he explici form of he second kind Chebyshev polynomials [] reads U n x) = [ n 2 ] k= ) k n k)!2x) n 2k k!n 2k)!..) In a previous paper [2] we have saed for hese polynomials an inegral represenaion of he ype: where: U n x) = n! e n H n 2x, ) d,.2) [ n 2 ] y k x n 2k H n x, y) = n! k!n 2k)! k=.3) are he wo-variable Hermie polynomials of Kampé de Férie [3], [4] ype, wih generaing funcion given by he formula exp x + y 2) = n= n n! H nx, y).

2 I is also possible o sae a differen represenaion for he second kind Chebyshev polynomials U n x) by rearranging he argumen of he H n x, y) polynomials. In fac, by noing ha n H n 2x, [ n 2 ] = n! and, from he fac ha k= ) = n! n [ n 2 ] k= ) k k 2x) n 2k k!n 2k)! n k)! = we can immediaely conclude wih U n x) = n! ) k 2x) n 2k k k!n 2k)! = H n 2x, ) e n k d, e H n 2x, )d..4) The use of he above inegral represenaions for he second kind Chebyshev polynomials can be used o inroduce furher generalized polynomial ses, including he wo-variable Chebyshev polynomials [5] and he wo-variable Gegenbauer polynomials. = 2 Two-variable generalized Chebyshev polynomials Before o proceed, we premise some relevan operaional relaions involving he generalized Hermie polynomials. Proposiion The polynomials H m x, y) solve he following parial differenial equaion: 2 x 2 H mx, y) = y H mx, y). 2.) Proof By deriving, separaely wih respec o x and o y, in he.3), we obain: x H mx, y) = mh m x, y), 2.2) y H mx, y) = H m 2 x, y). 2

3 From he firs of he above relaions, by deriving again wih respec o x and by noing he second relaion in 2.2), we end up wih he 2.). The above resuls help us o derive he imporan operaional rule. In fac, by considering he differenial equaion 2.) as a linear ordinary one in he variable y and by noing ha H n x, ) = x n, we can immediaely sae ha H n x, y) = e y 2 x 2 x n. 2.3) Proposiion 2 The wo-variable Hermie polynomials saisfy he following relaion x + 2y ) n ) = x n 2y) s n )H n x, y) s ). 2.4) s xs s= Proof By muliplying he l.h.s. of he above equaion by n n! and hen summing up, we find: n x + 2y ) n = e x+2y x) ). 2.5) n! x n= To develop he exponenial in he r.h.s. of he 2.5) we need o apply he Weyl ideniy and hen we have o calculae he commuaor of he wo operaors: [ x, 2y ] = 2 2 y x which help us o wrie: n= n x + 2y ) n = e x+y2 e 2y x ). n! x Afer expanding and manipulaing he r.h.s. of he previous relaion and by equaing he like powers we find immediaely he 2.4). The above resul gives us anoher imporan operaional rule for he generalized Hermie polynomials. By using in fac he ideniy saed in equaion 2.3), we have e y 2 x 2 x n = n 2y) s n )H n x, y) s ) 2.6) s xs s= and by noing ha he r.h.s. of he above relaion is no zero only for s =, we can immediaely obain e y 2 x 2 x n = x + 2y ) n. 2.7) x 3

4 Finally, we can sae Proposiion 3 The Hermie polynomials H n x, y) solve he following differenial equaion: 2y 2 x 2 H nx, y) + x x H nx, y) = nh n x, y) 2.8) Proof By using he resuls derived from he Proposiion 2, we can easily find ha x + 2y ) H n x, y) = H n+ x, y) 2.9) x and from he firs of he recurrence relaions saed in 2.2): x H nx, y) = nh n x, y) we have: x + 2y ) ) H n x, y) = nh n x, y) 2.) x x which is he hesis. From his saemen can be also derived an imporan recurrence relaion. By exploiing, in fac, he relaion 2.9), we obain: and hen we can conclude wih H n+ x, y) = xh n x, y) + 2y x H nx, y) 2.) H n+ x, y) = xh n x, y) + 2nyH n x, y). 2.2) Definiion Le x, y real variables and le α a real parameer, we say generalized Chebyshev polynomials of second kind, he polynomials defined by he following relaion: U n x, y; α) = n! e α H n 2x, y)d. 2.3) By using he recurrence relaions relevan o he wo-variable Hermie polynomials, proved above, we can sae he following Proposiion 4 The generalized Chebyshev polynomials U n x, y; α) saisfy he following recurrence relaions y U nx, y; α) = α U n 2x, y; α) 2.4) x U nx, y; α) = 2 α U n x, y; α). 4

5 Proof By deriving respec o y in he relaion 2.3), we ge: and since: we obain: y U nx, y; α) = n! e α y H n2x, y)d y H n2x, y) = )nn )H n 2 2x, y) y U nx, y; α) = n! which gives he firs of he 2.4). e α )nn )H n 2 2x, y)d The second relaion can be obained in he same way, by noing ha: x H n2x, y) = 2)nH n 2x, y). Proposiion 5 The generalized Chebyshev polynomials U n x, y; α) saisfy he follow Cauchy problem: 2 x 2 U nx, y; α) = 4 2 αy U nx, y; α) U n x, ; α) = 2x)n α n+. 2.5) Proof By deriving wih respec o x in he second ideniy of 2.4), we find: 2 x 2 U nx, y; α) = 4 ) α α U n 2x, y; α) and hen, since: we obain: 2 α U n 2x, y; α) = y U nx, y; α) 2 x 2 U nx, y; α) = 4 αy U nx, y; α). 2.6) By seing y = in he relaion 2.3), we have: and since U n x, ; α) = n! H n 2x, ) = 2x) n e α H n 2x, )d 5

6 we find ha is U n x, ; α) = 2x)n n! e α n d U n x, ; α) = 2x)n. 2.7) αn+ The parial differenial equaion, saed in 2.6), can be viewed as a firs order ordinary differenial equaion for he variable y; and hen by using he iniial condiion founded hrough he 2.7), we can sae he soluion: U n x, y; α) = e y 4 which compleely prove he proposiion. The symbol D x D α 2 2x)n x 2, 2.8) αn+ denoes he inverse of he derivaive, defined by D x fx) = x fξ) dξ. We have inroduced he generalized Chebyshev polynomials U n x, y; α) by using a differen inegral form of he sandard second kind Chebyshev polynomials, defined in he equaion 2.3). By using he same procedure, i is possible o obain similar inegral represenaions for he firs kind Chebyshev polynomials. In fac, since heir explici form is: T n x) = n 2 [ n 2 ] k= we can immediaely derive ha T n x) = 2n )! ) k n k )!2x) n 2k k!n 2k)!, 2.9) e n H n 2x, ) d. 2.2) We have also inroduced [] a Chebyshev-like polynomials by using he mehod of inegral represenaion: W n x) = 2 n + )! e n+ H n 2x, ) d. 2.2) We can now generalize he above Chebyshev polynomials. Definiion 2 Le x, y real variables and le α a real parameer, we define he following hree polynomials ses: U n x, y; α) = n! e α n H n 2x, y ) d, 2.22) 6

7 and: T n x, y; α) = 2n )! W n x, y; α) = n + )! e α n H n 2x, y ) d, 2.23) e α n+ H n 2x, y ) d. 2.24) Proposiion 6 The generalized Chebyshev polynomials saisfy he following recurrence relaions α U nx, y; α) = 2 n + )W nx, y; α) 2.25) α T nx, y; α) = n 2 U nx, y; α). Proof By deriving wih respec o α in he relaion 2.3), we find: α U nx, y; α) = n! e α n+ H n 2x, y ) d and hen he firs of equaions 2.25), immediaely, follows. In he same way by following a similar procedure by using he ideniy 2.23), we have: and hen he hesis. α T nx, y; α) = 2n )! e α n H n 2x, y ) d 3 Generalized Gegenbauer polynomials I is worh noing ha he Chebyshev polynomials can be viewed as a paricular case of he Gegenbauer polynomials. Definiion 3 Le x and µ real variables, we say n h order Gegenbauer polynomials, he polynomials defined by he follow relaion: C n µ) x) = [ n 2 ] ) k 2x) n 2k Γ n k + µ) Γµ) k!n 2k)! k= where Γµ) is he Euler funcion. By recalling he inegral represenaion of he above Euler funcion: 3.) Γµ) = 7 e µ d 3.2)

8 and by using he same argumens exploied for he Chebyshev case, we can sae he inegral represenaion for he Gegenbauer polynomials C n µ) + x) = e n+µ H n 2x, ) d. 3.3) n!γµ) We can also generalized he Gegenbauer polynomials by using heir inegral represenaion. Definiion 4 Le x, y real variables and le α a real parameer, we say generalized Gegenbauer polynomials, he polynomials defined by he following relaion: C n µ) x, y; α) = n!γµ) e α n+µ H n 2x, y ) d. 3.4) The above inegral represenaion is a very flexible ool; in fac i can be exploied o derive ineresing relaions regarding he Gegenbauer polynomials and also he Chebyshev polynomials [6]. Proposiion 7 Le ξ R, such ha ξ <, µ. The generaing funcion of he polynomials C n µ) x, y; α) is given by: n= ξ n C n µ) x, y; α) = [α 2xξ + yξ 2 ] µ. 3.5) Proof By muliplying boh sides of he ideniy 3.4), by ξ n and by summing up over n, we ge: n= and by noing ha: we can wrie: ξ n C n µ) x, y; α) = n= n= ξ) n n! n= ξ n n n!γµ) e α µ H n 2x, y ) d H n 2x, y ) = exp [ ξ 2x) + ξ 2 y) ] ξ n C n µ) x, y; α) = Γµ) e α e ξ2x)+ξ 2 y) µ d. 3.6) Finally, by inegraing over, by using he inegral represenaion of he Euler funcion, we obain he hesis. Proposiion 8 The generalized second kind Chebyshev polynomials and he generalized Gegenbauer polynomials saisfy he following recurrence relaion: 8

9 ) m m α m U nx, y; α) = m!c m+) n x, y; α). 3.7) Proof By deriving wih respec o α in he relaion 2.22), m-imes, we ge: m α m U nx, y; α) = )m n! e α n+m H n 2x, y ) d. The r.h.s. of he above ideniy can be wrien in he form: ) m n! e α n+m H n 2x, y ) d = )m m! n!m! and hen he hesis follows. e α n+m H n 2x, y ) d By using he recurrence relaions relaed o he Hermie polynomials saed in Proposiion 3, i is easy o noe ha: [ 2x) + y ) ] H n 2x, y ) = H n+ 2x, y ) x which can be used o derive he following resuls: 3.8) Theorem The generalized Gegenbauer polynomials C n µ) x, y; α) saisfy he recurrence relaions: and: n + 2µ Cµ) n+ x, y; α) = xcµ+) n x, y; α) yc µ+) n x, y; α) 3.9) y Cµ) n x, y; α) = µc µ+) n 2 x, y; α). 3.) Proof By using he relaion 3.8), we can wrie he generalized Gegenbauer polynomial of order n +, in he form: = n + )!Γµ) C µ) n+ x, y; α) = 3.) e α [2x) n+µ + y ) ] H n 2x, y ) d. x Afer exploiing he r.h.s of he above ideniy, we ge: = n + )!Γµ) C µ) n+ x, y; α) = 3.2) [ e α n+µ 2x)H n 2x, y ) d e α n +µ y2n)h n 2x, y 9 ) ] d

10 and hen: 2x = n + )!Γµ) 2yn n + )!Γµ) C µ) n+ x, y; α) = 3.3) e α n+µ H n 2x, y ) d e α n +µ H n 2x, y ) d. We can rearrange he above relaion in he form: and finally: = x n!γµ) y n )!Γµ) = x n!γµ + ) n + C µ) n+ x, y; α) = 2 n + y n )!Γµ + ) which proves he 3.9). 2µ Cµ) n+ e α n+µ H n 2x, y ) d e α n +µ H n 2x, y ) d x, y; α) = e α n+µ H n 2x, y ) d e α n +µ H n 2x, y ) d To show he recurrence relaion in he 3.), i is imporan o noe ha: y H n 2x, y ) nn ) = H n 2 2x, y ). 3.4) In fac, by deriving respec o y in equaion 3.4), we ge: y Cµ) n x, y; α) = n!γµ) and by using he 3.4), we can wrie: nn ) y Cµ) n x, y; α) = n!γµ) which immediaely gives he hesis. e α n+µ y H n 2x, y ) d e α n 2+µ H n 2 2x, y ) d

11 References [] P. Davis, Inerpolaion and Approximaion, Dover, New York, 975. [2] C. Cesarano, Inegral represenaions and new generaing funcions of Chebyshev polynomials, Rev. Ma. Complu., submied). [3] P. Appell J. Kampé de Férie, Foncions Hypergéomériques e Hypersphériques. Polynômes d Hermie, Gauhier-Villars, Paris, 926. [4] H.W. Gould, A.T. Hopper, Operaional formulas conneced wih wo eneralizaions of Hermie polynomials, Duke Mah. J., 29, 962, [5] P.E. Ricci, I polinomi di Tchebycheff in più variabili, Rend. Ma., 6) 978), [6] H.W. Srivasava, H.L. Manocha, A reaise on generaing funcions, Wiley, New York, 984.