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A Note on the Two-Variable Chebyshev Polynomials

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A Note on the Two-Variable Chebyshev Polynomials A note on the two-variable Chebyshev polynomials Clemente Cesarano Faculty of Engineering, International Telematic University UNINETTUNO Corso Vittorio Emanuele II, 39 – 00186 – Roma, Italy email: [email protected] ABSTRACT – In this paper we study the properties of the generalized Chebyshev polynomials; by using the integral representation of the first and second kind Cheby- shev polynomials, we define the symmetric two-variable Chebyshev polynomials. For this class of polynomials we derive integral representations, generating functions, recur- rence relations and partial differential equations by using the tool of the two-variable Hermite polynomials and related properties. Keywords: Two-variable Chebyshev polynomials, Integral representations. 2000 Mathematics Subject Classification: 33C45, 33D45. 1 Introduction In a previous paper [?] we have introduced the second kind Chebyshev polyno- mials by [ n ] X2 (¡1)k(n ¡ k)!(2x)n¡2k U (x) = : n k!(n ¡ 2k)! k=0 By exploiting the method of the integral representation, we have defined the two-variable, one-parameter, second kind Chebyshev polynomials Un(x; y; ®): Z +1 1 ¡®t Un(x; y; ®) = e Hn (2xt; ¡yt) dt n! 0 by using the related formula of the one-variable case: Z +1 1 ¡t Un(x) = e Hn (2xt; ¡t) dt : n! 0 1 In this article, we will introduce further generalizations of the Chebyshev polynomials by using again the method of the integral representation. Let us introduce the two-variable Chebyshev polynomials of the second kind: Definition 1 Let x and y two real variables, we say generalized two-variable Chebyshev polynomials of the second kind, the polynomials defined by the following rela- tion: [ n ] X2 (n ¡ k)!xn¡2kyk U (x; y) = : (1:1) n k!(n ¡ 2k)! k=0 It is easy to note that the above polynomials can be derived directly from the explicit form of the standard Chebyshev polynomials, or by using the integral representation of the polynomials Un(x; y; ®). We have, in fact 2 Two-variable Hermite polynomials The two-variable Hermite polynomials can be defined by using the relation stated in (1.2), after noting that: X+1 yn eyDf(x) = f(x + y) = f (n)(x) (2.1) n! n=0 and then: f(x) = xm implies eyDxm = (x + y)m P+1 m yD P+1 m f(x) = m=0 amx implies e f(x) = m=0 am(x + y) : The previous procedure can be easily generalized to exponential operators containing higher derivatives. In fact by considering the second derivative: +1 n 2 X y eyD f(x) = f (2n)(x) (2.2) n! n=0 and by noting that: m! D2nxm = xm¡2n (2.3) (m ¡ 2n)! we have: 2 m [ 2 ] n 2 X y m! eyD xm = xm¡2n: (2.4) n! (m ¡ 2n)! n=0 Definition 1 (2) The two-variable Hermite Polynomials Hm (x; y) of Kamp´ede F´erietform are defined by the following formula: [ m ] X2 m! H(2)(x; y) = ynxm¡2n (2.5) m n!(m ¡ 2n)! n=0 P+1 m It is important to note that, assuming f(x) = m=0 amx , the general identity reads: X+1 yD2 (2) e f(x) = amHm (x; y): (2.6) m=0 From the above definition we can state an elementary form of this kind of Hermite polynomials; in fact by the general identity (1.7), we immediately obtain: (1) m Hm (x; y) = (x + y) (2.7) which can also be recast in the form: X+1 yD (1) e f(x) = amHm (x; y): (2.8) m=0 In the following we will indicate the two-variable Hermite polynomials of (2) Kamp´ede F´erietform by using the symbol Hm(x; y) instead than Hm (x; y). The two-variable Hermite polynomials Hm(x; y) are linked to the ordinary Hermite polynomials by the following relations: µ ¶ 1 H x; ¡ = He (x) (2.9) m 2 m where: [ m ] X2 (¡1)rxn¡2r He (x) = m! (2.10) m r!(n ¡ 2r)!2r r=0 and: 3 Hm (2x; ¡1) = Hm(x) (2.11) where: [ m ] X2 (¡1)r(2x)n¡2r H (x) = m! (2.12) m r!(n ¡ 2r)! r=0 and it is also important to note that the Hermite polynomials Hm(x; y) satisfy the relation: m Hm(x; 0) = x : (2.13) Proposition 1 – The polynomials Hm(x; y) solve the following partial differ- ential equation: @2 @ H (x; y) = H (x; y) (2.14) @x2 m @y m Proof – By deriving, separately with respect to x and to y, in equation (1.21), we obtain: @ H (x; y) = mH (x; y) (2.15) @x m m¡1 @ H (x; y) = H (x; y): @y m m¡2 From the first of the above relation, by deriving again with respect to x and by noting the second relation in (1.23), we end up with the (1.22). The Proposition 1 help us to derive an important operational rule for the Hermite polynomials Hm(x; y). In fact, by considering the differential equation (1.22) as linear ordinary in the variable y and by remanding the (1.19) we can immediately state the following relation: 2 y @ m Hm(x; y) = e @x2 x (2.16) The generating function of the above Hermite polynomials can be state in many ways, we have in fact: Proposition 2 – The polynomials Hm(x; y) satisfy the following differential difference equation: 4 d Y (z) = a nY (z) + b n(n ¡ 1)Y (z) (2.17) dz n n¡1 n¡2 Yn(0) = ±n;0 where a and b are real numbers. Proof – By using the generating function method, by putting: X+1 tn G (z; t) = Y (z) (2.18) n! n n=0 with t continuous variable,we can rewrite the (2.23) in the form: d ¡ ¢ G (z; t) = at + bt2 G (z; t) (2.19) dz G (0; t) = 1 that is a linear ordinary differential equation and then its solution reads: ¡ ¢ G (z; t) = exp xt + yt2 (2.20) where we have putted az = x and bz = y. Finally, by exploiting the r.h.s of the previous relation we find the thesis and also the relation linking the Hermite polynomials and their generating function: ¡ ¢ X+1 tn exp xt + yt2 = H (x; y): (2.21) n! n n=0 3 Operatorial identities for the Hn(x; y): The use of operational identities (see Appendix A), may significantly simplify the study of Hermite generating functions and the discovery of new relations, hardly achievable by conventional means. Before entering in the two-variable case of the Hermite polynomials, we will introduce some identities related to the Hermite polynomials of the type Hn(x) (see (1.18)) that will be largely exploited in this section. By remanding that the following identity: µ ¶ 2 n ¡ 1 d n d e 4 dx2 (2x) = 2x ¡ (1) (3.1) dx 5 is linked to the standard Burchnall identity (see Appendix A), we can immedi- ately state the following relation. Proposition 3 – The operational definition of the polynomials Hn(x) reads: 2 ¡ 1 d n e 4 dx2 (2x) = Hn(x) (3.2) Proof – By exploiting the r.h.s of the (2.1), we immediately obtain the Burchnall identity: µ ¶ d n Xn 1 ds 2x ¡ = n! (¡1)s H (x) (3.3) dx (n ¡ s)!s! n¡s dxs s=0 after using the decoupling Weyl identity (see Appendix A), since the commu- tator of the operators of l.h.s. is not zero. The derivative operator of the (2.3) gives a not trivial contribution only in the case s = 0 and then we can conclude with: µ ¶ d n 2x ¡ (1) = H (x) (3.4) dx n which prove the statement. The relation (2.2) can be also derived from the explicit form of Hermite polynomials as in the case of the Hermite polynomials of the type Hn(x; y), that has been proved in the previous section (see (1.22)). The Burchnall identity can be also inverted to give another important rela- tion for the Hermite polynomials Hn(x). We find in fact: Proposition 4 – The polynomials Hn(x) satisfy the following operational iden- tity: µ ¶ µ ¶ 1 d Xn n ds H x + = (2x)n¡s (3.5) n 2 dx s dxs s=0 tn Proof – By multiplying the l.h.s. of the above relation by n! and then summing up, we obtain: µ ¶ X+1 n t 1 d 2 x+ 1 d t¡t2 H x + = e ( 2 )( dx ) : (3.6) n! n 2 dx n=0 By using the Weyl identity, the r.h.s. of the equation (2.6) reads: 2 x+ 1 d t¡t2 2xt t d e ( 2 )( dx ) = e e dx (3.7) 6 and from which the (2.5) immediately follows, after expanding the r.h.s and by equating the like t¡powers. The previous results can be used to derive some addition and multiplication relations for the Hermite polynomials. Proposition 5 – The polynomials Hn(x) satisfy the following identity 8 n; m 2 N: min(n;m) µ ¶µ ¶ X n m H (x) = (¡2)s s!H (x)H (x): (3.8) n+m s s n¡s m¡s s=0 Proof – By using the identity (2.4), contained in the Proposition 2, we can write: µ ¶ µ ¶ µ ¶ d n d m d n H (x) = 2x ¡ 2x ¡ = 2x ¡ H (x) (3.9) n+m dx dx dx m and by exploiting the r.h.s.
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