Chebyshev Polynomials and Generalized Complex

Chebyshev Polynomials and Generalized Complex Numbers

D. Babusci, G. Dattoli, E. Di Palma & E. Sabia

Advances in Applied Clifford Algebras ISSN 0188-7009 Adv. Appl. Clifford Algebras DOI 10.1007/s00006-013-0419-z

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Author's personal copy Adv. Appl. Clifford Algebras © 2013 Springer Basel DOI 10.1007/s00006-013-0419-z

Advances in Applied Clifford Algebras

Chebyshev Polynomials and Generalized Complex Numbers D. Babusci, G. Dattoli, E. Di Palma and E. Sabia∗ Abstract. The generalized complex numbers can be realized in terms of 2 × 2 or higher-order matrices and can be exploited to get different ways of looking at the trigonometric functions. Since Chebyshev polynomials are linked to the power of matrices and to trigonometric functions, we take the quite natural step to discuss them in the context of the theory of generalized complex numbers. We also briefly discuss the two-variable Chebyshev polynomials and their link with the third-order Hermite polynomials. Keywords. Generalized Complex Numbers, Generalized Trigonometry, Chebyshev Polynomials, Two-variable Chebyshev Polynomials.

1. Introduction The notion of complex number is one of the most versatile tools in pure and applied mathematics. The associated concepts can be merged with geometric and group theoretic point of view and have provided the backbone of the interpretation of physical theories either in classical, quantum and relativistic mechanics [1]. The idea of complex number evolved from the algebraic context, in which it initially flourished, to become the pivoting point of the understanding of rotations in the plane and later, with the introduction of quaternions, in the space [2],[3]. The link with the Pauli and Dirac matrices [4] and more in general with Clifford algebras disclosed an entire new world in relativistic quantum theory [5], [6], and its generalizations are opening new scenarios in the theory of differential equations with fractional derivatives [7],[8] as well as in the study of anomalous diffusion [9]. Once again most of the ancient flavor contained in the Euler work [10] still provides a powerful stimulus to get further results concerning e.g. the link ∗ Corresponding

author.

Author's personal copy D. Babusci, G. Dattoli, E. Di Palma and E. Sabia

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between exponentials and generalized “trigonometry” [11, 12, 13, 14, 15, 16]. Just to give an example we consider the identity h3 = 3h − 2a

(1)

which, if viewed as a third degree equation, has roots expressible in trigonometric form [17] α + 2 (n − 1) π hn = 2 sin 3 a = sin (α) ,

n = 0, 1, 2

(2)

The quantity h can also be considered an “imaginary unit”, whose properties can be obtained by recursion. The Euler formula, associated with such a unit, allows the definition of the “new” trigonometric functions ehβ = c0 (β) + hc1 (β) + h2 c2 (β) .

(3)

The relevant properties can be obtained by expanding the l. h. s. of eq. (3) and by exploiting the associated recurrences, implicit in eq. (1). On the other side, since each one of the roots of eq. (1) satisfies the identity (3), we obtain the definition ⎞ ⎛ hβ ⎞ ⎛ e 1 c0 (β) ⎝ c1 (β) ⎠ = Aˆ−1 ⎝ eh2 β ⎠ , c2 (β) eh3 β ⎛ ⎞ 1 h1 h21 Aˆ = ⎝ 1 h2 h22 ⎠ (4) 1 h3 h23 Furthermore they are easily shown to satisfy the following system of ODE

⎞ ⎛ ⎞ ⎛ ⎞⎛ c (β) 0 0 −2a c0 (β) d ⎝ 0 c1 (β) ⎠ = ⎝ 1 0 −3 ⎠ ⎝ c1 (β) ⎠ dβ c2 (β) c2 (β) 0 1 0

(5)

with initial conditions fixed by eq. (4) itself. This type of problems have been explored in ref. [15] where the link of this procedure with the Eisenstein numbers [17] has also been discussed, along with their link to the pseudo trigonometric functions [18]. The close relationship of Chebyshev polynomials and trigonometric functions is well known [19]; in the forthcoming section we expand such a link within the context of generalized imaginary units.

2. Chebyshev Polynomials and Generalized Imaginary Unit In this section we formulate the theory of Chebyshev polynomials as a byproduct of the generalized complex numbers. In analogy to the third order

Author's personal copy Chebyshev Polynomials and Generalized Complex Numbers

imaginary unit introduced in the previous section, we define its second order counterpart1 h2 = bh + a (6) whose higher-order powers are obtained by iteration as hn = bn−2 h + an−2

(n ∈ N )

(7)

with bn , an are functions of the parameters b, a. By multiplying both sides of eq. (7) by h and then by equating the “real” and “imaginary” parts (namely the coefficients without and with h), we obtain the recursion an+1 = a bn ,

bn+1 = an + b bn ,

which can also be rewritten in the matrix form an 0 an+1 ˆ b) ˆ b) = = Q(a, , Q(a, bn+1 bn 1

(8) a b

,

(9)

ˆ satisfies identity (7). where the matrix Q From Eq. (7) it also follows that eh φ = with C(φ) =

∞ φn n h = C(φ) + h S(φ), n! n=0

∞ φn an , n! n=0

S(φ) =

∞ φn bn , n! n=0

(10)

(11)

The previous equations provide an Euler-like identity, from which, by taking into account Eq. (8), the following differential equations for the cos- and sin-like functions C(φ), S(φ) can be derived d C(φ) = a S(φ), dφ

d S(φ) = C(φ) + b S(φ). dφ

(12)

The complex unit h is characterized by the two conjugated forms (the solutions of Eq. (6)) √ b ± b2 + 4 a h± = (13) 2 and, therefore, hn± = an + h± bn , (14) from which we obtain h+ hn− − h− hn+ a (hn−1 =√ − hn−1 (15) an = + − ) h+ − h − b2 + 4 a hn − hn− hn − hn− = √+ . bn = + h+ − h− b2 + 4 a For analogous reasons we define the following Euler-like identity eh± φ = C(φ) + h± S(φ) 1h

reduces to the ordinary imaginary unit for a2 = −1, b2 = 0.

(16)


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and define the following sin- and cos-like functions C(φ) =

−h− eh+ φ + h+ eh− φ , h+ − h −

S(φ) =

eh+ φ − eh− φ . h+ − h−

(17)

Let us now consider the recurrence Ln+1 = a Ln−1 + b Ln .

(18)

which, if considered as a difference equation, can be solved by the use of the Binet method [20], i.e., by setting Ln = hn in eq. (18) provides the relevant solution in the form (19) Ln = c1 hn+ + c2 hn− , where the constants c1,2 depend on the values assigned to L0 and L1 . The Chebyshev polynomials satisfy the recurrence (18) with a = 2 x and b = −1 [19], and, therefore we can associate the unit H to the corresponding difference equation defined by H 2 = 2 x H − 1, H± = x ± x2 − 1 (20) and

n H± = An + H ± B n , (21) where An and Bn are functions of the variable x , satisfying recurrences that can be obtained by applying the same method used before (see Eqs. (8), (9)), which yield An+1 0 −1 An = . (22) Bn+1 Bn 1 2x Since H+ + H− = 2 x, H+ H− = 1 we can derive from Eq. (21) the identities n n H+ + H− = An + x B n 2 n n H− = A2n + 2 x An Bn + Bn2 = 1. H+

(23)

By the use of Eqs. (15), (20) one obtains n−1 n n H n−1 − H− H+ H− − H − H+ = − +√ H+ − H− 2 x2 − 1 n n H − H− Bn = + = −An+1 . H+ − H− Furthermore, since H± dH± = ±√ dx x2 − 1 we find that functions Bn verify the following differential equation

(1 − x2 ) ∂x2 − 3 x ∂x + n2 − 1 Bn (x) = 0

An =

(24)

(25)

(26)

By replacing n → n + 1, we argue that the previous equation is that satisfied by the Chebyshev polynomials of second kind Un (x) [19], we can therefore set n+1 H n+1 − H− Un (x) = + √ . (27) 2 i 1 − x2


while the first kind counterpart is given by Tn (x) =

n n H+ + H− . 2

(28)

ˆ for the Chebyshev polyBy specializing Eq. (7) to the case of matrix Q nomials and by taking into account Eqs. (24), (21), one has −Un−1 (x) −Un (x) ˆ n+1 (−1, 2 x) = An+1 ˆ ˆ Q 1 + Q(−1, 2 x) Bn+1 = , Un+1 (x) Un (x) (29) ˆ from which, since det Q(−1, 2 x) = 1, we obtain the well-known identity Un2 (x) − Un−1 (x) Un+1 (x) = 1.

(30)

An interesting by-product of this method is the derivation of a general formula for the n-th power of a generic 2 × 2 matrix. The use of the Pauli matrices allows to set any 2 × 2 matrix in the form Pˆ = α ˆ 12 +

3

βk σ ˆk .

(31)

k=1

σi , σ ˆj } = 2 δij ), As a consequence of the anti-commutation properties of σ ˆk ( {ˆ we obtain 3 2 2 ˆ ˆ ˆ P = γ 12 + 2 α P , γ = −α + βk2 , (32) k=1

Therefore the matrix Pˆ can be considered as a kind of imaginary unit, according to the notion developed in this paper. Its powers can be easily evaluated using the following identity involving Chebyshev polynomials [21],[22] Pˆ n = Un−1 (α) Pˆ + Un−2 (α) ˆ12 .

(33)

Further properties could be obtained using the present formalism, but we will not dwell on this aspect of the problem. The paper has been aimed at providing a link between the concept of imaginary unit, according to the definition given in this paper, and special polynomials. It is interesting to note that the definition of the Chebyshev polynomials, given in terms of the imaginary unit H + and of its conjugate H− = (H+ )−1 allows further degrees of freedom in defining e.g. Chebyshev polynomials with negative index, namely U−n (x) =

−n+1 −n+1 n−1 H+ H n−1 − H+ − H− √ = −√ = −Un−2 (x) 2i 1 − x2 2i 1 − x2

(34)

or even fractional, but we will not further dwell on this aspect of the problem. More general questions may arise concerning the possibility of extending this method to other families of polynomials, this can in principle be done, even though the formulation of the problem requires deeper considerations concerning the nature of the polynomial under study.


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3. Final Comments We will provide a few examples regarding the “natural” inclusion of other families of polynomials within the context of the scheme developed in the previous section, we will then frame the relevant theory within a more general context. Fibonacci polynomials, characterized by the recurrence [23] Fn+1 (x)

=

xFn (x) + Fn−1 (x),

F0 (x)

=

0, F1 (x) = 1

(35)

can naturally be fit into the scheme and will be characterized by the matrix ˆ Q(1, x). The procedure can be extended to the Dickson polynomials [24] satisfying analogous recurrences and to other sets which can be reduced to Chebyshev forms. A further generalization is offered by the two variable Chebyshev polynomials satisfying the recurrence [25] (2)

(2)

(2)

Un+2 (u, v) = u Un+1 (u, v) − v Un(2) (u, v) + Un−1 (u, v)

(36)

Therefore the relevant theory can be associated with the (third order) imaginary unit defined by the equation Y 3 = uY 2 − vY + 1

(37)

and to the higher order trigonometry discussed in the introductory section [11, 12, 13, 14, 15, 16]. Just to provide a further hint of the possible ramifications of this type of (2) analysis we note that, being the generating function of the Un (u, v) provided by ∞ 1 (2) tn Un+1 (u, v) = (38) 1 − u t + v t2 − t3 n=0 by the use of the Laplace transform properties, we obtain ˆ ∞ 2 3 1 = ds e−s (1−u t+v t −t ) 1 − u t + v t2 − t3 0

(39)

and the generating function of third order Hermite polynomials [26], [27]2 ∞ tn (3) x t+y t2 +z t3 n=0 n! Hn (x, y, z) = e (3)

Hn (x, y, z) = n! (2)

n/3

Hn (x, y) = n! 2 The

(2)

r=0

n/2

(2)

z r Hn−3r (x,y) r!(n−3r)!

(40)

y r xn−2r r=0 r!(n−2r)!

polynomials Hn (x, y) are two variable Hermite polynomials and reduce to the standard form for y = 12 , it is also worth noting that they provide an orthogonal set only for negative values of the y parameter.


to write the two variable Chebyshev in terms of the following integral representation of the generalized Hermite polynomials ˆ ∞ 1 (2) Un+1 (u, v) = ds e−s Hn(3) (u s, −v s, s), (41) n! 0 This identity opens a new question: Can Hermite polynomials be accommodated within the same framework as Chebyshev? Or more in general: can other orthogonal polynomials, as e.g. Laguerre, be associated with the recurrences of some imaginary unit? The structure of the recurrences of Hermite polynomials (and Laguerre as well) does not allow such a possibility. In the case of two variable Hermite polynomials we have indeed the following identities connecting the nearest neighbor indices [26], [27] (2)

(2)

Hn+1 (x, y) = xHn(2) (x, y) + 2ynHn−1 (x, y) (2)

∂x Hn(2) (x, y) = nHn−1 (x, y) (42) They cannot be transformed into a difference equation, which can be then solved using the Binet method. The problem can be stated in more rigorous terms by noting that Hn (x, y) (and the Laguerre) polynomials belong to the family of quasi-monomial [26],[27], they can therefore “generated” through a one term iteration, namely ˆ n1 Hn (x, y) = M ˆ = x + 2y∂x M (43) ˆ is an operator containing a first order derivative. Within such a where M framework we find ˆ n+1 1 Hn+1 (x, y) = M (44) Even though containing an operator, the structure of the recurrence is very much simplified with respect to that defining the previous considered families and it is algebraically equivalent to that of ordinary monomials (hence the name). An interesting treatment of Chebyshev polynomials, which may open new perspectives even for the extension of the concepts associated with monomiality, has recently been discussed in ref. [28]. In that paper the Author proposes the concept of Chebyshev power, which henceforth will be called Chebyshev monomials (C-m) defined as (we have modified the notation of ref. [28]) [n] ¯n ¯n c X ± = H+ ± H− , ¯ +H ¯ −1 x=H (45) the following properties are worth to be stressed

[1] d [n] [n] X X X = n , c − c + c − dx

[1] d [n] [n] = n c X+ c X− c X− dx

Author's personal copy D. Babusci, G. Dattoli, E. Di Palma and E. Sabia [n] [m] c X+ c X+

[n+m]

= c X+

[n−m]

− c X+

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(46)

and can be usefully employed in the study of the theory of Chebyshev and of the associated families of polynomials. By defining e.g. the C-derivative ˆ =c X [1] d − dx

c Dx

(47)

we obtain from eq. (46) [n] ˆ2 c D x c X+

[n]

= n2 c X+

(48)

which, once expanded, yields a differential equation corresponding to that of first kind Chebyshev polynomials. Furthermore the C-m properties can be exploited to bridge between the imaginary unit formalism and that underlying monomiality. In this paper we have considered a complementary point of view to the theory of Chebyshev polynomials, it can be exploited to simplify most of the derivation of the associated properties. The possibility of extending the method to other family of polynomials like the multi-index bi-orthogonal forms, will be discussed elsewhere. Acknowledgments The Authors express their sincere appreciation to an anonymous referee for offering the possibility of extending the originally submitted note.

References [1] B. Jancewicz, Multivectors and Clifford Algebras in Electrodynamics. World Scientific (1988). [2] S. L. Altmann, Rotations, Quaternions and Double Groups. Clarendon Press, Oxford 1966. [3] D. Babusci, G. Dattoli and E. Sabia, Operational Methods and Lorentz Type Equations of Motion. J. Phys. Math. 3 (2011), 1-17. [4] D. Hestenes, Clifford Algebra and the interpretation of Quantum Mechanics. In: J. S. R. Chisholm/A.K. Commons (Eds.), Clifford Algebras and their Applications in Mathematical Physics. Reidel, Dordrecht/Boston (1986), 321–346. [5] R. Gerritsma, G. Kirchmair, F. Zahringer, E. Solano, R. Blatt C. F. Roots, Quantum simulation of the Dirac equation. Nature 463 (2010), 68. [6] D. Babusci, G. Dattoli and M. Quattromini, Relativistic equations with fractional and pseudo-differential operators. Phys. Rev. A 83 (2011) 062109. [7] R. Herrmann, Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore 2011. [8] D. Babusci, G. Dattoli, M. Quattromini, and P. E. Ricci, Appl. Math. Comput. 218 (2011), 1495. [9] K. Gorska, K. Penson, D. Babusci, G. Dattoli, and G. E. Duchamp, Phys. Rev. E 85 (2012), 031138.

Author's personal copy Chebyshev Polynomials and Generalized Complex Numbers [10] G. Dattoli and M. Del Franco, The Euler Legacy to Modern Physics. Lecture Notes of Seminario Interdisciplinare di Matematica 9 (2010), 1-24. [11] R. M. Yamaleev, Adv. Appl. Clifford Al. 10 (1) (2005), 15 and references therein. [12] D. Babusci, G. Dattoli, E. Di Palma, and E. Sabia, Adv. Appl. Clifford Al. 22 (2012), 271. [13] G. Dattoli, M. Migliorati, P. E. Ricci, [arXiv:1010.1676v1[math-ph]]. [14] G. Dattoli, E. Sabia, M. Del Franco, [arXiv:1003.2698v1[math-ph]]. [15] G. Dattoli and M. Del Franco arXiv:1002.4728v1 [math-ph]. [16] R. M. Yamaleev, Geometrical and physical interpretation of evolution governed by general complex algebra, J. Math. Anal. Appl. 10 (2008), doi:10.1016/j.jmaa.2007.09.018. [17] G. Dattoli, E. Sabia and M. Del Franco, The Pseudo Hyperbolic Functions and the Matrix Representation of Eisenstein Complex Numbers, arXiv: 1003.2698. [18] P. E. Ricci, Le funzioni Pseudo Iperboliche e Pseudo Trigonometriche. Pubblicazioni dell’Istituto di Matematica Applicata, N 192 (1978). [19] L. C. Andrews, Special Functions for Engineers and Appled Mathematicians. Mc Millan, New York (1985). [20] D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books 1986, p. 62. [21] P. E. Ricci, Atti Accad. Sc. Torino 109 (1974-75), and references therein. [22] G. Dattoli, C. Mari, and A. Torre, Nuovo Cimento B 108 (1993), 61. [23] T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, New York, 2001. [24] L. E. Dickson, Ann. Of Math. 11 (1897), 65–120; 161–183. [25] P. E. Ricci, Rendiconti di Matematica, Serie VI, vol. 11 (1978), 295. [26] D. Babusci, G. Dattoli, and M. Del Franco, Lectures on Mathematical Methods for Physics, Internal Report ENEA RT/2010/5837 (available at www.frascati.enea.it/biblioteca/). [27] G. Dattoli, P L. Ottaviani, A. Torre, L. V´ azquez Evolution operator equations: Integration with algebraic and finite difference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory. La Rivista del Nuovo Cimento vol. 20, issue 2 (1997), pp. 3-133. [28] G. W. Smith, The Chebyshev exponent arXiv:1209/60v1 [math. NT] 13 Sep. 2012.

D. Babusci INFN – Laboratori Nazionali di Frascati via E. Fermi, 40 IT 00044 Frascati (Roma) Italy e-mail: [email protected]

Author's personal copy D. Babusci, G. Dattoli, E. Di Palma and E. Sabia G. Dattoli, E. Di Palma and E. Sabia ENEA - Centro Ricerche Frascati via E. Fermi, 45 IT 00044 Frascati (Roma) Italy e-mail: [email protected] [email protected] [email protected] Received: April 3, 2013. Revised: July 15, 2013. Accepted: August 8, 2013 .

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