Chebyshev Polynomials with Applications to Two-Dimensional Operators

A new application of Chebyshev polynomials of second kind ( ) U n x to functions of two-dimensional operators is derived and discussed. It is related to the Hamilton-Cayley identity for operators or matrices which allows to reduce powers and smooth functions of them to superpositions of the first 1 N − powers of the considered operator in N-dimensional case. The method leads in two-dimensional case first to the recurrence relations for Chebyshev polynomials and due to initial conditions to the application of Chebyshev polynomials of second kind ( ) U n x . Furthermore, a new general class of Generating functions for Chebyshev polynomials of first and second kind ( ) U n x comprising the known Generating function as special cases is constructed by means of a derived identity for operator functions ( ) f A of a general two-dimensional operator A . The basic results are Formulas (9.5) and (9.6) which are then specialized for different examples of functions ( ) f x . The generalization of the theory for three-dimensional operators is started to attack and a partial problem connected with the eigenvalue problem and the Hamilton-Cayley identity is solved in an Appendix. A physical application of Chebyshev polynomials to a problem of relativistic kinematics of a uniformly accelerated system is solved. All operator calculations are made in coordinate-invariant form.


Introduction
The main purpose of this article is to examine an application of the Chebyshev How to cite this paper: Wünsche, A. as its polynomial case. By this way we consider selected aspects of the Chebyshev polynomials and find also some little known properties and relations, for example, a relation to an integral operator formed from the Bessel functions with the variable substituted by the operator of differentiation which generates a transformed variant of the Ultraspherical polynomials. The methods can be generalized to three-dimensional functions of operators and one partial problem for this is solved in Appendix B.
The Jacobi polynomials and their special case of the Ultraspherical and Gegenbauer polynomials are used in the form in which they were introduced by Szegö [1] (with citations of the "old" original papers) and which became now standard in many monographs about Special functions and Orthogonal polynomials, e.g., [2] [3] [4], as well as [5] in the NIST Handbook [6] and [7]. A special work about the Chebyshev polynomials is the monograph of Rivlin [8] where the approximation theory of functions takes on a great space.
In present article we investigate the general two-dimensional case of reduction of operator functions via the Hamilton-Cayley identity that seems to be new. This gives also some hints on the three-and higher-dimensional cases which may lead to an approximate conjecture for these forms. In two-dimensional case it leads essentially to an application of Chebyshev polynomials

Chebyshev and Legendre Polynomials as Special Cases of Hypergeometric Function and Ultraspherical and Gegenbauer Polynomials
We compile in this Section without proof some known basic relations for Chebyshev polynomials of first kind A relation connected with an argument transformation in the Jacobi polynomials of the form [1] ( ) ( ) ( ) together with some modifications using the symmetry of the Jacobi polynomials (2.3) is generally possible. The Jacobi polynomials and all their special cases belong to the classical orthogonal polynomials in a finite interval as which in their standard form is chosen the interval 1 1 x − ≤ ≤ + .
Two essentially different expansions of the Jacobi polynomials are which is the possible application to (2.2) of a quadratic transformation of Gauss They are a consequence of the quadratic transformations of the Hypergeome- and Legendre polynomials by However, this cannot successfully be continued to upper index 0 They possess unique properties among all Ultraspherical polynomials. The limiting transition used in (2.15) including the number This shows that they are the same for both kinds of Chebyshev polynomials.
With these recurrence relations the polynomials can be continued to arbitrary negative indices 1, 2, n = − −  that finds its explanation after transition to trigonometric polynomials (Section 4).

Series Expansion of Ultraspherical and Gegenbauer, Chebyshev and Legendre Polynomials and Fibonacci and Lucas Numbers
As special cases α β = or equivalently for Gegenbauer polynomials ( ) The second representations of the (pure) series in powers of z follows also directly from the representation (2.7) by the Hypergeometric function and its Taylor series expansion. From the discussed expansions follow in the most important special cases the expansions for: Chebyshev polynomials of second kind The first written expansions are the direct specializations from (2.5).
The representations for the Chebyshev polynomials T n z , e.g., [5] [7] [8] ( ) ( ) ( and for ( ) According to (2.9) the Gegenbauer polynomials with higher upper parameter ν can be obtained by differentiation Inserting this substitution one obtains immediately from (3.2) the following One may see that the first of the two expansions in (3.13) can be expressed in the following way Thus the polynomials We consider the simplest special cases. If we make the substitution of the argument of the Chebyshev polynomials of first kind corresponding to 1 2 The case to the Legendre polynomials corresponding to 0 α = cannot be represented in simple way in analogy to (3.7) and (3.8) or (3.17) and we write down the two series expansions obtained by specialization from (3.13) that played an important role in our previous papers to this subject, e.g. [12]. This suggests to use also other entire functions and to substitute their variable by the differentiation operator and then to apply this to the sequence of basic monomials but this makes only sense if one may find applications or interesting aspects of the obtained sequence of (usually, non-orthogonal) polynomials.
By specialization of the arguments in the derived sequences of polynomials one may obtain sequences of numbers. In certain cases one obtains only integers.
To get sequences of positive increasing integers one has to specialize the arguments by complex numbers since the considered polynomials (and also many here not considered polynomials) possess alternating coefficients. In particular, the well-known Fibonacci numbers n F can be obtained in the following way from the here considered polynomials, series and functions (e.g., [13] for last The also well-known Lucas numbers n L can be obtained analogously by ( [13] for last representation) The and they are related, among others (multiplicative ones), by [13] ( ) For convenience we give a short table of the Fibonacci and the Lucas numbers (Table 1). One may construct "similar" kinds of number sequences by changing the arguments of the functions, for example, ( ) with arbitrary fixed natural numbers N which, obviously, provide sequences of increasing integers (sometimes under omission of a few initial terms) which in some cases are reducible by divisions. Using other initial values in the same recurrence relations we also get new number sequences (in such cases the sequences are no more described by the here written formulae).
We will give yet the following analogous examples of sequences of increasing integers constructed from the Legendre polynomials A short table of these sequences of numbers is ( Table 2). The importance of such sequences of numbers rises if one finds applications, for example, in combinatorics.

Ultraspherical and Gegenbauer Polynomials with Integer and Semi-Integer Parameter
Almost all up to now written relations are true for arbitrary real and even complex variable z. We now consider properties which are only true or possible for real variable x in the basic interval 1  One may choose another normalization of the Chebyshev and Legendre polynomials according to  T n x then we find from the expansions in (3.6) This is well known and can be easily proved by complete induction. Since Clearly the index n, usually the degree of the polynomial within a sequence of polynomials, is here no more true as such. The inversion of relation (4.1) is for all integer n from which immediately follows The inversion of the relation (4.5) is with the correct special case for 0 One may transform these results into a series over Chebyshev polynomials of first kind as follows The more general relation of this kind takes on its simplest form expressed by where it is possible to use the inverse substitution The basic monomials n x can be represented by the Legendre polynomials according to 5 The presence of this integral in tables admits the conjecture that someone already went the way via fractional integration but there in [14] ( §15.321) we find (now old) citations of Dirichlet and of Mehler. However, they provide no other closed result or representation than the Legendre polynomials ( ) P n x whereas we started from it, possibly in the hope, to find other representations similar to (4.1) and (4.5).
The more general formula expressed in arbitrary Gegenbauer polynomials from which (4.17) is the special case 1 2 ν = . As hint for attention we mention that a separated factor within the sums in (4.17) and in (4.18) is not a factorial.
In Section 9 we derive a whole class of generating functions for the Chebyshev polynomials. For convenience and to be self-contained we give here the well-known basic generating functions for Chebyshev and Legendre polynomials Another kind of generating functions is (e.g., [5] and [11]) The more general result is representable by (modified entire) Bessel functions as distinguishing part.
It is sometimes favorable to consider another normalization of the Ultras-  that is sometimes advantageous and the graphics to different parameter α become more similar to each other (see Figure 1 and Figure 2). The recurrence relation becomes with only positive sign on the right-hand side as advantage. This is true for the general case of ( ) ( ) P n x α with integer and semi-integer parameter α . 6 We made a bracket around the upper parameter α to distinguish them from the different Associated Legendre polynomials

Unique Properties of the Chebyshev Polynomials
We discuss in this Section unique properties of Chebyshev polynomials of first kind ( ) T n z and write as variable z when the obtained relations are true for arbitrary complex z. First, we consider composite indices n pq = in Chebyshev polynomials of first kind. From (4.1) follows T cos cos T cos T T cos .
As example for illustration of this relation we choose and, in particular  In next section we investigate the expansion in Chebyshev polynomials of first kind and consider a mapping onto 2π-periodic functions that leads to Fourier series with an additional symmetry.

( ) T n x to Fourier Series of 2π-Periodic Functions
We consider the expansion of a sufficiently well-behaved function ( )   with arbitrary 0 ϕ . The expansion (6.9) together with (6.11) represents the Fourier decomposition of a general 2π-periodic and symmetric function together with the formula for the coefficients with integration limits which must go only over an arbitrary 2π-interval (Example in Figure 3). Due to additional symmetry (6.6) the formula for the coefficients can be written in the special form (6.10) where the integration limits over a half-period cannot be arbitrarily displaced but only over full 2π-periods. One obtains then a Fourier series instead of (6.9) including also sum terms con-

Application of Chebyshev Polynomials of Second Kind to Reduction of Powers of Two-Dimensional Operators
The Chebyshev polynomials of second kind possess an important application in the theory of functions of two-dimensional operators in connection with the Hamilton-Cayley identity. We deal with this in coordinate-invariant form and give the most important informations and basic formulae in Appendix A.
In this section we consider arbitrary two-dimensional operators A that means operators which satisfy the following two-dimensional Hamilton-Cayley Any two neighbored pairs from these relations can be taken as the initial conditions for the recurrence relations (7.8).
The recurrence relations (7.8) Graphical illustrations for the first four polynomials ( ) U n x are given in Figure 1.
For the same function We see from this formula that for the final calculation of this reduction to a linear combination of the operators I and A for a given function

Solution of Eigenvalue Problem for Two-Dimensional Operators and Arbitrary Functions of Operators
In this section, we calculate functions ( ) f A of two-dimensional operators A and represent these as superposition of the two linear independent operators A and 0 = I A . As preparation we consider the solution of the eigenvalue problem of the operator A in coordinate-invariant form.
The solution of the eigenvalue problem of two-dimensional operators A , , consists of the determination of the eigenvalues α by means of the secular equation and the determination of right-hand eigenvectors a and left-hand eigenvectors a  to the eigenvalues α . Instead of the eigenvectors, we determine below projection operators to these eigenvectors. We denote the two, in general, different solutions of the eigenvalue Equation ( where the substitutions ( It is easy to see that ± Π are projection operators for the determination of eigenvectors to the eigenvalues α ± and that they satisfy the relations 2 , 0, 1, for arbitrary vectors x and x  . This means that ± x Π is either a right-hand eigenvector ± e of A to eigenvalue α ± or is vanishing and ± x Π is either proportional to ± e or is vanishing, correspondingly. The identity operator I , By inserting in (8.6) the explicit form of the eigenvalues α ± given in (8.3), we find the following representation of ± Π [ ] where again the substitutions (7.3) are used. In the same way, we find 1 , where t and x are defined in (7.3) as parameters from the invariants of the operator A . This has the same form as the representation in (7.13) and the identification of the functions in front of I and A provides Generating functions for the Chebyshev polynomials of second kind ( ) U n x . We discuss this in the next section.
As first example for the reduction of a function of the operator A to a superposition of the operators I and A we find from (8.11) The case of ( ) where we used the identity ( ) ( )

A Whole Class of Generating Functions for the Chebyshev Polynomials of Both Kinds
With (7.13) and (8.11) we derived in Sections 7 and 8 two different representations of functions ( ) f A of an arbitrary two-dimensional operator A expressed by the two independent basic operators I and A . These two representations have to be equal. If we separate the parts proportional to the identity operator I and to the operator A we obtain first from terms proportional to I the identity and second from terms proportional to A the identity The relation which follows from (9.1) is a linear combination of these identities.
Next we consider an exponential function By separation of the even and odd parts with respect to variable t one may gain further Generating functions.

Exponential Function of a General Two-Dimensional Operator
In this section we consider in detail the exponential function ( ) The operator ( ) exp A is here decomposed into a product of two commuting operators. The first operator is proportional to the identity operator I and its determinant is the exponential of the trace A of the operator A . The last is a general property for the determinant of an exponential function of an arbitrary operator and follows almost immediately from the eigenvalue decomposition of the operator A . The second is the exponential of an operator here abbreviated ′ A with vanishing trace. Its determinant is therefore equal to 1. If we denote in analogy to (7.3) the parameters of the operator ′ A by t′ and x′ where x′ is vanishing due to vanishing trace then we find for the reduction of the exponential of the operator ′ A This is identical to the more specialized representation of the operator of the operator in the exponent in braces in (10.1).
A vanishing trace of an operator is usually obtained from the assumption of its antisymmetry according to where the superscript ' T ' means the transposition. The problem is that in a general linear or in an affine space this cannot be defined and that it requires an Euclidean or Pseudo-Euclidean space with definition of a symmetrical scalar product and thus of a symmetrical metric tensor 9 .
We mention that a two-dimensional operator

Degenerate Cases
The two-dimensional case of operators does not admit many degenerate cases.
We now make some short remarks about the case of degeneracy of the eigenvalues α ± that means about the coincidence α α The operator belongs in case of 0 ≠ A then to a Jordan normal form with only one non-vanishing element in one of the off-diagonals and is quadratically nilpotent.

Conclusions
A main result of this article was to show that the Chebyshev polynomials in connection with the two-dimensional Hamilton-Cayley identity can solve the problem of reduction of functions of two-dimensional operators to superpositions of this operator itself and of the identity operator in coordinate-invariant form. In Appendix C this is applied to an interesting problem of relativistic kinematics of a step-wise accelerated space-ship with final transition to a uniformly accelerated space-ship seen from the inertial systems of earth and of the space-ship. The solution of this problem uses in an intermediate step Chebyshev polynomials of first and of second kind. An aim was to generalize the application to functions of three-dimensional operators which need a generalization of the Chebyshev polynomials to polynomials which essentially depend on two continuous variables. The derived recurrence relations are 4-term relations instead of 3-term relation for the usual Chebyshev polynomials. The solution of this programme seems to be interesting for three-dimensional operators, in particular, in group theory. This programme is a difficult one and is not yet accomplished with present article. However, we could explicitly obtain the (here not presented) necessary polynomials but some properties and interesting relations, in particular, the desirable Generating functions for these polynomials are not obtained up to now.
In the introductory sections we discussed some properties of the Chebyshev polynomials, and tried to consider them within the more general sets of the Ultraspherical and of the widely equivalent Gegenbauer polynomials and included also the Legendre polynomials which take on an intermediate place between the Chebyshev polynomials of first and of second kind. We compiled mainly the formulae which are connected with explicit representation in form of expansions in power series and discussed trigonometric forms. Clearly, much is known but we obtained also here some new shades. For example, after a variable transformation within the Ultraspherical polynomials we obtained in Section 3 a set of polynomials which could be generated from the basic monomials by an operator which essentially uses the Bessel functions with the variable substituted by the operator of differentiation, and which does not depend on the degree of the polynomial and which was earlier applied in analogous form with success to Hermite polynomials. We mentioned the connection of Chebyshev polynomials to Fibonacci and Lucas members and showed possibilities to obtain other increasing sequences of integers from Ultraspherical polynomials. In many ways the Chebyshev polynomials of fist kind take on a peculiar position which does not fit to the general classes of Ultraspherical or Gegenbauer polynomials. At the end of Section 4 it is shown that this can be removed by another normalization of the Ultraspherical polynomials with some attractive properties but also with some less attractive properties. The exceptional position of the Chebyshev polynomials of first kind within the family of Ultraspherical polynomials is underlined by the short discussion of two properties. Similar to the role of prime numbers for all (composite) numbers the Chebyshev polynomials of first kind need only those with prime degree as building stones which allow the construction of all other Chebyshev polynomials of first kind by nested inclusions. The second exceptional property of Chebyshev polynomials of first kind is that in power series expansions within a given finite interval (which can be managed by transformations) in each degree they provide the best approximation by some criteria compared with the other sequences of Ultraspherical polynomials. This is in analogy to Fourier series in comparison to expansions of periodic functions in other complete sets of basic periodic functions. In Section 6 we mentioned shortly the mapping of the expansion in Chebyshev polynomials of first kind onto Fourier series and show that the obtained Fourier series possess an additional symmetry in comparison to general Fourier series.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.
with identity operator I and with the coefficients 0 appear also as coefficients in the following eigenvalue equation of the operator A to eigenvalues α ( ) ( ) The determinant ( ) The relation between the coefficients ( )  Clearly, all this is well known in one or the other form and serves here for the introduction of some of our notations.
To our experience, in coordinate-invariant calculations up to four-dimensional cases (in particular, three-dimensional case in optics of anisotropic media) it is very favorable to possess a notation which distinguishes the invariants from vectors and operators and is easily to recognize as such. We introduced the notation A for the trace of an operator A in arbitrary dimension and denote the other invariants with respect to similarity transformations as follows ( ) The great initiator of coordinate-invariant methods in optics of anisotropic media, in the theory of the Lorentz group and in elasticity theory was F.I. Fyodorov from Minsk [21] (he called this Covariant methods) and also we published in the seventies some papers to the optics of anisotropic media with application of coordinate-invariant methods (approximately 10 in "Ann. d. Physik") which we do not cite here. However, we hope that we find opportunity to represent much more about the very favorable coordinate-invariant methods in future.

Appendix B: Eigenvalue and Eigenvector Problem in Three-Dimensional Case in Coordinate-Invariant Form
We consider here the case of three-dimensional operators and sketch the solution of the problem to determine eigenvectors to eigenvalues in coordinate-invariant form.