_{1}

^{*}

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
*N*-1 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.

The main purpose of this article is to examine an application of the Chebyshev polynomials of both kinds

In the introductory sections we consider the most important properties of these polynomials for our aim. We embed the Chebyshev polynomials into the greater frame of Ultraspherical polynomials

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 [

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

The considerations are important for applications to functions of two-dimensional operators in physics illustrated in Appendix C by an example.

We apply there the Chebyshev polynomials to an interesting problem of relativistic kinematics which uses powers of Special Lorentz transformations for a uniformly accelerated system (space-ship) and which is connected with the application to basically two-dimensional operators. We work with coordinate-invariant methods which are often very advantageous and explain this in Appendix A.

We compile in this Section without proof some known basic relations for Chebyshev polynomials of first kind

The Rodrigues-type formula of the definition of Jacobi polynomials in the very successful form with notation

The Jacobi polynomials are the following special case of the Hypergeometric function

with the symmetry

A relation connected with an argument transformation in the Jacobi polynomials of the form [

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

Two essentially different expansions of the Jacobi polynomials are

In general, a simple form of the Taylor series of Jacobi polynomials in powers of z does not exist since the summations in formulae for the coefficients

Furthermore, in general, all powers of z from zero up to degree n are included with non-vanishing coefficients in

The special case ^{1}. This case admits the following new representation by the Hypergeometric function in comparison to (2.2)

which is the possible application to (2.2) of a quadratic transformation of Gauss and Kummer [

Sometimes, the Gegenbauer polynomials

where the coefficients on the right-hand side do not depend on the degree n of the polynomial that, however, is the case for differentiation of

in comparison to

for the Ultraspherical polynomials.

The Ultraspherical polynomials possess a transformation which for even

They are a consequence of the quadratic transformations of the Hypergeometric function

The Chebyshev polynomials of first kind

and Legendre polynomials by

However, this cannot successfully be continued to upper index

defined by the Ultraspherical polynomials

They possess unique properties among all Ultraspherical polynomials. The limiting transition used in (2.15) including the number

Since

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

The recurrence relations for

special case

As special cases

follows that the polynomials for even

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

Legendre polynomials

Chebyshev polynomials of first kind

The first written expansions are the direct specializations from (2.5).

The representations for the Chebyshev polynomials

and for

According to (2.9) the Gegenbauer polynomials with higher upper parameter

in particular, from

The case of Legendre polynomials leads to semi-integer fractional integration (e.g., [

corresponding to ^{2}

These integrals can be transformed to a representation by trigonometric functions (see next Section) but, apparently, they are not expressible in short closed form by well-introduced functions.

From the argument substitutions in the Ultraspherical polynomials

lead to new polynomials but after multiplication with certain powers of

Inserting this substitution one obtains immediately from (3.2) the following expansions

One may see that the first of the two expansions in (3.13) can be expressed in the following way

Thus the polynomials

by the substitution ^{3}. Making the substitution

Ultraspherical polynomials

We consider the simplest special cases.

If we make the substitution of the argument of the Chebyshev polynomials of first kind corresponding to

The case to the Legendre polynomials corresponding to

With the same substitutions of the argument of the Chebyshev polynomials of second kind corresponding to

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

The also well-known Lucas numbers

The Fibonacci numbers possess a known relation to the Golden ratio and to the Chebyshev polynomials of second kind

and they are related, among others (multiplicative ones), by [

For convenience we give a short table of the Fibonacci and the Lucas numbers (

n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | (3.24) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

F_{n} | 0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | 89 | 144 | 233 | 377 | 610 | |

L_{n} | 2 | 1 | 3 | 4 | 7 | 11 | 18 | 29 | 47 | 76 | 123 | 199 | 322 | 521 | 843 | 1364 |

One may construct “similar” kinds of number sequences by changing the arguments of the functions, for example,

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 (

n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | (3.26) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Pn | 1 | 1 | 3 | 7 | 19 | 51 | 141 | 393 | 1107 | 3139 | 8953 | |

1 | 1 | 5 | 13 | 49 | 161 | 581 | 2045 | 7393 | 26,689 | 97,285 | ||

1 | 1 | 9 | 25 | 145 | 561 | 2841 | 12,489 | 60,705 | 281,185 | 1,353,769 |

The importance of such sequences of numbers rises if one finds applications, for example, in combinatorics.

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 ^{4} and polynomials that leads to a unique property of Chebyshev polynomials of first kind

One may choose another normalization of the Chebyshev and Legendre polynomials according to

These graphics and those for

If we make the substitution

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

or using variable x

that is also easily to prove by complete induction. In the representation of this relation we have already taken into account the continuation of the polynomials

For the Chebyshev polynomials of second kind

Using

for all integer n from which immediately follows

The inversion of the relation (4.5) is

or by variable x

Both Formulas (4.4) and (4.9) for the inversion of the Chebyshev polynomials take on their simplest form with the extension of the polynomials to negative indices and, astonishingly, both formula are identical after exchange of the kind of Chebyshev polynomials.

We now calculate which trigonometric functions represent the Gegenbauer polynomials

This and the corresponding relations for other integer and semi-integer parameter

They are symmetric or antisymmetric with respect to reflection of the index at

polynomials

We now investigate the Gegenbauer polynomials with semi-integer upper parameters

semi-integer integration from ^{5}

with the correct special case for

A closed relation similar to the kind in case of the Chebyshev polynomials is hardly to find for

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 Gegenbauer polynomials

where it is possible to use the inverse substitution

The more general formula expressed in arbitrary Gegenbauer polynomials

from which (4.17) is the special case

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 (e.g., [

and, more, generally for the Gegenbauer polynomials

Another kind of generating functions is (e.g., [

The more general result is representable by (modified entire) Bessel functions as distinguishing part.

It is sometimes favorable to consider another normalization of the Ultraspherical polynomials ^{6}

We find then independently of the upper parameters

that is sometimes advantageous and the graphics to different parameter

and the formula for the differentiation is

This means, however, that not all formulae become very simple in the form of the polynomials

special cases of parameters

It is favorable that the normalization coefficients in

indices the sign in the relation

(4.11) using relations between factorials of negative and positive numbers then follows

with only positive sign on the right-hand side as advantage. This is true for the general case of

We discuss in this Section unique properties of Chebyshev polynomials of first kind

This means that for composite indices

Thus the Chebyshev polynomial

It is easy to construct similar relations, for example, for

We now consider the Chebyshev polynomials of second kind

or expressed in variable

As example for illustration of this relation we choose

With notation

(2.12) are a generalization in other direction as the here considered nested relations for the Chebyshev polynomials.

There exist also many interesting relations between Chebyshev polynomials of second and first kind. An interesting relation between Ultraspherical or Gegenbauer polynomials with Chebyshev polynomials of first kind is given in (4.15) and (4.16). All these relations possess a full counterpart in trigonometric identities using (4.1) and (4.5) and this is well known. For example, for the Chebyshev polynomials of second kind

and, in particular

Other forms of identities for

and for

Many relations for Chebyshev polynomials of both kinds and between them which are connected with recurrence relations and with differentiations one may find in tables (e.g., [

Another unique property of the Chebyshev polynomials which is restricted to the polynomials of first kind

The expansion of functions in series of Chebyshev and, more generally, of Ultraspherical or Gegenbauer polynomials and even Jacobi polynomials is connected with the completeness and orthogonality of these function sets. The completeness for continuous and infinitely continuously differentiable functions

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.

We consider the expansion of a sufficiently well-behaved function

Due to orthogonality relations

the coefficients

The similarity of expansions of functions ^{7}

with the new interval

however, with a peculiarity. This peculiarity is the additional symmetry

which is already repeated of the same kind after the half of the full 2π-period of the function

and due to 2π periodicity around

The chosen function is

Using now the relation

then due to symmetry of

with the coefficients given by integrals not over the full 2π period and with fixed limits

If we use the symmetry (6.6) then the formula for the coefficients (6.10) can be also represented by

with arbitrary

One may displace the whole picture of

The reason for the additional symmetry in the mapping of functions

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

where

which are two independent invariants of the operator

First we make a simplification under the supposition of nonvanishing determinant (^{8}

Furthermore, we introduced the abbreviations

From this relation follows for higher powers of

After making some few iterations of the elimination of higher powers of

where

that proves (7.6) and we find the necessary recurrence relations for the polynomials

In the special cases

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) are satisfied by both the Chebyshev polynomials

Graphical illustrations for the first four polynomials

For an arbitrary function

and with separation of the two parts proportional to the identity operator

For the same function

We see from this formula that for the final calculation of this reduction to a linear combination of the operators

In this section, we calculate functions

The solution of the eigenvalue problem of two-dimensional operators

consists of the determination of the eigenvalues

and the determination of right-hand eigenvectors

where the substitutions (7.3) are used. It does not make a restriction of the generality to suppose nondegeneracy of the eigenvalues

Using the Hamilton-Cayley identity (7.1) we now define the complementary operator

First of all, the complementary operator serves for the determination of the inverse operator to

Then one may determine projection operators

It is easy to see that

for arbitrary vectors

By inserting in (8.6) the explicit form of the eigenvalues

where again the substitutions (7.3) are used. In the same way, we find

According to (8.8), an arbitrary operator function

where

As first example for the reduction of a function of the operator

The case of

Another interesting example is the function

where we used the identity

The important case of an exponential function

With (7.13) and (8.11) we derived in Sections 7 and 8 two different representations of functions

and second from terms proportional to

Both identities possess the form of generating functions for the Chebyshev polynomials of second kind

where we used the relation (provable by complete induction or by trigonometric equivalent)

Using

and the identity (9.2) using

Apart from the monomials

We may check for the functions

Then from (9.5) and (9.6) easily follows

The relation which follows from (9.1) is a linear combination of these identities.

Next we consider an exponential function

Then from (9.5) and (9.6) follows (compare with (4.21))

These generating function are also known and are affirmed by program “Mathematica”.

We consider a third example with analytic modified Bessel functions at

For this function follows from (9.5) and (9.6)

Another interesting example is related to the function

For this example one finds

By separation of the even and odd parts with respect to variable t one may gain further Generating functions.

In this section we consider in detail the exponential function

The operator

is proportional to the identity operator

is the exponential of an operator here abbreviated

This is identical to the more specialized representation of the operator

A vanishing trace of an operator is usually obtained from the assumption of its antisymmetry according to

where the superscript '^{9}.

We mention that a two-dimensional operator

where

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

Substituting

This shows that

In case of

In case of

We consider now the special case if the determinant

This means that the operator

The second eigenvalue is equal to

operator for the determination of right-hand and left-hand eigenvectors to the eigenvalue

that results from the Hamilton-Cayley identity (7.1) under the supposition

If in addition to

and either

or it is non-vanishing and from (11.6) follows

The operator belongs in case of

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.

The author declares no conflicts of interest regarding the publication of this paper.

Wünsche, A. (2019) Chebyshev Polynomials with Applications to Two-Dimensional Operators. Advances in Pure Mathematics, 9, 990-1033. https://doi.org/10.4236/apm.2019.912050

Let

with identity operator

The determinant

The eigenvalues

The relation between the coefficients

The determinant

where s is an arbitrary permutation of N elements and

If

Using now the fully antisymmetric unit pseudo-tensor

where

with s an arbitrary permutation according to (A.4). For the determinant according to definition (A.5) one finds then

where

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

These notations are compatible concerning the dimension. For three-dimensional operators

Formally, the descent by one dimension is the division of the Hamilton-Cayley identity by the operator

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 [

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.

An operator

The meaning of the invariants is given in (A.11) where

The complementary operator

The inverse operator can be expressed by the complementary operator as follows

For the complementary operator

We consider first the special case of an eigenvalue

An arbitrary vector

Therefore, the operator

We consider now an arbitrary eigenvalue

It has to satisfy the eigenvalue equation

For the complementary operator

and its trace is

Therefore, the projection operator

With the three, in general, different eigenvalues

In this way, the functions

Notation: Vectors

We deal with here an interesting example where the application of Chebyshev polynomials of first and of second kind plays a role. It is connected with powers of Special Lorentz transformation which are, essentially, two-dimensional operators although we calculate with four-dimensional operators and the results are interesting for a uniformly accelerated space-ship.

We consider two inertial systems I and I'. In the inertial system I which we consider as resting (say earth) a body (say space-ship) starts with a velocity

It is well known that the Special Lorentz transformation from of a space vector

with the abbreviations (c is light velocity)

The inversion of (C.1) to

The Special Lorentz transformations of wave vectors

or by separation of the wave vector

In four-dimensional wave-vector-frequencies k and space-time vectors r according to

one has to require the invariance

The Special Lorentz transformation ^{10}

It is now evident that according to

the required invariance (C.7) is satisfied.

The transformation from inertial system I after n described steps to

and due to the same direction of the velocity

The general result is

Therefore the n-th power (C.10) of the Lorentz transformation

With

as the correct result.

With the two identities (see also (3.7) and (3.8))

and using it in (C.12) with the substitution

where we used

The coefficients

We now make the limiting transition from discrete steps of addition of a velocity

In the system

where we used the well-known limiting transition

the meaning of

The limiting transition for

in proper time

The transformation of the time T' from system of the space-ship to corresponding T of the system I of earth can be made by using the inversion of (C.1) to

The integration of both sides provides

with the inversion

This is the transition of the time T' from the space-ship to the corresponding time T in the system of earth and means that the time up to arrival to an object is for the space-ship travelers smaller than for the earth residents.

The way

The integration from

To find the way which takes the space-ship to the proper time T in the system I of earth one has to substitute T' according to (C.24) by T but this is not directly controllable since we cannot have an instant connection with the system of the space-ship and the times in each of the two systems are synchronized before. With the Formulae (C.4) one may discuss the change of wave vectors

We do not discuss the formulae here more in detail and mention that the transition to a continuous acceleration (no more an inertial system) is also not without problems^{11}.