Chebyshev Polynomials with Applications to Two-Dimensional Operators ()
1. Introduction
The main purpose of this article is to examine an application of the Chebyshev polynomials of both kinds
and
to the reduction of two-dimensional operators and its possible generalization to three-dimensional operators with application of the corresponding Hamilton-Cayley identities. This is made in coordinate-invariant form which is shortly sketched in Appendix A.
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
which are orthogonal in the finite interval
and give the connection to the Hypergeometric function
with the Jacobi polynomials
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
and to a general case of Generating functions. The problem connected with the application of the reduction to different operator functions and the calculation of corresponding Generating functions of the polynomials is solved. This is the second great problem for application which we deal with for the two-dimensional case. We attacked but could not finish up to now the solution of some problems which are connected, in particular, with the calculation of Generating functions for the new polynomials in the three-dimensional case. The results may find application in the group theory but many groups with basically three-dimensional operators do not need the general case of three-dimensional operators and the corresponding problems are solved already by more special approaches.
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.
2. 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
and of second kind
including for rationality also Legendre polynomials
as intermediate case. This illuminates their position within the Hypergeometric function
and their polynomial cases which all are representable as Jacobi polynomials
with their special case of Ultraspherical polynomials
or, almost fully equivalently to the last, the Gegenbauer polynomials
in the standard notations [1] [2] .
The Rodrigues-type formula of the definition of Jacobi polynomials in the very successful form with notation
introduced by Szegö [1] is [2] - [8]
(2.1)
The Jacobi polynomials are the following special case of the Hypergeometric function ![]()
(2.2)
with the symmetry
(2.3)
A relation connected with an argument transformation in the Jacobi polynomials of the form [1]
(2.4)
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
(2.5)
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
cannot be calculated in closed form. The differentiation of Jacobi polynomials leads again to Jacobi polynomials but with changed parameters ![]()
(2.6)
Furthermore, in general, all powers of z from zero up to degree n are included with non-vanishing coefficients in
. This changes radically in the special case
with powers only in steps of two from the maximal one downwards.
The special case
of the Jacobi polynomials is called the Ultraspherical polynomials1. This case admits the following new representation by the Hypergeometric function in comparison to (2.2)
(2.7)
which is the possible application to (2.2) of a quadratic transformation of Gauss and Kummer [9] (Chapter 2.1.5). There are two almost but not fully equivalent forms of Ultraspherical polynomials
and
where the lasts are called Gegenbauer polynomials and which are related to each other by
![]()
(2.8)
Sometimes, the Gegenbauer polynomials
possess advantages in comparison to Ultraspherical polynomials
, for example, in case of differentiation
(2.9)
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
(see (2.6)). Despite their equivalence the recurrence relations for the Gegenbauer polynomials
possess a simpler form than that for
and are
(2.10)
in comparison to
(2.11)
for the Ultraspherical polynomials.
The Ultraspherical polynomials possess a transformation which for even
and odd
leads to special Jacobi polynomials with transformed argument and unequal upper parameters as follows (Szegö [1] )
![]()
(2.12)
They are a consequence of the quadratic transformations of the Hypergeometric function
in case of
with a result which cannot be expressed by the Gegenbauer polynomials
alone.
The Chebyshev polynomials of first kind
and of second kind
and the Legendre polynomials
are important special cases of the Ultraspherical polynomials
. In particular, Chebyshev polynomials of second kind are equivalently defined by
(2.13)
and Legendre polynomials by
(2.14)
However, this cannot successfully be continued to upper index
since these polynomials are
that means they are different from zero only for
. Instead of this the Chebyshev polynomials of first kind
are
defined by the Ultraspherical polynomials
as
(2.15)
They possess unique properties among all Ultraspherical polynomials. The limiting transition used in (2.15) including the number
provides
(2.16)
Since
is not regularly defined as special case
of the Gegenbauer polynomials
one has to expect peculiarities in this special case which do not follow from the general case of Gegenbauer polynomials and must be separately derived. For example, they do not satisfy the recurrence relations (2.10) if the polynomials
are included. The recurrence relations for the three considered special series of polynomials can be written
![]()
![]()
(2.17)
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
that finds its explanation after transition to trigonometric polynomials (Section 4).
The recurrence relations for
and for
are specializations
and
of (2.10) where that for
was additionally divided by the common factors
. The recurrence relation for
arises from
special case
in (2.11) using the definition (2.15) and division by common factors.
3. Series Expansion of Ultraspherical and Gegenbauer, Chebyshev and Legendre Polynomials and Fibonacci and Lucas Numbers
As special cases
of Jacobi polynomials
the Ultraspherical polynomials
and equivalently the Gegenbauer polynomials
possess series expansions which follow from (2.5) by corresponding specialization. From the symmetry
(3.1)
follows that the polynomials for even
can only contain even powers of z and for odd
only odd powers of z. From the two series representations of the Hypergeometric function in (2.5) follow then by reordering of the arising double sums and evaluating the inner sum for the Ultraspherical polynomials ![]()
(3.2)
or equivalently for Gegenbauer polynomials ![]()
(3.3)
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
(3.4)
Legendre polynomials
(3.5)
Chebyshev polynomials of first kind
(3.6)
The first written expansions are the direct specializations from (2.5).
The representations for the Chebyshev polynomials
and
given at second place in (3.6) and (3.4) lead to the following known interesting representation for
, e.g., [5] [7] [8]
(3.7)
and for ![]()
(3.8)
According to (2.9) the Gegenbauer polynomials with higher upper parameter
can be obtained by differentiation
(3.9)
in particular, from
by
(3.10)
The case of Legendre polynomials leads to semi-integer fractional integration (e.g., [10] ) from the Chebyshev polynomials
and
corresponding to
where
are not polynomials but functions which we take from (3.8)2
(3.11)
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
, in particular, the substitution
is interesting. At first, it does not
lead to new polynomials but after multiplication with certain powers of
one obtains new polynomials. In [11] we denoted with
the following series of (non-orthogonal) polynomials
(3.12)
Inserting this substitution one obtains immediately from (3.2) the following expansions
(3.13)
One may see that the first of the two expansions in (3.13) can be expressed in the following way
(3.14)
Thus the polynomials
may be generated by application of an integral operator onto powers of
. This integral operator is built by the entire function formed from the Bessel functions
according to (see, e.g., [7] [9] )
(3.15)
by the substitution
of the variable u by the differentiation operator
. It is important that this operator is independent of index n of the generated polynomials3. Making the substitution
, in representations of the
Ultraspherical polynomials
, for example in (2.7) or in (2.2) with
, one may find expressions of the polynomials by the Hypergeometric function. This leads to the following possible expressions (we omit now again the primes)
(3.16)
We consider the simplest special cases.
If we make the substitution of the argument of the Chebyshev polynomials of first kind corresponding to
we find
(3.17)
The case to the Legendre polynomials corresponding to
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)
(3.18)
With the same substitutions of the argument of the Chebyshev polynomials of second kind corresponding to
we find
(3.19)
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
can be obtained in the following way from the here considered polynomials, series and functions (e.g., [13] for last representation)
(3.20)
The also well-known Lucas numbers
can be obtained analogously by ( [13] for last representation)
(3.21)
The Fibonacci numbers possess a known relation to the Golden ratio and to the Chebyshev polynomials of second kind
and the Lucas numbers a relation to the Chebyshev polynomials of first kind
. They play an important role in combinatorics due to their recurrence relations which are the same for both types
and
but with different initial numbers
(3.22)
and they are related, among others (multiplicative ones), by [13]
(3.23)
For convenience we give a short table of the Fibonacci and the Lucas numbers (Table 1).
![]()
Table 1. Fibonacci and Lucas numbers.
One may construct “similar” kinds of number sequences by changing the arguments of the functions, for example,
and
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
![]()
![]()
(3.25)
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.
4. 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
and which are related to Trigonometric functions4 and polynomials that leads to a unique property of Chebyshev polynomials of first kind
. To these properties belong also the orthogonality relations of the Ultraspherical and Gegenbauer polynomials with their special cases within the basic interval
(Figure 1 and Figure 2).
One may choose another normalization of the Chebyshev and Legendre polynomials according to
from which results
that leads to more similarity of all Figures. The polynomials
and
corresponding to
and
are then directly defined by
and the amplitude of the polynomials
corresponding to
is reduced by the factors
and we have
(see Figure 2). However, some formulae as, for example, the differentiation of the polynomials
become then more difficult than for Gegenbauer polynomials
. The same factor
could be also excluded in the definition of (3.14).
![]()
Figure 2. The first 4 modified Chebyshev polynomials of second kind
and their transforms
.
These graphics and those for
with higher parameter
are more similar to the graphics in the first two lines in Figure 1.
If we make the substitution
in the Chebyshev polynomial of first kind
then we find from the expansions in (3.6)
(4.1)
This is well known and can be easily proved by complete induction. Since
one can tentatively define
(4.2)
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
(4.3)
or using variable x
(4.4)
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
to negative indices n.
For the Chebyshev polynomials of second kind
one obtains by the substitution
in corresponding way from (3.4) the following well-known relations
(4.5)
Using
one may continue also in this case the polynomials formally to negative indices n by defining
(4.6)
for all integer n from which immediately follows
(4.7)
The inversion of the relation (4.5) is
(4.8)
or by variable x
(4.9)
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
after the substitution
. According to (2.9) we find the already complicated expressions
(4.10)
This and the corresponding relations for other integer and semi-integer parameter
allows to extend the Gegenbauer polynomials to negative indices with
(4.11)
They are symmetric or antisymmetric with respect to reflection of the index at
in dependence on m an odd or an even number. However, the
polynomials
make here an exception since they are not determined by the Gegenbauer polynomial
which themselves are only determined by a limiting transition.
We now investigate the Gegenbauer polynomials with semi-integer upper parameters
. It is hardly possible to find a closed representation for them similar to
or to
. By
semi-integer integration from
we obtain from the already prepared intermediate result by substitution of the integration variable
(Gradshteyn, and Ryzhik [7] , 3.675, with hint to Whittaker and Watson [14] )5
(4.12)
with the correct special case for
, ![]()
(4.13)
A closed relation similar to the kind in case of the Chebyshev polynomials is hardly to find for
but one may write down expansions. From (3.5), for example, follows
(4.14)
One may transform these results into a series over Chebyshev polynomials of first kind as follows
(4.15)
The more general relation of this kind takes on its simplest form expressed by Gegenbauer polynomials
and possesses the form
(4.16)
where it is possible to use the inverse substitution
. The basic monomials
can be represented by the Legendre polynomials according to
(4.17)
The more general formula expressed in arbitrary Gegenbauer polynomials
possesses the form
(4.18)
from which (4.17) is the special case
. 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 (e.g., [2] [3] [5] [7] )
![]()
![]()
(4.19)
and, more, generally for the Gegenbauer polynomials
(4.20)
Another kind of generating functions is (e.g., [5] and [11] )
![]()
![]()
(4.21)
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
or Gegenbauer polynomials
as follows6
(4.22)
We find then independently of the upper parameters ![]()
(4.23)
that is sometimes advantageous and the graphics to different parameter
become more similar to each other (see Figure 1 and Figure 2). The recurrence relation becomes
(4.24)
and the formula for the differentiation is
(4.25)
This means, however, that not all formulae become very simple in the form of the polynomials
. For example, the formula for the differentiation takes on the most simple form using the Gegenbauer polynomials
(see (2.9)) but the last fail to act for the Chebyshev polynomials of first kind
. The
special cases of parameters
are
![]()
![]()
(4.26)
It is favorable that the normalization coefficients in
are the same as this would be for the favorable choice of the normalization of
such as proposed in (3.14). Furthermore, it is favorable that in the extension to negative
indices the sign in the relation
in (4.6) changes to a positive sign for
according to
. More generally, from
(4.11) using relations between factorials of negative and positive numbers then follows
(4.27)
with only positive sign on the right-hand side as advantage. This is true for the general case of
with integer and semi-integer parameter
.
5. Unique Properties of the Chebyshev Polynomials
We discuss in this Section unique properties of Chebyshev polynomials of first kind
and write as variable z when the obtained relations are true for arbitrary complex z. First, we consider composite indices
in Chebyshev polynomials of first kind. From (4.1) follows
(5.1)
This means that for composite indices
the Chebyshev polynomials
possess the nested forms
(5.2)
Thus the Chebyshev polynomial
for composite indices n can be found from the Chebyshev polynomials
and
by the simple nested construction (5.2). This is very similar to the unique decomposition of a natural number into a product of prime numbers and, in principal, the Chebyshev polynomials of first kind need only to be given for prime-number indices and one can build all others in simple way by the nested construction with the possibility of variations by the number of permutations of the prime numbers of the composite index. For example, if we take
we can represent the even Chebyshev polynomials
in the following two ways
![]()
(5.3)
It is easy to construct similar relations, for example, for
and to consider many other examples.
We now consider the Chebyshev polynomials of second kind
for odd indices
. Then from (4.5) follows
(5.4)
or expressed in variable ![]()
(5.5)
As example for illustration of this relation we choose
and find by specialization of relations in (5.5)
(5.6)
With notation
and
these polynomials are sometimes separately taken into account in tables. We notice here also a great similarity of relations (5.3) and (5.6) to the relations (2.12) which for
or
, respectively, are identical. For arbitrary
the relations
(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
with composite indices
one may derive in analogous way to (5.5) plus using addition theorems for trigonometric functions the relation
(5.7)
and, in particular
(5.8)
Other forms of identities for
are
(5.9)
and for ![]()
(5.10)
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., [2] [3] [5] [7] ).
Another unique property of the Chebyshev polynomials which is restricted to the polynomials of first kind
is that for a given function
in the interval
they provide in approximations of each degree of the polynomial the best approximation by some criteria. One criterium for this is that within the mentioned interval the maximal modulus of the deviation of the approximation from the values of the function
is minimal. This was initiated by Chebyshev and further developed by many authors (e.g., Akhieser [15] , Nikolski [16] in [17] , Rivlin [18] ). Via a substitution in the Chebyshev polynomial of first kind this is related to a known similar property of Fourier series in the expansion of periodic functions.
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 the neighborhood of a considered point
is connected with the presence of a polynomial of each degree
within the set of polynomials. The known orthogonality relations for the Ultraspherical polynomials within the interval
are (e.g., [1] [2] [3] [5] )
(5.11)
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.
6. Relation of Expansions in Chebyshev Polynomials
to Fourier Series of 2π-Periodic Functions
We consider the expansion of a sufficiently well-behaved function
within the interval
in a series of Chebyshev polynomials of first kind
(6.1)
Due to orthogonality relations
(6.2)
the coefficients
of the expansion are then determined by the formula
(6.3)
The similarity of expansions of functions
within the interval
in Chebyshev polynomials of first kind
to Fourier series of 2π-periodic functions
can be established by an argument transformation in the expansion (6.1), for example7
(6.4)
with the new interval
corresponding to the primary interval
. Then one obtains a 2π-periodic function
(6.5)
however, with a peculiarity. This peculiarity is the additional symmetry
(6.6)
which is already repeated of the same kind after the half of the full 2π-period of the function
, in our choice of the transformation (6.4), around
according to
(6.7)
and due to 2π periodicity around
. In general,
due to only 2π-periodicity of the whole function
but the symmetries (6.6) and (6.7) are repeated in each further 2π-period (Figure 3).
The chosen function is
corresponding to
.
Using now the relation
(6.8)
then due to symmetry of
the expansion (6.1) can be transformed according to
(6.9)
with the coefficients given by integrals not over the full 2π period and with fixed limits
(6.10)
If we use the symmetry (6.6) then the formula for the coefficients (6.10) can be also represented by
(6.11)
with arbitrary
. 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 may displace the whole picture of
to the right by a value
on the
-axis by choosing a mapping
, in particular with
by the mapping
. The function
is then in last case only symmetric around the values
and
but not around
. One obtains then a Fourier series instead of (6.9) including also sum terms containing
with coefficients only specialized in last case by the symmetries around
. The mapping to arbitrary period lengths of functions
also does not make difficulties but in all cases we obtain only Fourier series with additional symmetries. Thus the expansion of a function
in a series over Chebyshev polynomials
to functions
is not fully equivalent to a general expansion of an arbitrary periodic function in a Fourier series.
The reason for the additional symmetry in the mapping of functions
onto periodic functions from the basic interval
onto the basic period of 2π is that the function
(and all other similar functions but not Tangent-like functions) is not monotonically increasing but repeats decreasingly its values in the second half of the period 2π. The decomposition of functions into series of the higher Ultraspherical polynomials
should provide after the transformation to
alternative decompositions of periodic function in comparison to Fourier series with lower importance and, apparently, are not interesting enough up to now.
7. 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
that means operators which satisfy the following two-dimensional Hamilton-Cayley identity
(7.1)
where
and
denote the trace and the determinant of the operator
according to
(7.2)
which are two independent invariants of the operator
with respect to similarity transformations. Our first aim is to reduce powers
by means of the Hamilton-Cayley identity (7.1) to linear combinations of the operator
and of the identity operator
with coefficients which are functions (polynomials) of the invariants of the operator
.
First we make a simplification under the supposition of nonvanishing determinant (
, non-degenerate case) and introduce a new operator
with determinant equal to 1 as follows8
(7.3)
Furthermore, we introduced the abbreviations
and
which play the role of variables in the following considerations. The case of vanishing determinant
is either essentially the one-dimensional case or a Jordan normal form with zeros in the main diagonal and one nonvanishing number in the off-diagonal and can be dealt with as a limiting case. We come back to this later. With the introduced variable x the Hamilton-Cayley identity for the operator
may be written in the form
(7.4)
From this relation follows for higher powers of ![]()
(7.5)
After making some few iterations of the elimination of higher powers of
from this equation by means of (7.4) one finds that
can be represented in the following form of the superposition of the operators
and ![]()
(7.6)
where
is a polynomials of
of the degree n and one sees that the polynomials in front of
and of
are essentially the same if one first introduces different ones. By complete induction
follows with application of (7.4)
(7.7)
that proves (7.6) and we find the necessary recurrence relations for the polynomials ![]()
(7.8)
In the special cases
follows from (7.6) by comparison with (7.4)
(7.9)
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
of first kind and by Chebyshev polynomials
of second kind (see 2.17) but only the Chebyshev polynomials of second kind obey the initial conditions (7.9) and, therefore, the solution is
(7.10)
Graphical illustrations for the first four polynomials
are given in Figure 1.
For an arbitrary function
of the operator
which can be defined by a Taylor series
in a neighborhood of
one obtains using (7.10) together with (7.6)
(7.11)
and with separation of the two parts proportional to the identity operator
and the operator ![]()
(7.12)
For the same function
of the more general operator
this means
(7.13)
We see from this formula that for the final calculation of this reduction to a linear combination of the operators
and
for a given function
one should possess the result for the corresponding sums in (7.13) containing the Chebyshev polynomials of second kind. They may be considered as Generating functions to these polynomials in a wide sense.
8. Solution of Eigenvalue Problem for Two-Dimensional Operators and Arbitrary Functions of Operators
In this section, we calculate functions
of two-dimensional operators
and represent these as superposition of the two linear independent operators
and
. As preparation we consider the solution of the eigenvalue problem of the operator
in coordinate-invariant form.
The solution of the eigenvalue problem of two-dimensional operators ![]()
(8.1)
consists of the determination of the eigenvalues
by means of the secular equation
(8.2)
and the determination of right-hand eigenvectors
and left-hand eigenvectors
to the eigenvalues
. Instead of the eigenvectors, we determine below projection operators to these eigenvectors. We denote the two, in general, different solutions of the eigenvalue Equation (8.2) by
according to
(8.3)
where the substitutions (7.3) are used. It does not make a restriction of the generality to suppose nondegeneracy of the eigenvalues
because the degenerate case
can be dealt with by a limiting procedure that, however, is not necessary to this moment for our purpose.
Using the Hamilton-Cayley identity (7.1) we now define the complementary operator
to an arbitrary two-dimensional operator
by
(8.4)
First of all, the complementary operator serves for the determination of the inverse operator to
according to
(8.5)
Then one may determine projection operators
and
to the eigenvalues
and
by
(8.6)
It is easy to see that
are projection operators for the determination of eigenvectors to the eigenvalues
and that they satisfy the relations
![]()
(8.7)
for arbitrary vectors
and
. This means that
is either a right-hand eigenvector
of
to eigenvalue
or is vanishing and
is either proportional to
or is vanishing, correspondingly. The identity operator
, the operator
and arbitrary operator functions
can now be represented by means of the projection operators
and
in the following way
![]()
![]()
(8.8)
By inserting in (8.6) the explicit form of the eigenvalues
given in (8.3), we find the following representation of ![]()
(8.9)
where again the substitutions (7.3) are used. In the same way, we find
(8.10)
According to (8.8), an arbitrary operator function
can be represented in the following way by a linear combination of the operators
and ![]()
(8.11)
where
and
are defined in (7.3) as parameters from the invariants of the operator
. This has the same form as the representation in (7.13) and the identification of the functions in front of
and
provides Generating functions for the Chebyshev polynomials of second kind
. We discuss this in the next section.
As first example for the reduction of a function of the operator
to a superposition of the operators
and
we find from (8.11)
(8.12)
The case of
with arbitrary
is similar but with few possibilities for simplifications in comparison to the general Formulae (8.11).
Another interesting example is the function
for which we find from (8.11)
(8.13)
where we used the identity
(8.14)
The important case of an exponential function
is separately discussed in Section 10.
9. 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
of an arbitrary two-dimensional operator
expressed by the two independent basic operators
and
. These two representations have to be equal. If we separate the parts proportional to the identity operator
and to the operator
we obtain first from terms proportional to
the identity
(9.1)
and second from terms proportional to
the identity
(9.2)
Both identities possess the form of generating functions for the Chebyshev polynomials of second kind
. They have a very general form for arbitrary functions
for which the derivatives
at
are well defined and thus the function can be expanded in a Taylor series around this point. By a certain linear combination of these identities of a kind which can be seen from the separated initial terms one obtains the identity
(9.3)
where we used the relation (provable by complete induction or by trigonometric equivalent)
(9.4)
Using
and
the identity (9.3) can be written
(9.5)
and the identity (9.2) using ![]()
(9.6)
Apart from the monomials
in the Taylor series of
itself we do not know other complete sets of polynomials for which Generating functions are derived up to now in such generality.
We may check for the functions
that the Formulas (9.5) and (9.6) lead directly to (known) representations of the Chebyshev polynomials of second and first kind (see (3.7) and (3.8)). As a first other function we consider
(9.7)
Then from (9.5) and (9.6) easily follows
(9.8)
The relation which follows from (9.1) is a linear combination of these identities.
Next we consider an exponential function
(9.9)
Then from (9.5) and (9.6) follows (compare with (4.21))
(9.10)
These generating function are also known and are affirmed by program “Mathematica”.
We consider a third example with analytic modified Bessel functions at
and with the parameter
(compare with similar function (3.15))
(9.11)
For this function follows from (9.5) and (9.6)
![]()
(9.12)
Another interesting example is related to the function
(9.13)
For this example one finds
![]()
(9.14)
By separation of the even and odd parts with respect to variable t one may gain further Generating functions.
10. Exponential Function of a General Two-Dimensional Operator
In this section we consider in detail the exponential function
of a two-dimensional operator
. It is important for applications, for example, in group theory. From general case (8.11) we specialize
(10.1)
The operator
is here decomposed into a product of two commuting operators. The first operator
![]()
(10.2)
is proportional to the identity operator
and its determinant is the exponential of the trace
of the operator
. 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
. The second operator in braces
![]()
(10.3)
is the exponential of an operator here abbreviated
with vanishing trace. Its determinant is therefore equal to 1. If we denote in analogy to (7.3) the parameters of the operator
by
and
where
is vanishing due to vanishing trace then we find for the reduction of the exponential of the operator ![]()
![]()
(10.4)
This is identical to the more specialized representation of the operator
with vanishing trace
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
(10.5)
where the superscript '
' 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 tensor9.
We mention that a two-dimensional operator
can be reduced to a superposition of the operators
and
using the Hamilton-Cayley identity for
in the exponent according to
(10.6)
where
can be dealt with as the operator
by corresponding substitution.
11. 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
that means about the coincidence
. A necessary and sufficient condition is the vanishing of the root in (8.3) that is the condition
(11.1)
Substituting
in the Hamilton-Cayley identity (1), we find
(11.2)
This shows that
is a quadratic nilpotent operator in case of degeneracy of the eigenvalues. We have to distinguish two subcases of different volume
and
.
In case of
, the projection operators
in (9) are no more defined. It can immediately be seen from (11.2) that
for arbitrary vectors
is either a right-hand eigenvector of
or is vanishing and analogously for
with regard to left-hand eigenvectors of
. Due to
, the left-hand and right-hand eigenvectors are orthogonal to each other. This is the case where the matrix to the whole operator
forms a two-dimensional cell in the Jordan normal form and where it cannot be diagonalized by means of a similarity transformation.
In case of
, the operator
is proportional to the identity operator
with the eigenvalue
as proportionality factor.
We consider now the special case if the determinant
of the operator
is vanishing. Then the operator has a vanishing eigenvalue and due to Hamilton-Cayley identity we have
(11.3)
This means that the operator
is idempotent in this case with trace equal to 1 if ![]()
(11.4)
The second eigenvalue is equal to
and the operator
is projection
operator for the determination of right-hand and left-hand eigenvectors to the eigenvalue
according to
(11.5)
that results from the Hamilton-Cayley identity (7.1) under the supposition
. For this case follows from (11.3)
(11.6)
If in addition to
also the trace
is vanishing then due to the Hamilton-Cayley identity (7.1)
is vanishing (nilpotent)
(11.7)
and either
itself is vanishing and
(11.8)
or it is non-vanishing and from (11.6) follows
(11.9)
The operator belongs in case of
then to a Jordan normal form with only one non-vanishing element in one of the off-diagonals and is quadratically nilpotent.
12. 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.
Appendix A: Hamilton-Cayley Identity in General N-Dimensional Case
Let
be a linear operator in an N-dimensional linear space. This operator satisfies the Hamilton-Cayley identity (e.g., [19] [20] [21] )
(A.1)
with identity operator
and with the coefficients
which appear also as coefficients in the following eigenvalue equation of the operator
to eigenvalues ![]()
(A.2)
The determinant
of an arbitrary N-dimensional operator
is here denoted by
and later its trace by
for arbitrary dimension.
The eigenvalues
and the related coefficients
are invariants of the operator
with respect to similarity transformations
by arbitrary nonsingular operators ![]()
(A.3)
The relation between the coefficients
and the eigenvalues
up to their order is reversibly unique and is simple for the traces
and for the determinants
and is more complicated for the other invariants.
The determinant
arises primarily as the coefficient of the transformation of the completely antisymmetric volume product ![]()
![]()
(A.4)
where s is an arbitrary permutation of N elements and
the sign of the permutation (perm.), when transforming the N linearly independent vectors
into N other vectors
by the linear operator operator
according to
(A.5)
If
is a set of N basis vectors
then an arbitrary vector
and
may be represented by (sum convention)
(A.6)
Using now the fully antisymmetric unit pseudo-tensor
(or Levi-Civita symbol) for the representation of the volume product V by the vector components in the basis
we find
(A.7)
where
is the volume product of the basis vectors (volume of elementary cell) and the Levi-Civita symbol is defined by
(A.8)
with s an arbitrary permutation according to (A.4). For the determinant according to definition (A.5) one finds then
(A.9)
where
is defined in fully equal way to
in (A.8) only written with upper indices. The determinant tensor
can be represented by the Kronecker symbol
according to
(A.10)
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
for the trace of an operator
in arbitrary dimension and denote the other invariants with respect to similarity transformations as follows
![]()
![]()
![]()
(A.11)
These notations are compatible concerning the dimension. For three-dimensional operators
we have
and
is the determinant and for two-dimensional operators
we have
and
together and
is the determinant but all other relations remain the same. The Hamilton-Cayley identity in four-, three- and two-dimensional case are (
)
![]()
![]()
(A.12)
Formally, the descent by one dimension is the division of the Hamilton-Cayley identity by the operator
. The inverse operator
to a given operator
can easily be calculated in coordinate-invariant way using the Hamilton-Cayley identity of the corresponding dimension.
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.
An operator
is three-dimensional if it satisfies the three-dimensional Hamilton-Cayley identity
(B.1)
The meaning of the invariants is given in (A.11) where
and all higher invariants are also vanishing in three-dimensional case. Due to the Hamilton-Cayley identity (B.1) all powers
and functions
can be reduced to superpositions of the operators
and
.
The complementary operator
to the operator
is defined in three-dimensional case as follows
![]()
(B.2)
The inverse operator can be expressed by the complementary operator as follows
(B.3)
For the complementary operator
using the three-dimensional Hamilton-Caylex identity (B.1) for the reduction of
follows generally
(B.4)
We consider first the special case of an eigenvalue
of
and then the general case. For eigenvalue
it is necessary that the determinant of
is vanishing that means
and according to (B.4) we have then
(B.5)
An arbitrary vector
is right-hand eigenvector and an arbitrary vector
is left-hand eigenvector of
to eigenvalue
if ![]()
(B.6)
Therefore, the operator
is projection operator to the determination of right-hand and left-hand eigenvectors of the operator
to eigenvalue
. We consider here only the non-degenerate cases
and
and do not normalize the eigenvectors. All this can be managed.
We consider now an arbitrary eigenvalue
of a three-dimensional operator
that means
(B.7)
It has to satisfy the eigenvalue equation
(B.8)
For the complementary operator
to the operator
we find
(B.9)
and its trace is
(B.10)
Therefore, the projection operator
for the determination of eigenvectors to the (non-degenerate) eigenvalue
of the operator
is
![]()
(B.11)
With the three, in general, different eigenvalues
(non-degenerate case) of a three-dimensional operator
one may represent functions of this operator in the following form
![]()
![]()
(B.12)
In this way, the functions
of the operator
are reduced to superpositions of the three operators
with coefficients which are functions of the invariants
. Mainly for lack of place we do not consider here degenerate cases. Clearly, it is difficult to write all this explicitly together with the solutions of the eigenvalue Equation (B.8) by the Cardano formulae. On the other side, one may make the reduction of powers
also by introduction of two new sets of polynomials of two variables leading to new identities similar to the case of two-dimensional operators which leads to Chebyshev polynomials of second kind and to a general kind of Generating functions as demonstrated.
Appendix C: An Application of Chebyshev Polynomials to Powers of Special Lorentz Transformations
Notation: Vectors
bold types,
scalar product,
vector product,
dyadic product of two vectors
and
with trace
.
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
and remains with this velocity in the inertial system
meaning that it rests there and after a certain time starts from this inertial system
again with the same velocity
to a new inertial system and moves there with a new velocity
considered in the primary system I. We repeat this in n steps and ask for the velocity
with which the space-ship moves in
relatively to I. It does not play a role that in each new inertial system the velocity is enlarged by a finite velocity
in comparison to the preceding inertial system that is not really makable since at the end we make a limiting transition to a constant acceleration by smaller steps in smaller times and go to the limit of infinitely small steps. This is a problem of kinematics of Special Relativity theory where one has to calculate the product of n Special Lorentz transformations and may consider then the limiting case
.
It is well known that the Special Lorentz transformation from of a space vector
and a time t from inertial system I to inertial system I' moving with velocity
in I possesses the form
(C.1)
with the abbreviations (c is light velocity)
(C.2)
The inversion of (C.1) to
in dependence on
can be made by the substitution
in (C.1). In separation of the space vector
in parts parallel and perpendicular to
the Lorentz transformation (C.1) takes on the form
![]()
(C.3)
The Special Lorentz transformations of wave vectors
and frequencies
is
(C.4)
or by separation of the wave vector
in analogy to (C.3)
![]()
(C.5)
In four-dimensional wave-vector-frequencies k and space-time vectors r according to
(C.6)
one has to require the invariance
(C.7)
The Special Lorentz transformation
can be represented then in four-dimensional coordinate-invariant form as, e.g. [22] (§16)10
(C.8)
It is now evident that according to
(C.9)
the required invariance (C.7) is satisfied.
The transformation from inertial system I after n described steps to
is made by the n-th power
of
. It is clear that it must possess the same structure as (C.8) that means
(C.10)
and due to the same direction of the velocity
in each step we have
(C.11)
The general result is
(C.12)
Therefore the n-th power (C.10) of the Lorentz transformation
can be written explicitly
(C.13)
With
one easily finds that
leads to
given in (C.8) and also the case
using
and
leads to the identity operator I according to
(C.14)
as the correct result.
With the two identities (see also (3.7) and (3.8))
![]()
(C.15)
and using it in (C.12) with the substitution
(see (C.2)) one finds for the velocity
expressed by
in relation to the light velocity c
(C.16)
where we used
(see (C.2)) and introduced coefficients
and obtain
(C.17)
The coefficients
are factors which characterize how near the modulus of the velocity after n described steps in inertial system
comes in comparison to the primary inertial system I of the earth.
We now make the limiting transition from discrete steps of addition of a velocity
in every step in inertial systems
to a continuous function under the assumption that this increase happens to constant time intervals
and introduce a constant acceleration
by definition
(C.18)
In the system
connected with the space-ship the last is to every time
in its coordinate origin
if it was at time
in the coordinate origin of the inertial system I that means at
to the time
. We now consider a time
in the systems
of the space-ship and make with
in Formula (C.17) the following limiting transition
(C.19)
where we used the well-known limiting transition
. With
the meaning of
one obtains then the velocity
of the space-ship in the inertial system I expressed by the time
in the space-ship or what is the same the negatively taken velocity of the earth in the proper time of the space-ship with simple result
(C.20)
The limiting transition for
provides
(C.21)
in proper time
in the system of the spaceship.
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
in dependence on
and setting
in the space ship (time dilatation) and due to dependence of
on time
we have to start from the differential form of this relation
(C.22)
The integration of both sides provides
(C.23)
with the inversion
(C.24)
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
which the space-ship travels in the inertial system I expressed by the time
of the space-ship or negatively taken the way of the earth seen from the space-ship in its proper time can be found by integration of
(C.25)
The integration from
up to a time T' provides
(C.26)
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
and frequency
of signals sent from the system of space-ship to earth or inversely that we will not do here.
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 problems11.
NOTES
1The name results from the Spherical harmonics
which can be represented by the following formula
, where
depends only on
as continuous variable. Among others the polynomials
possess the following representation by Ultraspherical polynomials (positive sign of
is to choose)
and are called Associated Legendre polynomials (Legendre polynomials in case
).
2The lower limit
of the standard fractional integral is here substituted by
. With lower limit
the integral would become undetermined at
and would not provide correct values for
. The correct integrals after transformation to trigonometric functions correspond to the chosen lower limit (next Section).
3In analogous way the Hermite polynomials
can be generated by application of the integral operator
onto the basic monomials
according to
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.
4Outside this basic interval they are related to corresponding Hyperbolic functions.
5The 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
whereas we started from it, possibly in the hope, to find other representations similar to (4.1) and (4.5).
6We made a bracket around the upper parameter
to distinguish them from the different Associated Legendre polynomials
where the upper indices m can take on only integer values (see also footnote in Section 2).
7We choose the Minus in front of
since we want that the interval
is mapped onto the part
of the whole interval
by a monotonically increasing function.
8Properly speaking, we get two related operators
as possible results of the substitution depending on the choice of the sign of
that does not possess influence on the following considerations.
9By means of a symmetrical metric tensor
one may define using covariant and contravariant indices a tensor
from which follows
having used the symmetry of
and now the antisymmetry of
means
.
10The transformation of space and time for the part parallel to the velocity
is essentially a two-dimensional one.
11In addition, we are also not sure what is already calculated and discussed in the literature of astrophysics and space travels and bring this as an application of Chebyshev polynomials.