1. Introduction
The following two differential equations are known as the Chebyshev differential equations:
and
For
, the Pochhammer symbol
is defined by
The classical generalized hypergeometric series
is defined by the formula [
1]
The Chebyshev polynomials are sequences of orthogonal polynomials, usually distinguished between Chebyshev polynomials of the first kind, denoted by
, which obey
and Chebyshev polynomials of the second kind,
, given by
Equations (
5) and (
6) are indeed polynomials due to the definition of the Pochhammer symbol, and they also turn out to be solutions to Equations (
1) and (
2), respectively.
Both sets of Chebyshev polynomials are sequences of orthogonal polynomials: the
are orthogonal with weight function
integrated over the interval
and the
are orthogonal with weight function
over
. Both sequences of polynomials obey the same three-term recurrence relation
but with different initial conditions. The following relationship between
and
is known [
2] (2.48) for
:
The following formula relates the difference of two Chebyshev polynomials of the second kind to a Chebyshev polynomial of the first kind [
3] (p. 9):
If
, then the forward difference of
f, written
, is defined by the formula
and we define a “backwards shift operator” by the formula
. In this article, we investigate solutions of the families of second order difference equations with polynomial coefficients
and
where
, which we call the Chebyshev difference equations of the first and second kind, respectively.
There has been recent interest in discrete analogues of special functions, by which we mean a function
that obeys some qualitatively similar properties to a related well-known function
. For instance, a Bessel difference equation was investigated in [
4], whose solutions were shown to be generalized hypergeometric series with variable parameters. Such “discrete Bessel functions” were applied in [
5] to solve discrete wave and diffusion equations. We define the
operator by
. In [
6], the Bessel difference equation was generalized to the discrete hypergeometric difference equation,
where
denotes a certain operator containing a forward difference. We shall solve Equation (
10) and (
11) in terms of solutions of Equation (
12), and we will develop some of their properties that justify calling these Chebyshev difference equations.
The phrase “Chebyshev difference equation” sometimes appears in the literature, e.g., in the recent article [
7] (40), in reference to the Equation (
7) and in [
8] (5.2) which refers to a scaled version of Equation (
7) for monic Chebyshev polynomials. We do not use the terminology in this way. Instead, we call Equation (
7) the “three-term recurrence” for classical Chebyshev polynomials, and we will find a discrete analogue for it in the sequel.
Other similar sounding functions include an existing “Chebyshev polynomial of a discrete variable” which can be found in [
9] (p. 33) as a special case of the Hahn polynomials. In Ref. [
10,
11], the “
rth discrete Chebyshev polynomial of order
N” is defined. These polynomials are also distinct from the polynomials appearing in this article.
We define the discrete monomials
as “falling factorials”, i.e.,
. Of particular interest is that the falling factorial obeys a “discrete power rule”
. We contrast Equation (
4) with the discrete hypergeometric series,
, defined by
which solves Equation (
12). Discrete special functions defined by an instance of Equation (
13) import the same parameter set
as its analogous continuous special function defined by Equation (
4). Many representations of special functions replace
t in Equation (
4) with an expression of the form “
”, for some constant
. For instance, the sine function is
and the discrete sine is given by the formula
[
6] (Proposition 20). Since given an arbitrary
and
,
, it is not possible to always naively map what appears in the independent variable arguments of functions defined by Equation (
4) to their discrete analogues (
13) in general, explaining the extra parameters. The final argument
appearing in Equations (
5) and (
6) acts as a barrier to a simple importation of Chebyshev polynomials to the discrete case from the continuous case, but we resolve this dilemma in the sequel.
2. Chebyshev Difference Equation
The natural way suggested by prior work to find the discrete analogue of a polynomial is to replace each monomial in it with . We now demonstrate in the following example that this method fails for the Chebyshev polynomials.
Example 1. The first few classic Chebyshev polynomials of the first kind (5) appear in the following: These polynomials obey the recurrence (7). Naively replacing with , we obtain the following possible discrete analogues:and we would obtain the an analogue of (7) by replacing all terms of the form with : However, this fails even in the case :and so the well-known method of finding discrete analogues fails in this case. The problem we have highlighted in Example 1 is caused by the appearance of “
” in the final argument of Equation (
5), and the example demonstrates that the discrete hypergeometric series (
13) cannot create a discrete analogue of a function whose classical hypergeometric representation contains horizontal shifts in the independent variable. To fix this problem, we can simply replace
t with
in Equation (
1) to get
and we do the same in Equation (
2) to get
Of course,
and
solve Equations (
14) and (
15), but they are now in a proper form for obtaining the discrete analogues. We obtain the difference Equations (
10) and (
11) by replacing all terms of the form
from Equations (
14) and (
15) with
.
The discrete Chebyshev polynomials of the first kind,
, are defined by
By applying [
6] (Proposition 2), we see that Equation (
16) may be written in terms of a classical
with a variable parameter as
The discrete Chebyshev polynomials of the second kind,
, are defined by
and similarly applying [
6] (Proposition 2) here yields
Both of these functions are finite sums due to Equation (
3), since
is zero for all
.
The following lemma will be useful in deriving the difference equations for and .
Lemma 1. The following formulas hold:
- 1.
, and
- 2.
.
Proof. For 1. in Lemma 1, calculate
For 2., use 1. and the discrete product rule to calculate
completing the proof. □
The following theorem is a discrete analogue of (
14).
Theorem 1. The polynomials (16) solve Equation (10). Proof. By Equations (
12) and (
16), we know that
satisfies
Apply Lemma 1 and multiply by 2 to get
Expanding yields
and, by algebra, we obtain
Divide by
t and then replace
t with
to arrive at
completing the proof. □
We now establish the discrete analogue of the three-term-recurrence for the discrete Chebyshev polynomials of the first kind.
Theorem 2. The polynomials (16) obey the recurrence relation Proof. Let
. Apply Equation (
16) to each term of Equation (
20) to get
and
Define
and compute
define
and compute
and finally define
and compute
Therefore,
completing the proof. □
In light of Equations (
18) and (
20), the following classical hypergeometric relation is yielded.
Corollary 1. The following formula holds for all and for all : We now demonstrate the difference equation that the discrete Chebyshev polynomials of the second kind solve.
Theorem 3. The polynomials (18) solve Equation (11). Proof. By Equations (
12) and (
18), we know that
satisfies
yielding
Multiply by
, replace
t with
, and expand to get
completing the proof. □
We now prove the discrete analogue of the three-term recurrence for the discrete Chebyshev polynomials of the second kind.
Theorem 4. The polynomials (18) obey the recurrence relation Proof. Let
. Apply Equation (
18) to each term of Equation (
21) to get
and
Define
and compute
define
and compute
and finally define
and compute
Therefore,
completing the proof. □
Using Equation (
19), we immediately obtain a corollary that gives us an interesting identity for
.
Corollary 2. The following formula holds for all and for all : The following formula is a discrete analogue of (
8).
Theorem 5. The following difference formula holds for all : Proof. Taking the difference of Equation (
16) yields
completing the proof. □
The following theorem is a discrete analogue of (
9).
Theorem 6. The polynomials (16) and (18) obey the following recurrence relation for all Proof. Take the difference of (
20) to obtain
By Equation (
22), we get
which simplifies to
but the second term is identically zero by Equation (
21) with
n replaced with
, completing the proof. □
As with many previous results, we obtain a result for here as well.
Corollary 3. The following formula holds for all and for all : Thus far, we have seen multiple properties of the classical Chebyshev polynomials that have direct discrete analogues. We now present an example of a property that is not sustained by the discrete analogue:
Example 2. Given a sequence of orthogonal polynomials , meaning each is of degree n and there is an inner product such that whenever and , it is known [12] (Theorem 3.2.1) that they obey a three-term-recurrence, i.e., there are constants and such that Considering the list of polynomials in Table 1, we will use simple algebra and the contrapositive of [12] (Theorem 3.2.1) to show that the sequence of discrete Chebyshev polynomials (of either kind) does not form a sequence of orthogonal polynomials. First, suppose that Equation (23) holds for the discrete Chebyshev polynomials of the first kind for . This would mean that there exist constants , , and such that This yields the system of equationswhich has no solution. Therefore, the sequence of discrete Chebyshev polynomials of the first kind does not form a sequence of orthogonal polynomials. Now suppose that Equation (23) holds for the discrete Chebyshev polynomials of the second kind for . This means there would exist constants , , and such thatleading towhich similarly has no solution. Hence, the sequence of discrete Chebyshev polynomials of the second kind do not form a sequence of orthogonal polynomials.