Some properties of Chebyshev polynomials

In this paper we obtain some new bounds for Chebyshev polynomials and their analogues. They lead to the results about zero distributions of certain sums of Chebyshev polynomials and their analogues. Also we get an interesting property about the integrals of certain sums of Chebyshev polynomials.

Let and be the Chebyshev polynomials of the first kind and of the second kind, respectively. These polynomials satisfy the recurrence relations

Chebyshev polynomials are of great importance in many areas of mathematics, particularly approximation theory. Many papers and books [3,4] have been written about these polynomials. Chebyshev polynomials defined on are well understood, but the polynomials of complex arguments are less so. Reported here are several bounds for Chebyshev polynomials defined on including zero distributions of certain sums of Chebyshev polynomials. Moreover, we will introduce certain analogues of Chebyshev polynomials and study their properties. Also we get an interesting property about the integrals of certain sums of Chebyshev polynomials.

Other generalized Chebyshev polynomials (known as Shabat polynomials) have been introduced in [5] and they are studied in the theory of graphs on surfaces and curves over number fields. For a survey in this area, see [6].

For and , i.e., , we let

In detail,

and it is easy to show that for an odd integer n,

Since , we may apply results about to those about . But and ϵ was a nonnegative real number less than , and so properties of will be investigated separately from those of .

In this section we list some new results related to the bounds of Chebyshev polynomials , and their analogues , defined on including zero distributions of certain sums of Chebyshev polynomials and their analogues. And we will get an interesting property about the integrals of certain sums of Chebyshev polynomials. We first begin with properties about bounds of , , and . We may compute that for ,

and

Proposition 1Suppose thatzis a complex number satisfying. Then for,

and

Also

and

Proposition 1 will be used in the proofs of Theorems 4 and 6.

Remarks For a complex number z with , we may follow the procedure of the proof of (2) to obtain

and

that is best possible since . There seem to be larger lower bounds than 1 for in (2). First, we observe that

because . Deciding in some general situations exactly where the minimum occurs seems to be extremely difficult. For example, machine calculation suggests that for , takes its minimum 3.91735… in at four modulus 1 roots of the polynomial

But we may conjecture that, by numerical computations, the value

occurs in and lies between and n, where can be replaced by something larger. We now ask naturally what the minimum is for . If one simply looks at the case , it seems that is close to its minimum at . But

is the coefficient of in the power series expansion of . In fact,

For , by (2). In the following proposition, we obtain an upper bound for arbitrary , .

Proposition 2Let, whereand. Then, for,

Remarks With the same notations with Proposition 2, it follows from

that, for large, is large. But and

These imply that the upper bound

is greater than , but for large, it is close to . Also by machine computations (e.g., and ), we may check that the inequality (5) is sharp.

It is natural to ask about the bounds on the unit circle.

Proposition 3For, we have

Remarks The right inequality in (6) will be shown in Section 3 by using (2). So obtaining a better lower bound than 1 in (2) can improve this inequality. The left inequality in (6) is best possible in the sense that

For , it is easy to see

by (4), and it seems to be true that

The proof of (6) will be given in Section 3 by using a well-known identity . But does not hold. So we cannot use this to prove (7) if it is true.

All zeros of the polynomial lie in . More generally the convex combination of and has all its zeros in . This will be proved in Proposition 5 below. So one might ask: where are the zeros of polynomials like or around ? The next theorem answers this for .

Theorem 4Letfor positive integersnandk. Thenhas all its zeros in. Furthermore, forkeven, has at leastnreal zeros, and forkodd, has at leastreal zeros.

Remarks Let for positive integers n and k. We can use the same method as in the proof of Theorem 4 to show that has at least n real zeros in . Furthermore, for k even, there is no real zero outside , and for k odd, there is one more real zero on .

Proposition 5The polynomial

has all zeros in.

Using analogues of and , we consider analogues of and investigate their zero distributions. Define

Theorem 6has all its zeros in.

Finally, we get an interesting property about the integrals of sums of Chebyshev polynomials. Observe that

and

equals

For example, from and we can calculate

and we see that these two integrals are different. But for and , the integrals have the same value.

Proposition 7For,

Remark It seems to be true that for k large, ,

but

These remain open problems.

Proof of Proposition 1 Suppose that z is a complex number satisfying . Using (1), for , we have

and

Then by recurrence,

By (8) and the identity

we have

Next we prove the results about and . For n odd and , it follows from the definition of and (8) that

This inequality

for other three cases (i.e., n odd and , n even and , n even and ) can be proved in the same way. Finally, for n odd, by and (9), we have

In the same way, we can check that for n even,

 □

Proof of Proposition 2 Using the identity , for we have

If we set , , then

Since , it suffices to consider the case . The above implies

So

Also

and

and so

Now we see with (10) and (11) that

 □

Proof of Proposition 3 Suppose that z is a complex number satisfying . First, it follows from (2) that

Using the identity , we have

and so it is enough to show that

We use induction on n. For , . Assume the result holds for k. Then

and

 □

Proof of Theorem 4 All zeros of and are real and lie in and for ,

For convenience, by removing ‘cos’ and the constant π, and can be identified with the ascending chain of rational numbers and , respectively. We may calculate that for n odd

and for n even

By using the above and denoting ■ a zero of , □ a zero of , we see that, for n even, all zeros between −1 and 1 listed in increasing order are of the form

where the center is the 0 that is the zero of . For n odd, all zeros listed in increasing order are of the form

where the center means that all those three numbers □, ■, □ are and one □ comes from the zero of . Now consider sign changes using the above two chains in increasing order so that for n odd or even we may check that, if k is even, has at least n real zeros in , and if k is odd, has at least real zeros in . On the other hand, the zeros z of satisfy

If , then , which contradicts (2). Thus all zeros of lie in . □

‘Bad pairs’ of polynomial zeros were defined in [2]. It is an easy consequence of Fell [1] that, if the all zeros of and form ‘good pairs’, their convex combination has all its zeros real.

Proof of Proposition 5 Following the proof of Theorem 4, we may see that for n even, all zeros and between −1 and 1 listed in increasing order are of the form

For n odd, all zeros listed in increasing order are of the form

where the center means that both numbers □, ■ are . Thus we can see that for n even, all zeros of and form good pairs, and for n odd, all pairs from integral polynomials and are good. It follows that, by Fell [1], all zeros of the convex combination are real and in . □

Proof of Theorem 6 The zeros z of satisfy

If , then , which contradicts (4). □

Proof of Proposition 7 Using , we have

and

So we only need to show that

and

But this equality follows from just replacing the variable θ by −θ. □

The author declares that they have no competing interests.

The author wishes to thank Professor Kenneth B. Stolarsky who let the author know some questions in this paper. The author is grateful to the referee of this paper for useful comments and suggestions that led to further development of an earlier version. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0011010).

1. Fell, HJ: On the zeros of convex combinations of polynomials. Pac. J. Math.. 89, 43–50 (1980)

2. Kim, S-H: Bad pairs of polynomial zeros. Commun. Korean Math. Soc.. 15, 697–706 (2000)

3. Mason, JC, Handscomb, DC: Chebyshev Polynomials, Chapman & Hall/CRC, Boca Raton (2003)

4. Rivlin, TJ: Chebyshev Polynomials. From Approximation Theory to Algebra and Number Theory, Wiley, New York (1990)

5. Shabat, GB, Voevodsky, VA: Drawing curves over number fields. The Grothendieck Festschrift, pp. 199–227. Birkhäuser, Basel (1990)

6. Shabat, GB, Zvonkin, A: Plane trees and algebraic numbers, Jerusalem combinatorics ’93, pp. 233–275. Am. Math. Soc., Providence (1994)