Some properties of Chebyshev polynomials
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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.
KeywordsChebyshev polynomials bounds sums zeros
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  and they are studied in the theory of graphs on surfaces and curves over number fields. For a survey in this area, see .
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 .
2 New results
Proposition 1 Suppose that z is a complex number satisfying. Then for,
Proposition 1 will be used in the proofs of Theorems 4 and 6.
For , by (2). In the following proposition, we obtain an upper bound for arbitrary , .
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.
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 4 Letfor positive integers n and k. Thenhas all its zeros in. Furthermore, for k even, has at least n real zeros, and for k odd, 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 .
has all zeros in.
Theorem 6 has all its zeros in.
and we see that these two integrals are different. But for and , the integrals have the same value.
These remain open problems.
If , then , which contradicts (2). Thus all zeros of lie in . □
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 , all zeros of the convex combination are real and in . □
If , then , which contradicts (4). □
But this equality follows from just replacing the variable θ by −θ. □
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).
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