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.
Keywords:Chebyshev 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 .
and it is easy to show that for an odd integer n,
2 New results
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 ,
Proposition 1 will be used in the proofs of Theorems 4 and 6.
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
Remarks With the same notations with Proposition 2, it follows from
These imply that the upper bound
It is natural to ask about the bounds on the unit circle.
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
by (4), and it seems to be true that
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 .
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
Finally, we get an interesting property about the integrals of sums of Chebyshev polynomials. Observe that
These remain open problems.
Then by recurrence,
By (8) and the identity
In the same way, we can check that for n even,
Now we see with (10) and (11) that
and so it is enough to show that
and for n even
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
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 , all zeros of the convex combination are real and in . □
So we only need to show that
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).