On linear combinations of Chebyshev polynomials
Dragan Stankov · Publications de l Institut Mathematique · 2015
We investigate an infinite sequence of polynomials of the form: a0Tn(x) + a1Tn?1(x) + ... + amTn?m(x) where (a0, a1,..., am) is a fixed m-tuple of real numbers, a0, am ? 0, Ti(x) are Chebyshev polynomials of the first kind, n = m, m+ 1, m+ 2,.... Here we analyze the structure of the set of zeros of such polynomial, depending on A and its limit points when n tends to infinity. Also the expression of envelope of the polynomial is given. An application in number theory, more precise, in the theory of Pisot and Salem numbers, is presented.