ON LINEAR COMBINATIONS OF CHEBYSHEV POLYNOMIALS

Abstract

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 6 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.