On Linear Combinations of Chebyshev Polynomials

Dragan Stankov

We investigate an infinite sequence of polynomials of the form: $$ a_0T_n(x)+a_1T_{n-1}(x)+\dots+a_mT_{n-m}(x) $$ where $(a_0,a_1,\ldots,a_m)$ is a fixed $m$-tuple of real numbers, $a_0,a_m\neq0$, $T_i(x)$ are Chebyshev polynomials of the first kind, $n=m,m+1,m+2,\ldots$ 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.