ON LINEAR COMBINATIONS OF CHEBYSHEV POLYNOMIALS

Dragan Stankov

Abstract: 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.

Keywords: Chebyshev polynomials, envelope, Pisot numbers, Salem numbers

Classification (MSC2000): 11B83; 11R09;12D10

Full text of the article: (for faster download, first choose a mirror)

• Compressed DVI file (24 kilobytes)
• Compressed PostScript file (444 kilobytes)
• PDF file (160 kilobytes)