Geodesic
On Linear Combinations of Chebyshev Polynomials
Publications de l'Institut Mathématique, _N_S_97 (2015) no. 111, p. 57

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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.
DOI : 10.2298/PIM150220001S
Classification : 11B83 11R09,12D10
Keywords: Chebyshev polynomials, envelope, Pisot numbers, Salem numbers 
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     author = {Dragan Stankov},
     title = {On {Linear} {Combinations} of {Chebyshev} {Polynomials}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {57 },
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     doi = {10.2298/PIM150220001S},
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Dragan Stankov. On Linear Combinations of Chebyshev Polynomials. Publications de l'Institut Mathématique, _N_S_97 (2015) no. 111, p. 57 . doi: 10.2298/PIM150220001S

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