arXiv subjects

Konstantin Shcherbakov

Publications and source records attributed to Konstantin Shcherbakov.

2 recordsLinked to original sources

arXiv2024

Chebyshev polynomials, their remarkable properties and connection with Catalan numbers

The main result of the article says that the formal power series equal to the ratio of two neighboring Chebyshev polynomials, after some renormalization, approximates the generating function of the Catalan numbers. We present a proof of this result using continued fractions. Also, for completeness of the presentation, we recall the basic properties of Chebyshev polynomials and formal power series that we use.

math.CO↗

arXiv2024

Roots of polynomials and tangents of circles

Given a real cubic function $f(x)$ with three roots, take an equilateral triangle $ABC$, the projections of which vertices are the roots of $f(x)$. There is a folklore fact that the vertical lines through the extrema of $f(x)$ are tangent to the inscribed circle of $ABC$. We generalise this fact to a regular $n$-gon and the corresponding degree $n$ polynomial.

math.CA↗