Searcharxiv⌕ Search

arXiv subjects

Alexander Stokolos

Publications and source records attributed to Alexander Stokolos.

11 recordsLinked to original sources
arXiv2026

Analytic summation of series involving higher-order derivatives of Chebyshev polynomials of the second kind and their applications to convolved linear recurrent sequences

This paper considers functional series whose terms are higher-order derivatives of Chebyshev polynomials of the second kind, where the degree of the polynomial is related to the order of the derivative. Analytic summation is used to determine the rational functions to which these series converge. These functions are expressed in terms of Chebyshev polynomials evaluated at a specific argument. Connections are established between derivatives of Chebyshev polynomials of the second kind and special numerical sequences generated by linear recurrence relations. New closed-form formulas are obtained for the sums of the series at various values of the argument. As consequences, combinatorial identities are derived for the Fibonacci, Lucas, and Pell numbers, for sections of the Fibonacci sequence, and for their convolutions. By means of analytic continuation, sums of formally divergent series are obtained, which in special cases correspond to the classical Euler formulas.

math.CV↗
arXiv2026

Convolved Numbers of $k$-sections of the Fibonacci Sequence: Properties, Consequences

One possible data encryption scheme is related to stream ciphers, which use a sufficiently long pseudo-random sequence. To increase the cryptographic strength of the cipher, linear shift algorithms (generated by linear recurrent sequences such as the Fibonacci sequence and its generalizations) are additionally used. Two such generalizations are convolved Fibonacci numbers $\{F_n^{(s)}\}_{n=1}^\infty$ and k-sections of the Fibonacci sequence $\{\Phi_{n,k}\}_{n=1}^\infty$ $( \Phi_{n,k}=F_{nk}/F_k).$ This article considers a further generalization of Fibonacci numbers, namely convolutions of k-sections of the Fibonacci sequence $\{\Phi_{n,k}^{(s)}\}_{n=1}^\infty$. These numbers are defined by the relations: $$ \Phi_{n,k}^{(1)}=\sum_{j=0}^{n-1}\Phi_{j+1,k}\Phi_{n-j,k\,},\qquad \Phi_{n,k}^{(s)}=\sum_{j=0}^{n-1}\Phi_{j+1,k}\Phi_{n-j,k}^{(s-1)}\,,\quad s=2,3,...$$Moreover, $\Phi_{n,1}=F_n, \Phi_{n,1}^{(s)}=F_n^{(s)}$. An explicit formula for the representation of convolutions of k-sections of the Fibonacci sequence and a Binet type formula is established:$$\Phi_{n,k}^{(s)}=5^{-s}(F_k)^{-2s-1}\sum_{j=0}^{s}(-1)^{(k-1)j}{n+2s\choose j}{n+s-1-j\choose n-1} F_{k(n+2s-2j)}.$$ Several consequences were also obtained for $F_n$ and $F_n^{(s)}$, based on the connection between the derivatives of Chebyshev polynomials of the second kind $U_n(z)$ and their derivatives, as well as the connection for convolutions of k-sections of the Fibonacci sequence with derivatives of Chebyshev polynomials of the second kind via Lucas numbers $L_k$. Note that the sequences $\{\Phi_{n,k}^{(s)}\}_{n=1}^\infty$ for $k=3,4,...$ and $s=1,2,..$ are not included in the OEIS encyclopedia.

math.CA↗
arXiv2026

Reciprocal Polynomials with Zeros on the Unit Circle and Derivatives of Chebyshev Polynomials of the Second Kind

In this article, we consider the reciprocal antisymmetric polynomial \[P(z) = \sum_{j = 0}^{s}(-1)^j\gamma_j\left(z^j - z^{N + s + 1 - j}\right), \ \gamma_0 = 1.\] It is shown that if all the zeros of $P(z)$ are located on the unit circle, that $\displaystyle\left|\gamma_j\right| \leq {s \choose j}\left({N + s + 1 \choose j}\right)^{-1}$, $j = 1,\ldots,s$; moreover, these estimates cannot be improved in the general case. Factorization formulas for extremal polynomials are given: \[ \begin{align} \phantom{a} & \sum_{j = 0}^{s}(-1)^j{s \choose j}\left({N + s + 1 \choose j}\right)^{-1}\left(z^j - z^{N + s + 1 - j}\right) \\ &= (1 - z)^{2s + 1} \prod_{j = 1}^{\left[\frac{N - s}{2}\right]} \left[z^2 + 1 + 2z(1 - 2(\nu_j)^2)\right] \begin{cases} (1 + z), & N - s \mbox{ is odd} \\ 1, & N - s \mbox{ is even} \end{cases} \end{align} \] where $\left\{\nu_j\right\}_{j = 1}^{\left[\frac{N - s}{2}\right]}$ is the set of positive zeros of the polynomial $U_N^{(s)}(z)$ given $\displaystyle U_N(z) = \sum_{j = 0}^{\left[\frac{N}{2}\right]} (-1)^j \frac{(N - j)!}{j!(N - 2j)!}(2z)^{N - 2j}$ are the Chebyshev Polynomials of the Second Kind and $U_N^{(s)}(z)$ is the $s$th derivative of $U_N(z)$. As an application of the results, formulas were obtained expressing the derivatives of Chebyshev polynomials of the second kind through linear combinations of Chebyshev polynomials of the second kind: \[\frac{2^s}{s!}(1 - z^2)^sU_N^{(s)}(z) = (-1)^s \sum_{j = 0}^{s}(-1)^j{N-j \choose N-s} {N+s+1 \choose j} U_{N + s - 2j}(z). \]

math.CV↗
arXiv2025

Extremal polynomials for the Rogosinski--Szeg\H{o} estimates of the third coefficient of nonnegative sine polynomials

In the class of normalized sine-polynomials $S(t),$ non-negative on $[0,\pi],$ W.Rogosinski and G.Szeg\H{o} 1950 considered a number of extremal problems and proved, among other things, sharp upper and lower estimates for the coefficient $a_3.$ Their proof is based on the Luk\'acs representation of non-negative algebraic polynomials. This method does not lead to the construction of polynomials attaining the extreme values. We consider the corresponding problem in the framework of normalized typically real polynomials $P(z)$ on the unit disc in $\mathbb C.$ By L.Fej\'er's method with the additional use of the Chebyshev polynomials of the second kind and their derivatives, we regain the sharp upper and lower estimates for $a_3$ and identify the extremal polynomials. The corresponding statements for sine polynomials follow by the observation $S(t)=\text{Im}\{P(e^{it})\}$. For odd $N$ the extremizers are unique, for even $N$ there is a one-parameter family of extremizers.

math.CV↗
arXiv2024

Some properties of the quadrinomials $p(z)=1+\kappa(z+z^{N-1})+z^N$ and $q(z)=1+\kappa(z-z^{N-1})-z^N$

We show that all the zeros of the quadrinomial $p(z)=1+\kappa(z+z^{N-1})+z^N$ lie on the unit circle if and only if the inequalities \[ -1\le\kappa\le 1\; (\mbox{ if $N$ is even}),\;\; -1\le\kappa\le N/(N-2)\; (\mbox{ if $N$ is odd}) \] hold. For the quadrinomial $q(z)=1+\kappa(z-z^{N-1})-z^N$, the corresponding inequalities are \[ -N/(N-2)\le\kappa\le 1\; (\text{ if $N$ is odd}),\;\; -N/(N-2)\le\kappa\le N/(N-2)\; (\text{ if $N$ is even}). \] In the cases of limiting values of the parameter $\kappa$, we provide factorization formulas for the corresponding quadrinomials. For example, when $N$ is odd and $\kappa=N/(N-2)$, the following representation is valid: \[ p(z)=(1+z)^3\prod_{j=1}^{(N-3)/2}[1+z^2-2z\gamma_j], \] where $\gamma_j=1-2\nu_j^2$ with $\{\nu_j\}_{j=1}^{(N-3)/2}$ being the collection of positive roots of the equation $U'_{N-2}(x)=0$; here \[ U_j(x)=U_j(\cos t)=\frac{\sin(j+1)t}{\sin t}=2^j x^j+\ldots \] are Chebyshev polynomials of the second kind and $U'_j(x)$ are their derivatives. Similar factorization formulas are also provided for $q(z)$. As an application of the obtained results, we give the factorization formulas for the derivative of the Fej\'er polynomial, as well as construct certain univalent polynomials related to the polynomials $p(z)$ and $q(z)$.

math.CA↗
arXiv2024

Extremizers for the Rogosinski-Szeg\"o estimate of the second coefficient in nonnegative sine polynomials

For the class of sine polynomials $b_1\sin t+b_2\sin2t+...+b_N\sin Nt,\; (b_N\not= 0),$ which are nonnegative on $(0,\pi)$, W. Rogosinski and G. Szeg\"o derived, among other things, exact bounds for $|b_2|$ via the Luk\'acs presentation of nonnegative algebraic polynomials and a variational type argument for exact bounds, but they did not find the extremizers. Within this algebraic framework, we construct explicit polynomials which attain these bounds and prove their uniqueness. The proof uses the Fej\'er -Riesz representation of nonnegative trigonometric polynomials, a 7-band Toeplitz matrix of arbitrary finite dimension, and Chebyshev polynomials of the second kind and their derivatives.

math.CA↗
arXiv2022

An extremal problem for odd univalent polynomials

For the univalent polynomials $F(z) = \sum\limits_{j=1}^{N} a_j z^{2j-1}$ with real coefficients and normalization \(a_1 = 1\) we solve the extremal problem \[ \min_{a_j:\,a_1=1} \left( -iF(i) \right) = \min_{a_j:\,a_1=1} \sum\limits_{j=1}^{N} {(-1)^{j+1} a_j}. \] We show that the solution is $\frac12 \sec^2{\fracπ{2N+2}},$ and the extremal polynomial \[ \sum_{j = 1}^N \frac{U'_{2(N-j+1)} \left( \cos\left(\fracπ{2N+2}\right)\right)}{U'_{2N} \left( \cos\left(\fracπ{2N+2}\right)\right)}z^{2j-1} \] is unique and univalent, where the $U_j(x)$ are the Chebyshev polynomials of the second kind and $U'_j(x)$ denotes the derivative. As an application, we obtain the estimate of the Koebe radius for the odd univalent polynomials in $\mathbb D$ and formulate several conjectures.

math.CV↗
arXiv2022

Koebe's theorem for trinomials with fold symmetry

The Koebe problem for univalent polynomials with real coefficients is fully solved only for trinomials, which means that in this case the Koebe radius and the extremal polynomial (extremizer) have been found. The general case remains open, but conjectures have been formulated. The corresponding conjectures have also been hypothesized for univalent polynomials with real coefficients and $T$-fold rotational symmetry. This paper provides confirmation of these hypotheses for trinomials $z + az^{T + 1} + bz^{2T + 1}$. Namely, the Koebe radius is $r=4\cos^2 \frac{π(1+T)}{2+3T}$, and the only extremizer of the Koebe problem is the trinomial \begin{gather*} B^{(T)}(z)=z+\frac2{2+3T}\left(-T+(2+2T)\cos\frac{πT}{2+3T}\right)z^{1+T}+\\ +\frac1{2+3T}\left(2+T-2T\cos\frac{πT}{2+3T}\right)z^{1+2T}. \end{gather*} Key words and phrases: Koebe one-quarter theorem, Koebe radius, univalent polynomial, trinomials with fold symmetry.

math.CV↗
arXiv2021

Search for Invariant Sets of the Generalized Tent Map

This paper describes a predictive control method to search for unstable periodic orbits of the generalized tent map. The invariant set containing periodic orbits is a repelling set with a complicated Cantor-like structure. Therefore, a simple local stabilization of the orbit may not be enough to find a periodic orbit, due to the small measure of the basin of attraction. It is shown that for certain values of the control parameter, both the local behavior and the global behavior of solutions change in the controlled system; in particular, the invariant set enlarges to become an interval or the entire real axis. The computational particularities of using the control system are considered, and necessary conditions for the orbit to be periodic are given. The question of local asymptotic stability of subcycles of the controlled system's stable cycles is fully investigated, and some statistical properties of the subset of the classical Cantor middle thirds set that is determined by the periodic points of the generalized tent map are described.

math.DS↗
arXiv2019

A theorem of Besicovitch and a generalization of the Birkhoff Ergodic Theorem

A remarkable theorem of Besicovitch is that an integrable function $f$ on $\mathbb{R}^2$ is strongly differentiable if and only if its associated strong maximal function $M_S f$ is finite a.e. We provide an analogue of Besicovitch's result in the context of ergodic theory that provides a generalization of Birkhoff's Ergodic Theorem. In particular, we show that if $f$ is a measurable function on a standard probability space and $T$ is an invertible measure-preserving transformation on that space, then the ergodic averages of $f$ with respect to $T$ converge a.e. if and only if the associated ergodic maximal function $T^*f$ is finite a.e.

math.CA↗