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Dmitriy Dmitrishin

Publications and source records attributed to Dmitriy Dmitrishin.

At least 19 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 $\frac12 \sec^2{\frac{\pi}{2N+2}},$ and the extremal polynomial \[ \sum_{j = 1}^N \frac{U'_{2(N-j+1)} \left( \cos\left(\frac{\pi}{2N+2}\right)\right)}{U'_{2N} \left( \cos\left(\frac{\pi}{2N+2}\right)\right)}z^{2j-1} \] 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 $r=4\cos^2 \frac{\pi(1+T)}{2+3T}$, 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{\pi T}{2+3T}\right)z^{1+T}+\\ +\frac1{2+3T}\left(2+T-2T\cos\frac{\pi T}{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↗
arXiv2022

Extremal problems for trinomials with fold symmetry

The famous T. Suffridge polynomials have many extremal properties: the maximality of coefficients when the leading coefficient is maximal; the zeros of the derivative are located on the unit circle; the maximum radius of stretching the unit disk with the schlicht normalization $F(0)=0$, $F'(0)=1$; the maximum size of the unit disk contraction in the direction of the real axis for univalent polynomials with the normalization $F(0)=0$, $F(1)=1.$ However, under the standard symmetrization method $\sqrt[T]{F(z^T)}$, these polynomials go to functions, which are not polynomials. How can we construct the polynomials with fold symmetry that have properties similar to those of the Suffridge polynomial? What values will the corresponding extremal quantities take in the above-mentioned extremal problems? The paper is devoted to solving these questions for the case of the trinomials $F(z)=z+az^{1+T}+bz^{1+2T}$. Also, there are suggested hypotheses for the general case in the work.

math.CV↗
arXiv2021

Sharp weak type estimates for a family of Soria bases

Let $\mathcal{B}$ be a collection of rectangular parallelepipeds in $\mathbb{R}^3$ whose sides are parallel to the coordinate axes and such that $\mathcal{B}$ contains parallelepipeds with side lengths of the form $s, \frac{2^N}{s} , t $, where $s, t > 0$ and $N$ lies in a nonempty subset $S$ of the natural numbers. We show that if $S$ is an infinite set, then the associated geometric maximal operator $M_\mathcal{B}$ satisfies the weak type estimate $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) > \alpha\right\}\right| \leq C \int_{\mathbb{R}^3} \frac{|f|}{\alpha} \left(1 + \log^+ \frac{|f|}{\alpha}\right)^{2}$$ but does not satisfy an estimate of the form $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) > \alpha\right\}\right| \leq C \int_{\mathbb{R}^3} \phi\left(\frac{|f|}{\alpha}\right)$$ for any convex increasing function $\phi: \mathbb[0, \infty) \rightarrow [0, \infty)$ satisfying the condition $$\lim_{x \rightarrow \infty}\frac{\phi(x)}{x (\log(1 + x))^2} = 0\;.$$

math.CA↗
arXiv2020

On Suffridge polynomials

We consider some known and some new properties of the family of polynomials introduced by Ted Suffridge in 1969. We begin by giving a brief overview of their extremal properties in classic and more recent work. We also give a compact form for Suffridge polynomials which matches a general pattern discovered by Brandt. Our approach allows us to find the coefficients which Brandt's result was not giving explicitly. This new presentation provides us the tools to obtain an estimate of the rate of approximation of the generalized Koebe functions by univalent polynomials. Furthermore, we consider the presentation of Suffridge polynomials in Robertson's form and find the suiting Robertson measure. This suggests a new way to approximate step functions by continuous monotonic ones. We then study the lack of robustness of the univalency of these polynomials and suggest a new family of polynomials for which we conjecture the univalency of a subclass. Namely, we prove the quite surprising fact that by extending the family by letting the discrete argument in the polynomial coefficients become continuous one does not increase the set of univalent polynomials. Only the initial polynomials remain univalent. In this new one parameter family generalizing the Suffridge polynomials, it is remarkable that the Suffridge polynomials are already extremal as they correspond to the choice of the parameter set to 1; moreover the complex Fej\'er polynomials correspond the choice of the parameter set to 0, and the choice of the parameter set to -1 corresponds the polynomials $z+(z^N/N)$. Remarkably, computer simulations seem to clearly indicate that the image of the unit disc under these new polynomial mapping is a simply-connected region bounded by a simple curve. This justifies the conjectural univalency of these polynomials for the whole range of the parameters.

math.CV↗
arXiv2020

On C. Michel's hypothesis about the modulus of typically real polynomials

Extremal problems for typically real polynomials go back to a paper by W. W. Rogosinski and G. Szeg\H{o}, where a number of problems were posed, which were partially solved by using orthogonal polynomials. Since then, not too many new results on extremal properties of typically real polynomials have been obtained. Fundamental work in this direction is due to M.~Brandt, who found a novel way of solving extremal problems. In particular, he solved C. Michel's problem of estimating the modulus of a typically real polynomial of odd degree. On the other hand, D. K. Dimitrov showed the effectivity of Fej\'er's method for solving the Rogosinski--Szeg\H{o} problems. In this article, we completely solve Michel's problem by using Fej\'er's method.

math.CA↗
arXiv2019

On the Koebe Quarter Theorem for Polynomials

D. Dimitrov has posed the problem of finding polynomials that set the sharpness of the Koebe Quarter Theorem for polynomials and asked whether Suffridge polynomials are optimal. We disprove Dimitrov's conjecture for polynomials of degree 3, 4, 5 and 6. For polynomials of degree 1 and 2 the conjecture is obviously true. On the way we introduce a new family of polynomials that allows us to state a conjecture about the value of the Koebe radius for polynomials of a specific degree.

math.CV↗
arXiv2018

Univalent polynomials and Koebe's one-quarter theorem

The famous Koebe $\frac14$ theorem deals with univalent (i.e., injective) analytic functions $f$ on the unit disk $\mathbb D$. It states that if $f$ is normalized so that $f(0)=0$ and $f'(0)=1$, then the image $f(\mathbb D)$ contains the disk of radius $\frac14$ about the origin, the value $\frac14$ being best possible. Now suppose $f$ is only allowed to range over the univalent polynomials of some fixed degree. What is the optimal radius in the Koebe-type theorem that arises? And for which polynomials is it attained? A plausible conjecture is stated, and the case of small degrees is settled.

math.CV↗
arXiv2018

Dimitrov's question for the polynomials of degree 1,2,3,4,5,6

In 2002 Dimitar Dimitrov posted the problem of finding the optimal polynomials that provide the sharpness of Koebe Quarter Theorem for polynomials and asked whether Suffridge polynomials are optimal ones. We disproved Dimitrov's conjecture for polynomials of degree 3,4,5 and 6. For polynomials of degree 1 and 2 the conjecture is valid.

math.CV↗
arXiv2018

Estimating the Koebe radius for polynomials

For a pair of conjugate trigonometrical polynomials $C (t) = \sum_ { j = 1 } ^N { { a_j}\cos jt }, S(t) = \sum_ { j = 1 } ^N { { a_j}\sin jt }$ with real coefficients and normalization ${a_1} = 1 $ we solve the extremal problem \[ \sup_ {a_2,...,a_N} \left ({ \min_t \left\{ {\Re \left ({ F\left ({ { e^ {it} } } \right) } \right): \Im \left ({ F\left ({ { e^ {it} } } \right) } \right) = 0 } \right\} } \right) = -\frac14 \sec ^2\frac\pi{N + 2}. \] We show that the solution is unique and is given by \[ a_j^ {(0)} = \frac {1} { { { U'_N}\left ({\cos \frac{\pi } { { N + 2 } } } \right) } } { U' _ { N - j + 1 } }\left ({\cos \frac{\pi } { { N + 2 } } } \right) { U_ { j - 1 } }\left ({\cos \frac{\pi } { { N + 2 } } } \right), \] where the $U_j(x)$ are the Chebyshev polynomials of the second kind, and the $U'_j(x)$ are their derivatives, $j = 1, \ldots, N.$ As a consequence, we obtain some theorems on covering of intervals by polynomial images of the unit disc. We formulate several conjectures on a number of extremal problems on classes of polynomials.

math.CV↗
arXiv2018

Fejer Polynomials and Control of Nonlinear Discrete Systems

We consider optimization problems associated to a delayed feedback control (DFC) mechanism for stabilizing cycles of one dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizing $T$-cycles of a differentiable function $f: \mathbb{R}\rightarrow\mathbb{R}$ of the form $$x(k+1) = f(x(k)) + u(k)$$ where $$u(k) = (a_1 - 1)f(x(k)) + a_2 f(x(k-T)) + \cdots + a_N f(x(k-(N-1)T))\;,$$ with $a_1 + \cdots + a_N = 1$. Following an approach of Morg\"ul, we associate to each periodic orbit of $f$, $N \in \mathbb{N}$, and $a_1,\ldots,a_N$ an explicit polynomial whose Schur stability corresponds to the stability of the DFC on that orbit. We prove that, given any 1- or 2-cycle of $f$, there exist $N$ and $a_1,\ldots,a_N$ whose associated polynomial is Schur stable, and we find the minimal $N$ that guarantees this stabilization. The techniques of proof will take advantage of extremal properties of the Fej\'er kernels found in classical harmonic analysis.

math.DS↗