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.

In this article, we consider the reciprocal antisymmetric polynomial \begin{displaymath} P(z) = \sum_{j = 0}^{s}(-1)^jγ_j\pa{z^j - z^{N + s + 1 - j}}, \ γ_0 = 1. \end{displaymath} It is shown that if all the zeros of $P(z)$ are located on the unit circle, that $\displaystyle\abs{γ_j} \leq {s \choose j}\pa{N + s + 1 \choose j}^{-1}$, $j = 1,\ldots,s$; moreover, these estimates cannot be improved in the general case. Factorization formulas for extremal polynomials are given: \begin{eqnarray*} \lefteqn{\sum_{j = 0}^{s}(-1)^j{s \choose j}\pa{N + s + 1 \choose j}^{-1}\pa{z^j - z^{N + s + 1 - j}}} \\ &=& (1 - z)^{2s + 1} \prod_{j = 1}^{\br{\frac{N - s}{2}}} \br{z^2 + 1 + 2z\pa{1 - 2(ν_j)^2}} \begin{cases} (1 + z), & N - s \mbox{ is odd} \\ 1, & N - s \mbox{ is even} \end{cases} \end{eqnarray*} where $\cb{ν_j}_{j = 1}^{\br{\frac{N - s}{2}}}$ is the set of positive zeros of the polynomial $U_N^{(s)}(z)$ given $\displaystyle U_N(z) = \sum_{j = 0}^{\br{\frac{N}{2}}} (-1)^j \frac{(N - j)!}{j!(N - 2j)!}(2z)^{N - 2j}$ 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: \begin{displaymath} \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). \end{displaymath}

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^∞ and k-sections of the Fibonacci sequence {Φ_n,k}_n=1^∞ ( Φ_n,k=F_nk/F_k). This article considers a further generalization of Fibonacci numbers, namely convolutions of k-sections of the Fibonacci sequence {Φ_n,k^(s)}_n=1^∞. These numbers are defined by the relations: Φ_n,k^(1)=∑_j=0^n-1Φ_j+1,kΦ_n-j,k , Φ_n,k^(s)=∑_j=0^n-1Φ_j+1,kΦ_n-j,k^(s-1) , s=2,3,... Moreover, Φ_n,1=F_n, Φ_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: Φ_n,k^(s)=5^-s(F_k)^-2s-1∑_j=0^s(-1)^(k-1)jn+2s jn+s-1-j 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 {Φ_n,k^(s)}_n=1^∞ for k=3,4,... and s=1,2,.. are not included in the OEIS encyclopedia.

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 <= kappa <= {1, if N is even, N/(N-2), 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) <= kappa <= {1, if N is odd, N/(N-2), 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) Pi((N-3)/2)(j=1) [1 + z(2)-2z gamma(j)], where gamma(j) = 1-2 nu(2)(j) with {nu(j)}(j=1)((N-3)/2) being the collection of positive roots of the equation UN-2 '(x) = 0; here U-j(x) = U-j(cos t) = sin(j+1)t/sint = 2(j)x(j)+... are Chebyshev polynomials of the second kind, and Uj '(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 Fejer polynomial, as well as construct certain univalent polynomials related to the polynomials p(z) and q(z).

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 Fejér polynomial, as well as construct certain univalent polynomials related to the polynomials $p(z)$ and $q(z)$.

For the class of sine polynomials b_1sin t+b_2sin 2t+...+b_Nsin Nt, (b_N≠ 0), which are nonnegative on (0,π ) , W. Rogosinski and G. Szegő derived, among other things, exact bounds for |b_2| via the Lukács 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 Fejér - 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.

In the class of normalized sine-polynomials S(t), non-negative on [0,π], W.Rogosinski and G.Szegő 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 Lukács 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 ℂ. By L.Fejér'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)=Im{P(e^it)}. For odd N the extremizers are unique, for even N there is a one-parameter family of extremizers.

On the class of typically real odd polynomials of degree 2N-1 F(z)=z+∑_j=2^Na_jz^2j-1 we consider two problems: 1) stretching the central unit disc under the above polynomial mappings and 2) estimating the coefficient a_2. It is shown that |F(z)|≤1/2^2(π/2N+2), -1+4sin^2(π/2N+4)≤ a_2≤-1+4cos^2(π/N+2) for odd N, and -1+4(ν_N)^2≤ a_2≤ -1+4cos^2(π/N+2) for even N, where ν_N is a minimal positive root of the equation U'_N+1(x) = 0 with U'_N + 1(x) being the derivative of the Chebyshev polynomial of the second kind of the corresponding order. The above boundaries are sharp, the corresponding estremizers are unique and the coefficients are determined.

The direction of research is the development of principles and methods for making scientifically based decisions in the design and additive manufacturing of bone substitutes based on apatite-biopolymer composites with functional properties depending on the nature of the localization of the cavity bone defect and its size. The relevance is due to the fact that the development of an intelligent decision-making support system based on neural network modeling, the development of methods for their training, tabagato-criterion optimization of design processes, will allow the creation of three-dimensional solid models of defects taking into account their spatial structure and bone substitutes for the synthesis of biomaterials with controlled composition, porosity and mechanical strength, which are optimal for a specific area of bone replacement, which will increase the effectiveness of treatment and prosthetics in orthopedics and traumatology. A set of methods for analyzing images of bone tissue, taking into account its spatial structure, which are obtained by sensors of different physical nature, with the use of neural network models, development of methods of their design, optimization, and training is proposed. A modification of the method of learning neural networks based on gradient descent, based on the application of the theory of nonlinear dynamics, is proposed. Corresponding theoretical provisions have been developed.

In connection with the increase in the number and severity of various types of bone tissue injuries received as a result of wounds during military operations in Ukraine, an important issue in orthopedics and traumatology is making informed decisions about the possibility of restoring the integrity and functions of bone tissue when using different types of composition, porosity and strength of apatite-biopolymer composites. The scientific direction of research is the development of principles and methods for making scientifically based decisions in the design and additive manufacturing of bone substitutes based on apatite-biopolymer composites with functional properties depending on the nature of the localization of the cavity bone defect and its size. A set of methods for analyzing images of bone tissue, taking into account its spatial structure, which are obtained by sensors of different physical nature, with the use of neural network models, development of methods of their design, optimization and training is proposed. The new knowledge obtained as a result of the project will become the necessary basis for making optimal decisions in practice for the introduction of the latest methods of treatment and prosthetics in trauma surgery, oncology, cranio-maxillofacial surgery, dentistry, taking into account the risks of biocompatibility of apatite-biopolymer composites. Software development of an intelligent decision support system will be used to design bone substitutes with controlled composition, structure, porosity and mechanical strength for the further selection of additive technology for its production from apatite-polymer composites, which will contribute to increasing the efficiency of treatment and prosthetics in orthopedics and traumatology.

We show that all the zeros of the quadrinomial p(z)=1+κ(z+z^N-1)+z^N lie on the unit circle if and only if the inequalities -1≤κ≤ 1 (), -1≤κ≤ N/(N-2) () hold. For the quadrinomial q(z)=1+κ(z-z^N-1)-z^N, the corresponding inequalities are -N/(N-2)≤κ≤ 1 ( if N is odd), -N/(N-2)≤κ≤ N/(N-2) ( if N is even). In the cases of limiting values of the parameter κ, we provide factorization formulas for the corresponding quadrinomials. For example, when N is odd and κ=N/(N-2), the following representation is valid: p(z)=(1+z)^3∏_j=1^(N-3)/2[1+z^2-2zγ_j], where γ_j=1-2ν_j^2 with {ν_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)=sin(j+1)t/sin t=2^j x^j+… 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 Fejér polynomial, as well as construct certain univalent polynomials related to the polynomials p(z) and q(z).

For the univalent polynomials F(z) = ∑ _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( -iF(i) ) = min _a_j: a_1=1∑ _j=1^N(-1)^j+1 a_j. We show that the solution is 1/2 ^2( π/2N+2) , and the extremal polynomial ∑ _j = 1^N U'_2(N-j+1)( cos( π/2N+2) ) /U'_2N( cos( π/2N+2) ) z^2j-1 is unique and univalent, where U_j(x) is a Chebyshev polynomial of the second kind and U'_j(x) denotes the derivative. As an application, we obtain an estimate of the Koebe radius for odd univalent polynomials in 𝔻 and formulate several conjectures.

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 T -fold rotational symmetry. This paper provides confirmation of these hypotheses for trinomials z + a z T + 1 + b z 2 T + 1 z + az^{T + 1} + bz^{2T + 1} . Namely, the Koebe radius is r = 4 cos 2 ⁡ ( π ( 1 + T ) 2 + 3 T ) r=4\cos ^2\left (\frac {\pi (1+T)}{2+3T}\right ) , and the only extremizer of the Koebe problem is the trinomial B ( T ) ( z ) = z + 2 2 + 3 T ( − T + ( 2 + 2 T ) cos ⁡ ( π T 2 + 3 T ) ) z 1 + T + + 1 2 + 3 T ( 2 + T − 2 T cos ⁡ ( π T 2 + 3 T ) ) z 1 + 2 T . \begin{gather*} B^{(T)}(z)=z+\frac 2{2+3T}\left (-T+(2+2T)\cos \left (\frac {\pi T}{2+3T}\right )\right )z^{1+T}+\\ +\frac 1{2+3T}\left (2+T-2T\cos \left (\frac {\pi T}{2+3T}\right )\right )z^{1+2T}. \end{gather*}

Extremal problems for typically real polynomials go back to a paper by W. W. Rogosinski and G. Szegő, 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 efficiency of Fejér’s method for solving the Rogosinski–Szegő problems. In this article, we completely solve Michel’s problem by using Fejér’s method.

The emergence and subsequent popularization of lean manufacturing have become one of the most significant for improving the efficiency and productivity of operations. The use of lean manufacturing tools and methods leads to the elimination of waste in the organization. Traditional information systems that allow organizations to share information about resources while managing process performance and traceability have a number of disadvantages such as security, interoperability, and transparency. Currently, distributed ledger technology (block-chain) is widely used for this purpose. This article presents a study of decentralized management of the implementation of a distributed ledger infrastructure, which is selected based on the characteristics of the production system. This study proposes a framework that analyzes lean production methods using simulation and data envelopment analysis (DEA) to accommodate the underlying multi-objective decision-making problem. The current study examines the impact of the simultaneous application of RCA technology, lean manufacturing methods, and distributed ledger technology on the total time, costs, and time of production processes.

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.

The Koebe One Quarter Theorem states that the range of any Schlicht function contains the centered disc of radius 1/4 which is sharp due to the value of the Koebe function at −1. A natural question is finding polynomials that set the sharpness of the Koebe Quarter Theorem for polynomials. In particular, it was asked in [7] whether Suffridge polynomials [15] are optimal. For polynomials of degree 1 and 2 that is obviously true. It was demonstrated in [10] that Suffridge polynomials of degree 3 are not optimal and a promising alternative family of polynomials was introduced. These very polynomials were actually discovered earlier independently by M. Brandt [3] and D. Dimitrov [9]. In the current article we reintroduce these polynomials in a natural way and make a far-reaching conjecture that we verify for polynomials up to degree 6 and with computer aided proof up to degree 52. We then discuss the ensuing estimates for the value of the Koebe radius for polynomials of a specific degree.

For the polynomials F(z)=∑j=1Najzj with real coefficients and normalization a1=1 we solve the extremal problemsupa2,…,aN⁡(infz∈D⁡{Re(F(z)):Im(F(z))=0}). We show that the solution is −14sec2⁡πN+2, and the extremal polynomial1UN′(cos⁡πN+2)∑j=1NUN−j+1′(cos⁡πN+2)Uj−1(cos⁡πN+2)zj is unique and univalent, where the Uj(x) are the Chebyshev polynomials of the second kind, j=1,…,N. As an application, we obtain the estimate of the Koebe radius for the univalent polynomials in D and formulate several conjectures.

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=4cos^2 π(1+T)/2+3T, and the only extremizer of the Koebe problem is the trinomial B^(T)(z)=z+2/2+3T(-T+(2+2T)cosπ T/2+3T)z^1+T+ +1/2+3T(2+T-2Tcosπ T/2+3T)z^1+2T. Key words and phrases: Koebe one-quarter theorem, Koebe radius, univalent polynomial, trinomials with fold symmetry.