arXiv subjects

Daniel Gray

Publications and source records attributed to Daniel Gray.

9 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↗

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↗

arXiv2018

On majorization of closed walks vector of trees with given degree sequences

Let $C_{v}(k;T)$ be the number of the closed walks of length $k$ starting at vertex $v$ in a tree $T$. We prove that for a given tree degree sequence $\pi$, then for any tree with degree sequence $\pi$, the sequence $C(k;T)\equiv(C_{v}(k;T), v\in V(T))$ is weakly majorized by the sequence $C(k, T_{\pi}^*)\equiv C(k, T_{\pi}^*, v\in V(T^*))$, where $T_{\pi}^*$ is the greedy tree corresponding to $\pi$. In addition, for two trees degree sequences $\pi,~\pi'$, if $\pi$ is majorized by $\pi'$, then $C(k;T_{\pi}^*)$ is weakly majorized by $C(k;T_{\pi'}^*)$.

math.CO↗

arXiv2012

Trees with the most subtrees -- an algorithmic approach

When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in applications. The number of subtrees in the extremal cases constitute sequences which are of interest to number theorists. The structures which maximize or minimize the number of subtrees among general trees, binary trees and trees with a given maximum degree have been identified previously. Most recently, results of this nature are generalized to trees with a given degree sequence. In this note, we characterize the trees which maximize the number of subtrees among trees of a given order and degree sequence. Instead of using theoretical arguments, we take an algorithmic approach that explicitly describes the process of achieving an extremal tree from any random tree. The result also leads to some interesting questions and provides insight on finding the trees close to extremal and their numbers of subtrees.

math.CO↗

arXiv2012

The Number of Subtrees of Trees with Given Degree Sequence

This paper investigates some properties of the number of subtrees of a tree with given degree sequence. These results are used to characterize trees with the given degree sequence that have the largest number of subtrees, which generalizes the recent results of Kirk and Wang. These trees coincide with those which were proven by Wang and independently Zhang et al. to minimize the Wiener index. We also provide a partial ordering of the extremal trees with different degree sequences, some extremal results follow as corollaries.

math.CO↗