arXiv subjects

Zenan Šabanac

Publications and source records attributed to Zenan Šabanac.

9 recordsLinked to original sources

arXiv2026

Discrete Space-Time Wave Kernels and Trace Identities on Regular Graphs

We study the discrete space-time wave equation on a $(q+1)$-regular graph $X$ associated with the affine Laplace-type operator. For the forward time-difference scheme we derive explicit formulas for the two fundamental solutions (wave kernels) in terms of discrete modified Bessel functions and the non-backtracking walk counts on $X$ thus providing a direct and explicit link between wave propagation and combinatorial graph data. Utilizing uniqueness property of the wave kernel, we prove a new trace-type formula associated to the affine Laplace-type operator on $X$ and apply it to deduce many combinatorial identities. For example, we derive a closed-form expression for evaluation of some trigonometric sums twisted by an additive character as well as evaluations of finite sums of Chebyshev polynomials twisted by binomial coefficients.

math.CO↗

arXiv2026

Discrete Space-Time Wave Kernels on Regular Trees

We study the forward discrete space-time wave equation on the homogeneous $(q+1)$-regular tree $T_{q+1}$ associated with a two-parameter generalized Laplacian. Under the natural nonnegativity assumption on this operator, we derive explicit formulas for the two fundamental wave kernels. The formulas are given in terms of discrete $I$-Bessel functions and yield convolution representations for solutions with general initial conditions. In the boundary case corresponding to the bottom of the spectrum, we obtain another explicit representation of the wave kernel in terms of discrete $J$-Bessel functions. This representation leads to a discrete analogue of the classical $I\!\leftrightarrow\!J$ relation. We also perform both analytic and numerical studies of the asymptotic behavior of the wave kernels, including large radial distance, large time, and large degree of the tree. An important feature of our analysis is that the wave kernels are expressed as finite sums; hence, the propagation formulas remain finite for every discrete time.

math.SP↗

arXiv2024

Generalisation of an IMO Geometry Problem

In this paper, we generalise an interesting geometry problem from the 1995 edition of the International Mathematical Olympiad (IMO) using analytic geometry tools.

math.MG↗

arXiv2024

Relations between Chebyshev, Fibonacci and Lucas polynomials via trigonometric sums

In this paper we derive some new identities involving the Fibonacci and Lucas polynomials and the Chebyshev polynomials of the first and the second kind. Our starting point is a finite trigonometric sum which equals the resolvent kernel on the discrete circle with $m$ vertices and which can be evaluated in two different ways. An expression for this sum in terms of the Chebyshev polynomials was deduced in \cite{JKS} and the expression in terms of the Fibonacci and Lucas polynomials is deduced in this paper. As a consequence, we establish some further identities involving trigonometric sums and Fibonacci, Lucas, Pell and Pell-Lucas polynomials and numbers, thus providing a "physical" interpretation for those identities. Moreover, the finite trigonometric sum of the type considered in this paper can be related to the effective resistance between any two vertices of the $N$-cycle graph with four nearest neighbors $C_{N}(1,2)$. This yields further identities involving Fibonacci numbers.

math.NT↗

arXiv2023

Discrete Bessel functions and discrete wave equation

In this paper, we study discrete Bessel functions which are solutions to the discretization of Bessel differential equations when the forward and the backward difference replace the time derivative. We focus on the discrete Bessel equations with the backward difference and derive their solutions. We then study the transformation properties of those functions, describe their asymptotic behaviour and compute Laplace transform. As an application, we study the discrete wave equation on the integers in timescale $T=\mathbb{Z}$ and express its fundamental and general solution in terms of the discrete $J$-Bessel function. Going further, we show that the first fundamental solution of this equation oscillates with the exponentially decaying amplitude as time tends to infinity.

math.DS↗

arXiv2022

On the Hurwitz-type zeta function associated to the Lucas sequence

We study the theta function and the Hurwitz-type zeta function associated to the Lucas sequence $U=\{U_n(P,Q)\}_{n\geq 0}$ of the first kind determined by the real numbers $P,Q$ under certain natural assumptions on $P$ and $Q$. We deduce an asymptotic expansion of the theta function $θ_U(t)$ as $t\downarrow 0$ and use it to obtain a meromorphic continuation of the Hurwitz-type zeta function $ζ_{U}\left( s,z\right) =\sum\limits_{n=0}^{\infty }\left(z+U_{n}\right) ^{-s}$ to the whole complex $s-$plane. Moreover, we identify the residues of $ζ_{U}\left( s,z\right)$ at all poles in the half-plane $\Re(s)\leq 0$.

math.NT↗

arXiv2020

The heat kernel of a weighted Maass-Laplacian with real weights

We derive closed formula for the heat kernel $K_\mathbb{H}$ associated to Maass-Laplacian operator $D_k$ for any real $k$ and prove that heat kernel $K_\mathbb{H}$ is strictly monotone decreasing function. We also derive some important asymptotic formulae for the heat kernel $K_\mathbb{H}$.

math.SP↗

arXiv2017

Determination of a jump by Fourier and Fourier-Chebyshev series

By observing the equivalence of assertions on determining the jump of a function by its differentiated or integrated Fourier series, we generalize a previous result of Kvernadze, Hagstrom and Shapiro to the whole class of functions of harmonic bounded variation and without finiteness assumption on the number of discontinuities. Two results on determination of jump discontinuities by means of the tails of integrated Fourier-Chebyshev series are derived.

math.CA↗

arXiv2016

Strong boundedness, strong convergence and generalized variation

A trigonometric series strongly bounded at two points and with coefficients forming a log-quasidecreasing sequence is necessarily the Fourier series of a function belonging to all $L^{p}$ spaces, $1\leq p < \infty$. We obtain new results on strong convergence of Fourier series for functions of generalized bounded variation.

math.CA↗