arXiv subjects

Lejla Smajlović

Publications and source records attributed to Lejla Smajlović.

At least 19 recordsLinked to original sources

arXiv2026

On a generating function of the basis of weakly holomorphic functions on an elliptic curve

Let $\Gz$ be a cofinite Fuchsian group whose quotient $\Gz\backslash\HH$ has genus one and a single cusp, normalized to be at $\infty$ with width one. Let $\Xg$ be its smooth compactification, which is an elliptic curve over $\CC$. Let $x,y$ be canonical generators of the function field on $\Xg$, which have poles of order $2,3$ respectively at the cusp and which {satisfy} a generalized Weierstrass relation $y^2+(a_1x+a_3)y=x^3+a_2x^2+a_4x+a_6$. Let $f$ be the unique (up to scale) weight two cusp form for $\Gz$, normalized so that the associated holomorphic differential is $ω=dx/(2y+a_1x+a_3)$. With all this, we prove that the generating function identity \[ \frac{\bigl(y(τ)+y(z)+a_1x(z)+a_3\bigr)f(z)}{x(z)-x(τ)}=\sum_{m\ge0}Φ_m(τ)q_z^m \] defines a family $\{Φ_m\}_{m\ge0}$ of weakly holomorphic modular functions on $\Xg$ such that the set $\{Φ_0\}\cup\{Φ_m\}_{m\ge2}$ is a canonical basis of the space $M_{0,\Gz}^{!,\infty}$ of weakly holomorphic modular functions on $\Xg$ with poles supported only at the cusp. As an application, by comparing the generating function of the family $\{Φ_m\}_{m\ge0}$ with the generating function of the Niebur--Poincaré series, we show that the generating function of the special values of the Kloosterman zeta function at $1$ is the holomorphic part of a certain weight two harmonic Maass form up to an additive constant.

math.NT↗

arXiv2026

Resultants, Recursive Formulas and Binet Formulas for $\textrm{SU}(N)$ Verlinde Sums

We study Verlinde sums $V_{n}(N,m)$, which for our purposes are finite sums of trigonometric functions $\{r_{x}\}$, associated to $\textrm{ SU}(N)$. In this article we prove the following results: (i) the polynomial $D_N(w):=\prod_x(w-r_x)$ is equal to, up to an explicit and computable factor, the iterated resultant of $Q(u)=u^m-1$ with an explicit elementary polynomial $W_N$ which is independent of $n$ and $m$; (ii) for fixed $N$ and $m$, $V_n(N,m)$ satisfies a linear recursion of order at most $\binom{m-1}{N-1}$, together with arguments showing that one actually has a recursion of length equal to the number of distinct roots of $D_{N}$; (iii) for fixed $N$ and $m$, a Binet-type closed form expressing $V_n(N,m)$ as an explicit finite sum of $n$-th powers of algebraic numbers which are rescaled reciprocals of the distinct roots of $D_N(w)$ is derived. Several explicit examples are provided for $N=2,3,4,5,6$ and $8$.

math.CO↗

arXiv2026

Evaluating $\mathrm {SU}(3)$ Verlinde sums using spectral graph theory

We realize the $\mathrm{SU}(3)$ Verlinde sums $V_n(m)$, up to an explicit factor, as the values at $n$ of the spectral zeta function of a higher-order Laplace operator on the triangular discrete torus on $m^2$ vertices. For fixed $m$, we express their generating function in terms of the logarithmic derivative of an associated even spectral polynomial and express this polynomial as an explicit iterated resultant. Exploiting permutation symmetry, we prove that this polynomial is a cube over $\mathbb{Q}$, apart from an explicit quadratic factor when $3\mid m$. This factorization yields shorter linear recurrences satisfied by $V_n(m)$ with constant coefficients. We also derive Binet-type formulas expressing $V_n(m)$ as finite linear combinations of powers of rescaled inverse squares of the roots of the spectral polynomial. These results provide an efficient algorithm for computing these Verlinde sums. Several fully developed examples demonstrating computational efficiency of the method are given, including expressions in terms of Fibonacci and Lucas numbers.

math.CO↗

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↗

arXiv2026

Diffusion Computation versus Quantum Computation: A Comparative Model for Order Finding and Factoring

We study a hybrid computational model for integer factorization in which the only non-classical resource is access to an \emph{iterated diffusion process} on a finite graph. Concretely, a \emph{diffusion step} is defined to be one application of a symmetric stochastic matrix (the half-lazy walk operator) to an $\ell^{1}$--normalized state vector, followed by an optional readout of selected coordinates. Let $N\ge 3$ be an odd integer which is neither prime nor a prime power, and let $b\in(\mathbb{Z}/N\mathbb{Z})^\ast$ have odd multiplicative order $r={\rm ord}_N(b)$. We construct, without knowing $r$ in advance, a weighted Cayley graph whose vertex set is the cyclic subgroup $\langle b\rangle$ and whose edges correspond to the powers $b^{\pm 2^t}$ for $t\le \lfloor \log_2 N\rfloor+1$. Using an explicit spectral decomposition together with an elementary doubling lemma, we show that $r$ can be recovered from a single heat-kernel value after at most $O((\log_2 N)^2)$ diffusion steps, with an effective bound. We then combine this order-finding model with the standard reduction from factoring to order finding (in the spirit of Shor's framework) to obtain a randomized factorization procedure whose success probability depends only on the number $m$ of distinct prime factors of $N$. Our comparison with Shor's algorithm is \emph{conceptual and model-based}. We replace unitary $\ell^2$ evolution by Markovian $\ell^1$ evolution, and we report complexity in two cost measures: digital steps and diffusion steps. Finally, we include illustrative examples and discussion of practical implementations.

math.SP↗

arXiv2025

On a generating function of Niebur-Poincaré series

Let $Γ\subset PSL_2(\mathbb{R})$ be a Fuchsian group of the first kind which has a cusp $i\infty$ of width one. In this paper, we first consider a generating function formed with the Niebur--Poincaré series $\{F_{m,s}(τ)\}_{m\ge 1}$ associated to $i\infty$. We prove a relation between the continuation of this generating function to $s=1$ with the resolvent kernel associated to the hyperbolic Laplacian and the non-holomorphic Eisenstein series associated to $i\infty$, also at $s=1$. Secondly, we show that, for any $s\in \mathbb{N}$, the generating function equals Poincaré type series involving polylogarithms. We also consider a generating function formed with derivatives in $s$ of the Niebur--Poincaré series and prove that the continuation of the generating function at $s=1$ can be expressed in terms of $Γ$-periodization of a point-pair invariant involving the Rogers dilogarithm and the Kronecker limit function associated to the non-holomorphic Eisenstein series.

math.NT↗

arXiv2024

The discrete analogue of the Gaussian

This paper illustrates the utility of the heat kernel on $\mathbb{Z}$ as the discrete analogue of the Gaussian density function. It is the two-variable function $K_{\mathbb{Z}}(t,x)=e^{-2t}I_{x}(2t)$ involving a Bessel function and variables $x\in\mathbb{Z}$ and real $t\geq 0$. Like its classic counterpart it appears in many mathematical and physical contexts and has a wealth of applications. Some of these will be reviewed here, concerning Bessel integrals, trigonometric sums, hypergeometric functions and asymptotics of discrete models appearing in statistical and quantum physics. Moreover, we prove a new local limit theorem for sums of integer-valued random variables, obtain novel special values of the spectral zeta function of Bethe lattices, and provide a discussion on how $e^{-2t}I_{x}(2t)$ could be useful in differential privacy.

math-ph↗

arXiv2024

Constructing heat kernels on infinite graphs

Let $G$ be an infinite, edge- and vertex-weighted graph with certain reasonable restrictions. We construct the heat kernel of the associated Laplacian using an adaptation of the parametrix approach due to Minakshisundaram-Pleijel in the setting of Riemannian geometry. This is partly motivated by the wish to relate the heat kernels of a graph and a subgraph, or of a domain and a discretization of it. As an application, assuming that the graph is locally finite, we express the heat kernel $H_G(x,y;t)$ as a Taylor series with the lead term being $a(x,y)t^r$, where $r$ is the combinatorial distance between $x$ and $y$ and $a(x,y)$ depends (explicitly) upon edge and vertex weights. In the case $G$ is the regular $(q+1)$-tree with $q\geq 1$, our construction reproves different explicit formulas due to Chung-Yau and to Chinta-Jorgenson-Karlsson. Assuming uniform boundedness of the combinatorial vertex degree, we show that a dilated Gaussian depending on any distance metric on $G$, which is uniformly bounded from below can be taken as a parametrix in our construction. Our work extends in part the recent articles [LNY21, CJKS23] in that the graphs are infinite and weighted.

math.AP↗

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

The parametrix construction of the heat kernel on a graph

In this paper we develop the parametrix approach for constructing the heat kernel on a graph $G$. In particular, we highlight two specific cases. First, we consider the case when $G$ is embedded in a Eulidean domain or manifold $Ω$, and we use a heat kernel associated to $Ω$ to obtain a formula for the heat kernel on $G$. Second, we consider when $G$ is a subgraph of a larger graph $\widetilde{G}$, and we obtain a formula for the heat kernel on $G$ from the heat kernel on $\widetilde{G}$ restricted to $G$.

math.AP↗

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↗

arXiv2023

The resolvent kernel on the discrete circle and twisted cosecant sums

Let $X_m$ denote the discrete circle with $m$ vertices. For $x,y\in X_{m}$ and complex $s$, let $G_{X_m,χ_β}(x,y;s)$ be the resolvent kernel associated to the combinatorial Laplacian which acts on the space of functions on $X_{m}$ that are twisted by a character $χ_β$. We will compute $G_{X_m,χ_β}(x,y;s)$ in two different ways. First, using the spectral expansion of the Laplacian, we show that $G_{X_m,χ_β}(x,y;s)$ is a generating function for certain trigonometric sums involving powers of the cosecant function; by choosing $β$ or $s$ appropriately, the sums in question involve powers of the secant function. Second, by viewing $X_{m}$ as a quotient space of $\mathbb{Z}$, we prove that $G_{X_m,χ_β}(x,y;s)$ is a rational function which is given in terms of Chebyshev polynomials. From the existence and uniqueness of $G_{X_m,χ_β}(x,y;s)$, these two evaluations are equal. From the resulting identity, we obtain a means by which one can obtain explicit evaluations of cosecant and secant sums. The identities we prove depend on a number of parameters, and when we specialize the values of these parameters we obtain several previously known formulas. Going further, we derive a recursion formula for special values of the $L$-functions associated to the cycle graph $X_{m}$, thus answering a question from arXiv:2212.13687v1.

math.CO↗

arXiv2023

Discrete diffusion-type equation on regular graphs and its applications

We derive an explicit formula for the fundamental solution $K_{T_{q+1}}(x,x_{0};t)$ to the discrete-time diffusion equation on the $(q+1)$-regular tree $T_{q+1}$ in terms of the discrete $I$-Bessel function. We then use the formula to derive an explicit expression for the fundamental solution $K_{X}(x,x_{0};t)$ to the discrete-time diffusion equation on any $(q+1)$-regular graph $X$. Going further, we develop three applications. The first one is to derive a general trace formula that relates the spectral data on $X$ to its topological data. Though we emphasize the results in the case when $X$ is finite, our method also applies when $X$ has a countably infinite number of vertices. As a second application, we obtain a closed-form expression for the return time probability distribution of the uniform random walk on any $(q+1)$-regular graph. The expression is obtained by relating $K_{X}(x,x_{0};t)$ to the uniform random walk on a $(q+1)$-regular graph. We then show that if $\{X_{h}\}$ is a sequence of $(q+1)$-regular graphs whose number of vertices goes to infinity and which satisfies a certain natural geometric condition, then the limit of the return time probability distributions from $\{X_{h}\}$ is equal to the return time probability distribution on the tree $T_{q+1}$. As a third application, we derive formulas which express the number of distinct closed irreducible walks without tails on a finite graph $X$ in terms of moments of the spectrum of its adjacency matrix.

math.PR↗

arXiv2023

An integer factorization algorithm which uses diffusion as a computational engine

In this article we develop an algorithm which computes a divisor of an integer $N$, which is assumed to be neither prime nor the power of a prime. The algorithm uses discrete time heat diffusion on a finite graph. If $N$ has $m$ distinct prime factors, then the probability that our algorithm runs successfully is at least $p(m) = 1-(m+1)/2^{m}$. We compute the computational complexity of the algorithm in terms of classical, or digital, steps and in terms of diffusion steps, which is a concept that we define here. As we will discuss below, we assert that a diffusion step can and should be considered as being comparable to a quantum step for an algorithm which runs on a quantum computer. With this, we prove that our factorization algorithm uses at most $O((\log N)^{2})$ deterministic steps and at most $O((\log N)^{2})$ diffusion steps with an implied constant which is effective. By comparison, Shor's algorithm is known to use at most $O((\log N)^{2}\log (\log N) \log (\log \log N))$ quantum steps on a quantum computer. As an example of our algorithm, we simulate the diffusion computer algorithm on a desktop computer and obtain factorizations of $N=33$ and $N=1363$.

quant-ph↗

arXiv2022

On an approach for evaluating certain trigonometric character sums using the discrete time heat kernel

In this article we develop a general method by which one can explicitly evaluate certain sums of $n$-th powers of products of $d\geq 1$ elementary trigonometric functions evaluated at $\mathbf{m}=(m_1,\ldots,m_d)$-th roots of unity. Our approach is to first identify the individual terms in the expression under consideration as eigenvalues of a discrete Laplace operator associated to a graph whose vertices form a $d$-dimensional discrete torus $G_{\mathbf{m}}$ which depends on $\mathbf{m}$. The sums in question are then related to the $n$-th step of a Markov chain on $G_{\mathbf{m}}$. The Markov chain admits the interpretation as a particular random walk, also viewed as a discrete time and discrete space heat diffusion, so then the sum in question is related to special values of the associated heat kernel. Our evaluation follows by deriving a combinatorial expression for the heat kernel, which is obtained by periodizing the heat kernel on the infinite lattice $\mathbb{Z}^{d}$ which covers $G_{\mathbf{m}}$.

math.CO↗

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

Construction of Poincaré-type series by generating kernels

Let $Γ\subset \textrm{PSL}_2({\mathbb R})$ be a Fuchsian group of the first kind having a fundamental domain with a finite hyperbolic area, and let $\widetildeΓ$ be its cover in $\textrm{SL}_2({\mathbb R})$. Consider the space of twice continuously differentiable, square-integrable functions on the hyperbolic upper half-plane, which transform in a suitable way with respect to a multiplier system of weight $k\in{\mathbb R}$ under the action of $\widetildeΓ$. The space of such functions admits the action of the hyperbolic Laplacian $Δ_k$ of weight $k$. Following an approach of Jorgenson, von Pippich and Smajlović (where $k=0$), we use the spectral expansion associated to $Δ_k$ to construct a wave distribution and then identify the conditions on its test functions under which it represents automorphic kernels and further gives rise to Poincaré-type series. An advantage of this method is that the resulting series may be naturally meromorphically continued to the whole complex plane. Additionally, we derive sup-norm bounds for the eigenfunctions in the discrete spectrum of $Δ_k$.

math.NT↗