Linear Recurrence Research Articles

Higher Reciprocity Laws and Ternary Linear Recurrence Sequences

Higher Reciprocity Laws and Ternary Linear Recurrence Sequences

On the p-Adic Valuation of Third Order Linear Recurrence Sequences

On the p-Adic Valuation of Third Order Linear Recurrence Sequences

In search of a higher Bochner theorem

We initiate the study of a natural generalisation of the classical Bochner–Krall problem asking which linear ordinary differential operators possess sequences of eigenpolynomials satisfying linear recurrence relations of finite length; the classical case corresponds to the 3-term recurrence relations with real coefficients subject to some extra restrictions. We formulate a general conjecture and prove it in the first non-trivial case of operators of order 3.

Regularized dynamic mode decomposition algorithm for time sequence predictions

Dynamic Mode Decomposition (DMD) aims at extracting intrinsic mechanisms in a time sequence via linear recurrence relation of its observables, thereby predicting later terms in the sequence. Stability is a major concern in DMD predictions. We adopt a regularized form and propose a Regularized DMD (ReDMD) algorithm to determine the regularization parameter. This leverages stability and accuracy. Numerical tests for Burgers’ equation demonstrate that ReDMD effectively stabilizes the DMD prediction while maintaining accuracy. Comparisons are made with the truncated DMD algorithm.

Riemannian optimization of photonic quantum circuits in phase and Fock space

We propose a framework to design and optimize generic photonic quantum circuits composed of Gaussian objects (pure and mixed Gaussian states, Gaussian unitaries, Gaussian channels, Gaussian measurements) as well as non-Gaussian effects such as photon-number-resolving measurements. In this framework, we parametrize a phase space representation of Gaussian objects using elements of the symplectic group (or the unitary or orthogonal group in special cases), and then we transform it into the Fock representation using a single linear recurrence relation that computes the Fock amplitudes of any Gaussian object recursively. We also compute the gradient of the Fock amplitudes with respect to phase space parameters by differentiating through the recurrence relation. We can then use Riemannian optimization on the symplectic group to optimize MM-mode Gaussian objects, avoiding the need to commit to particular realizations in terms of fundamental gates. This allows us to “mod out” all the different gate-level implementations of the same circuit, which now can be chosen after the optimization has completed. This can be especially useful when looking to answer general questions, such as bounding the value of a property over a class of states or transformations, or when one would like to worry about hardware constraints separately from the circuit optimization step. Finally, we make our framework extendable to non-Gaussian objects that can be written as linear combinations of Gaussian ones, by explicitly computing the change in global phase when states undergo Gaussian transformations. We implemented all of these methods in the freely available open-source library MrMustard, which we use in three examples to optimize the 216-mode interferometer in Borealis, and 2- and 3-modes circuits (with Fock measurements) to produce cat states and cubic phase states.

On the Linear Recurrence of (Generalized) Hybrid Numbers Sequences and Moment Problems

On the Linear Recurrence of (Generalized) Hybrid Numbers Sequences and Moment Problems

Kepler sets of second-order linear recurrence sequences over Qp

Let [Formula: see text] be a second-order linear recurrence sequence with constant coefficients over the field of [Formula: see text]-adic numbers [Formula: see text]. We study the set of limit points of the sequence of consecutive ratios [Formula: see text] in [Formula: see text].

Simulating time delays and space–time memory interactions: An analytical approach

This study introduces a novel analytical framework to explore the effects of Caputo spatial and temporal memory indices combined with a proportional time delay on (non)linear (1+2)-dimensional evolutionary models. The solution is expressed as a Cauchy product of an absolutely convergent series that effectively captures the dynamics of these parameters. By extending the differential transform method into higher-dimensional fractional space, we reformulate the evolution equation as a (non)linear higher-order recurrence relation, which enables the precise determination of fractional series coefficients. Our findings show that Caputo derivatives and time delay significantly impact the system’s behavior, with graphical analysis revealing a continuous transition from a stationary to an integer state solution. The study also identifies a quantitative analogy between the Caputo-time fractional derivative and proportional time delay that highlights the role of Caputo derivatives as memory indices. This method has proven highly effective in deriving solutions for fractional higher-dimensional extensions of evolutionary equations.

Ruin Probabilities as Recurrence Sequences in a Discrete-Time Risk Process

The theory of linear recurrence sequences is applied to obtain an explicit formula for the ultimate ruin probability in a discrete-time risk process. It is assumed that the claims distribution is arbitrary but has finite support {0,1,…,m+1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\{0,1,\\ldots ,m+1\\}}$$\\end{document}, for some integer m≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{m\\ge 1}$$\\end{document}. The method requires finding the zeroes of an m degree polynomial and solving a system of m linear equations. An approximation is derived and some numerical results and plots are provided as examples.

A nonlinear Filbert-like matrix with three free parameters: From linearity to nonlinearity

Abstract Filbert and Lilbert matrices are defined by terms of linear Fibonacci-like sequences. By considering terms of such linear recurrences and additional free parameters, we define a nonlinear variant of these matrices via ratios of q-forms of terms of Fibonacci-like sequences whose indices are in nonlinear forms. We derive explicit formulae for the matrices L and U come from LU-decomposition, their inverses, inverse of the main matrices and their determinants.

On multiplicative dependent vectors with coordinates from certain higher order recurrences

In this paper, we prove that there are finitely many multiplicative dependent vectors with coordinates from non-degenerate linear recurrence sequences [Formula: see text] of a fixed order [Formula: see text] These sequences satisfy the recurrence relation [Formula: see text] with initials [Formula: see text]. Here, the coefficients of the recurrence relation are positive integers satisfying [Formula: see text] or [Formula: see text] In both these conditions, [Formula: see text]

On Robustness for the Skolem, Positivity and Ultimate Positivity Problems

The Skolem problem is a long-standing open problem in linear dynamical systems: can a linear recurrence sequence (LRS) ever reach 0 from a given initial configuration? Similarly, the positivity problem asks whether the LRS stays positive from an initial configuration. Deciding Skolem (or positivity) has been open for half a century: the best known decidability results are for LRS with special properties (e.g., low order recurrences). But these problems are easier for "uninitialized" variants, where the initial configuration is not fixed but can vary arbitrarily: checking if there is an initial configuration from which the LRS stays positive can be decided in polynomial time (Tiwari in 2004, Braverman in 2006). In this paper, we consider problems that lie between the initialized and uninitialized variants. More precisely, we ask if 0 (resp. negative numbers) can be avoided from every initial configuration in a neighborhood of a given initial configuration. This can be considered as a robust variant of the Skolem (resp. positivity) problem. We show that these problems lie at the frontier of decidability: if the neighbourhood is given as part of the input, then robust Skolem and robust positivity are Diophantine hard, i.e., solving either would entail major breakthroughs in Diophantine approximations, as happens for (non-robust) positivity. However, if one asks whether such a neighbourhood exists, then the problems turn out to be decidable with PSPACE complexity. Our techniques also allow us to tackle robustness for ultimate positivity, which asks whether there is a bound on the number of steps after which the LRS remains positive. There are two variants depending on whether we ask for a "uniform" bound on this number of steps. For the non-uniform variant, when the neighbourhood is open, the problem turns out to be tractable, even when the neighbourhood is given as input.

Deflection of light by a Reissner–Nordström black hole and Painlevé VI equation

We consider the bending angle of the trajectory of a photon incident from and deflected to infinity around a Reissner–Nordström black hole. We treat the bending angle as a function of the squared reciprocal of the impact parameter and the squared electric charge of the background normalized by the mass of the black hole. It is shown that the bending angle satisfies a system of two inhomogeneous linear partial differential equations with polynomial coefficients. This system can be understood as an isomonodromic deformation of the inhomogeneous Picard–Fuchs equation satisfied by the bending angle in the Schwarzschild spacetime, where the deformation parameter is identified as the background electric charge. Furthermore, the integrability condition for these equations is found to be a specific type of the Painlevé VI equation that allows an algebraic solution. We solve the differential equations both at the weak and strong deflection limits. In the weak deflection limit, the bending angle is expressed as a power series expansion in terms of the squared reciprocal of the impact parameter and we obtain the explicit full-order expression for the coefficients. In the strong deflection limit, we obtain the asymptotic form of the bending angle that consists of the divergent logarithmic term and the finite O(1) term supplemented by linear recurrence relations which enable us to straightforwardly derive higher order coefficients. In deriving these results, the isomonodromic property of the differential equations plays an important role. Lastly, we briefly discuss the applicability of our method to other types of spacetimes such as a spinning black hole.

(Almost) complete characterization of the stability of a discrete-time Hawkes process with inhibition and memory of length two

Abstract We consider a Poisson autoregressive process whose parameters depend on the past of the trajectory. We allow these parameters to take negative values, modelling inhibition. More precisely, the model is the stochastic process $(X_n)_{n\ge0}$ with parameters $a_1,\ldots,a_p \in \mathbb{R}$ , $p\in\mathbb{N}$ , and $\lambda \ge 0$ , such that, for all $n\ge p$ , conditioned on $X_0,\ldots,X_{n-1}$ , $X_n$ is Poisson distributed with parameter $(a_1 X_{n-1} + \cdots + a_p X_{n-p} + \lambda)_+$ . This process can be regarded as a discrete-time Hawkes process with inhibition and a memory of length p. In this paper we initiate the study of necessary and sufficient conditions of stability for these processes, which seems to be a hard problem in general. We consider specifically the case $p = 2$ , for which we are able to classify the asymptotic behavior of the process for the whole range of parameters, except for boundary cases. In particular, we show that the process remains stochastically bounded whenever the solution to the linear recurrence equation $x_n = a_1x_{n-1} + a_2x_{n-2} + \lambda$ remains bounded, but the converse is not true. Furthermore, the criterion for stochastic boundedness is not symmetric in $a_1$ and $a_2$ , in contrast to the case of non-negative parameters, illustrating the complex effects of inhibition.

Some infinite series summations involving linear recurrence relations of order 2 and 3

This paper extends known results of second and third order recursive sequences through extensive formulations of properties of the roots of their characteristic equations, some are old but most are new. They are applied to novel studies of $\sum_{n=0}^{\infty}{\frac{a_{mn}}{10^{n+1}}, \ m=1,2,3}$, including their convergence criteria, and applied to many standard sequences, as particular cases of a generic $\left\{a_n\right\}$. The detailed development of the algebra of the pertinent theorems, and their associated lemmas and corollaries, should open up new vistas for interested number theorists with the concluding results on series values.

Deciphering two decades of cellular reprogramming in cancer: A bibliometric analysis of evolving trends and research frontiers

Recent research has reevaluated the traditional view of cancer's linear progression and recurrence by introducing cellular reprogramming a process in which cancer cells can their state under certain conditions. This change is driven by a combination of genetic and epigenetic factors, with pivotal roles played by key genes, and pathways, notably Wnt and Notch. The complexity of cancer's behavior is further influenced by factors such as the epithelial-mesenchymal transition (EMT) and therapy-induced stress, both of which are significant contributors to cancer recurrence. In this context bibliometric analysis emerges as a crucial tool for evaluating the impacts and trends within scientific literature. Our study utilized bibliometrics to analysis the role of cellular reprogramming oncology over the past two decades, highlighting its potential to improve cancer treatment outcomes. In conducting this analysis, we searched for literature search on cellular reprogramming (CR) in the Web of Science database, covering the years 2002–2022. We employed visualization tools like Citespace, VOSviewer, and Bibliometrix to analyze the collected data resulting in a dataset of 3102 articles. The United States and China emerged as leading contributors to this field, with the University of Texas MD Anderson Cancer Center being the most prolific institution. Menendez was the most influential scholar in this research domain. Cancers was the journal with the most publications on this subject. The most local-cited document was the article titled “Hallmarks of Cancer: The Next Generation”. A comprehensive analysis has been conducted based on keywords and cited references. In recent years, the research emphasis has shifted to “extracellular vesicles,” “cancer therapy,” and “cellular plasticity”. Therefore, this analysis uses bibliometrics to chart cutting-edge progress in cancer's cellular reprogramming, aiding experts to quickly understand and innovate in this crucial area.

Spectral properties of a functional binomial matrix

In this article, we consider the definition of the Fibonacci polynomial sequence with the second-order linear recurrence relation, where coefficients and initial conditions depend on the variable $t$. And then, we introduce the functional binomial matrix depending on the coefficients of the second-order linear recurrence relation. In the following, we study the spectral properties of the functional binomial matrix using the Fibonacci polynomial sequence and we obtain a diagonal decomposition for it using the Vandermunde matrix. Finally, by applying some linear algebra tools we obtain a number of combinatorial identities involving the Fibonacci polynomial sequence.

Positivity-hardness results on Markov decision processes

This paper investigates a series of optimization problems for one-counter Markov decision processes (MDPs) and integer-weighted MDPs with finite state space. Specifically, it considers problems addressing termination probabilities and expected termination times for one-counter MDPs, as well as satisfaction probabilities of energy objectives, conditional and partial expectations, satisfaction probabilities of constraints on the total accumulated weight, the computation of quantiles for the accumulated weight, and the conditional value-at-risk for accumulated weights for integer-weighted MDPs. Although algorithmic results are available for some special instances, the decidability status of the decision versions of these problems is unknown in general. The paper demonstrates that these optimization problems are inherently mathematically difficult by providing polynomial-time reductions from the Positivity problem for linear recurrence sequences. This problem is a well-known number-theoretic problem whose decidability status has been open for decades and it is known that decidability of the Positivity problem would have far-reaching consequences in analytic number theory. So, the reductions presented in the paper show that an algorithmic solution to any of the investigated problems is not possible without a major breakthrough in analytic number theory. The reductions rely on the construction of MDP-gadgets that encode the initial values and linear recurrence relations of linear recurrence sequences. These gadgets can flexibly be adjusted to prove the various Positivity-hardness results.

Higher-degree Artin conjecture

ABSTRACT For an algebraic number α we consider the orders of the reductions of α in finite fields. In the case where α is an integer, it is known by the work on Artin’s primitive root conjecture that the order is ‘almost always almost maximal’ assuming the Generalized Riemann Hypothesis (GRH), but unconditional results remain modest. We consider higher-degree variants under GRH. First, we modify an argument of Roskam to settle the case where α and the reduction have degree two. Second, we give a positive lower density result when α is of degree three and the reduction is of degree two. Third, we give higher-rank results in situations where the reductions are of degree two, three, four or six. As an application we give an almost equidistribution result for linear recurrences modulo primes. Finally, we present a general result conditional to GRH and a hypothesis on smooth values of polynomials at prime arguments.

Fibonacci primes, primes of the form 2n − k and beyond

We speculate on the distribution of primes in exponentially growing, linear recurrence sequences (un)n≥0 in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we guess that either there are only finitely many primes un, or else there exists a constant cu>0 (which we can give good approximations to) such that there are ∼culog⁡N primes un with n≤N, as N→∞. We compare our conjecture to the limited amount of data that we can compile. One new feature is that the primes in our Euler product are not taken in order of their size, but rather in order of the size of the period of the un(modp).