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Asymptotic expansion of solutions of the 2nd order difference equations in an unbounded domain
Acta Applicandae Mathematicae. 2025. Vol. 195. Article 4.
Difference equations play a crucial role in a wide array of mathematical and physical tasks. In this article, we focus on the analysis of a second order linear homogeneous difference equation with smooth coefficients via WKB method. It is well-known that such equations exhibit two WKB solutions in a segment devoid of turning and singular points. We establish a theorem demonstrating the existence of these solutions in an unbounded domain under certain conditions regarding the smoothness and growth behavior of the coefficients at infinity. Furthermore, using this theorem, we derive the asymptotic expansion of Laguerre polynomials for large orders and values, yielding estimates that align with existing results.
Keywords: semiclassical approximationквазиклассическое приближениеLaguerre polynomialsdifference equationsразностные уравненияWKB methodполиномы Лагерра
Publication based on the results of:
Veretennikov A., Ахмярова А. Т., Теория вероятностей и ее применения 2024 Т. 70 № 2 С. 211–227
Предложен новый вариант усиленного закона больших чисел для попарно независимых случайных величин. Основная цель — ослабить требование существования математического ожидания каждого из слагаемых. Предположение о попарной независимости также ослаблено. ...
Added: July 17, 2026
Veretennikov A., Moscow Mathematical Journal 2024 Vol. 24 No. 1 P. 107–124
Second order recurrence are established for a d-dimensional diffusion with an additive Wiener process, with switching, and with one recurrent and one transient regime and constant switching intensities, under suitable conditions. As a corollary, the rate of convergence towards the invariant regime of order t^{−2} is claimed. The approach is based on embedded Markov chains ...
Added: July 16, 2026
Veretennikov A., Mathematics 2023 Vol. 11 No. 21 Article 4514
Positive recurrence for a single-server queueing system is established under generalized service intensity conditions, without the assumption of the existence of a service density distribution function, but with a certain integral type lower bound as a sufficient condition. Positive recurrence implies the existence of the invariant distribution and a guaranteed slow convergence to it in ...
Added: July 16, 2026
Veretennikov A., Markov Processes and Related Fields 2023 Vol. 23 No. 2 P. 259–294
The ergodic Bellman's (HJB) equation is proved for a one-dimensional controlled diffusion with switching with variable diffusion and drift coefficients both depending on control; the intensities of transitions of the discrete component are constant. Its existence and uniqueness is established. Also, the convergence of the reward iteration improvement algorithm is established to the cost constant ...
Added: July 16, 2026
On recurrence, convergence and mixing rate for generalised Wright - Fisher's diffusion with mutation
Veretennikov A., Sineokiy R., Markov Processes and Related Fields 2023 Vol. 23 No. 2 P. 241–258
Generalised one-dimensional Fisher -- Wright diffusion process with mutations is consiedered. This is a well-known model in populational genetics. The goal of the paper is an exponential recurrence of the process, which also implies exponential rate of convergence towards the invariant measure. ...
Added: July 16, 2026
Veretennikov A., Mathematics 2023 Vol. 11 No. 14 Article 3096
Polynomial recurrence bounds for a class of stochastic differential equations with a rotational symmetric gradient type drift and an additive Wiener process are established, as well as certain a priori moment inequalities for solutions. The key feature of this paper is that the approach does not use Lyapunov functions because it is not clear how ...
Added: July 16, 2026
Kuninets A., Malygina E., Leevik A. G. et al., Journal of Computer Virology and Hacking Techniques 2026 No. 22 Article 62
In this work, we investigate the application of Barnes–Wall lattices in post-quantum cryptographic schemes. We survey and analyze several constructions of Barnes–Wall lattices, including subgroup chains, the generalized k-ing construction, and connections with Reed-Muller codes, highlighting their equivalence over both Z[i] and Z. Building on these structural insights, we introduce a new algorithm for efficient ...
Added: July 16, 2026
Bolbachan V., / Series math "arxiv.org". 2024.
Chow polylogarithms are some special functions arising in explicit description of the Beilinson regulator map. The most interesting functional equation for this function reflects its vanishing on the boundary in the Bloch's cycle complex. We show that this functional equation formally follows from more simple ones, namely skew-symmetry, functoriality and multiplicativity.
To prove this, we study ...
Added: July 16, 2026
Bolbachan V., / Series math "arxiv.org". 2024.
Let K be a field of characteristic zero. We prove that its motivic cohomology in degree m−1 and weight m is rationally isomorphic to the cohomology of the polylogarithmic complex. This gives a partial extension of A. Suslin theorem describing the indecomposable K3 of a field. ...
Added: July 16, 2026
Zapryagaev A., Pahomov F., Logic Journal of the IGPL 2026 Vol. 34 No. 4 Article 12
We prove the linear orders first-order definable in the standard model (Z;<,+) of Presburger arithmetic are exactly those that are (Z;<,+)-definably embeddable into the lexicographic ordering on Z^n for some n. ...
Added: July 16, 2026
Veretennikov A., Veretennikova M., Reliability: Theory & Applications 2022 Vol. 17 No. 3(69) P. 273–291
A simple model of the new notion of ``Markov up'' processes is proposed; its positive recurrence and ergodic properties are shown under the appropriate conditions. ...
Added: July 16, 2026
Veretennikov A., Stochastics and Partial Differential Equations: Analysis and Computations 2022 Vol. 10 P. 1165–1179
Positive recurrence of a $d$-dimensional diffusion with an additive Wiener process, with switching and with one recurrent and one transient regimes and variable switching intensities is established under suitable conditions. The approach is based on embedded Markov chains. ...
Added: July 15, 2026
Veretennikov A., Queueing Systems 2022 Vol. 100 No. 3-4 P. 357–359
B.A. Sevastyanov's result about Erlang telephone station problem has been extended in several publications. In this short note one open question about this model has been discussed. The whole volume was devoted mainly to open problems related to the name of Erlang. ...
Added: July 15, 2026
Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2026 Vol. 114 No. 1 P. 014217–014217
This study investigates the dynamical origins and statistical properties of extreme events (EEs) in a diffusively coupled theoretical Brusselator system, extending from pairwise interactions to globally coupled networks. Statistically, the emergence of EEs is characterized by heavy-tailed probability density functions and exponential interevent interval distributions, alongside an analysis of the complementary cumulative distribution function and ...
Added: July 15, 2026
Prokofev V., Zabrodin A., Proceedings of the Steklov Institute of Mathematics 2020 Vol. 309 P. 225–239
We consider solutions of the matrix Kadomtsev-Petviashvili (KP) hierarchy that are trigonometric functions of the first hierarchical time t1 = x and establish the correspondence with the spin generalization of the trigonometric Calogero-Moser system at the level of hierarchies. Namely, the evolution of poles xi and matrix residues at the poles aαibβi of the solutions with respect to the kth hierarchical time of the ...
Added: July 14, 2026
Prokofev V., Zabrodin A., Journal of Physics A: Mathematical and Theoretical 2021 Vol. 54
We consider solutions of the Kadomtsev–Petviashvili hierarchy which are elliptic functions of x = t1. It is known that their poles as functions of t2 move as particles of the elliptic Calogero–Moser model. We extend this correspondence to the level of hierarchies and find the Hamiltonian Hk of the elliptic Calogero–Moser model which governs the dynamics of poles with respect to the kth ...
Added: July 14, 2026
Prokofev V., Zabrodin A., Theoretical and Mathematical Physics 2021 Vol. 208 P. 1093–115
We consider solutions of the 2D Toda lattice hierarchy which are elliptic functions of the zeroth time t_0=x. It is known that their poles as functions of t_1 move as particles of the elliptic Ruijsenaars-Schneider model. The goal of this paper is to extend this correspondence to the level of hierarchies. We show that the ...
Added: July 14, 2026
Prokofev V., Zabrodin A., Теоретическая и математическая физика 2023 Т. 217 № 2 С. 299–316
We continue the study of the B-Toda hierarchy (the Toda lattice with the constraint of type B), which can be regarded as a discretization of the BKP hierarchy. We introduce the tau function of the B-Toda hierarchy and obtain bilinear equations for it. Examples of soliton tau functions are presented in explicit form. ...
Added: July 14, 2026
Шиманогов И. Н., Vyalyi M., Дискретный анализ и исследование операций 2025 Т. 32 № 4(166) С. 213–230
A well-studied class of algorithmic problems is that of regular realizability: checking the non-emptiness of the intersection of a regular language with a given language. This problem has a natural algebraic interpretation: verifying whether an element of a Boolean algebra belongs to the kernel of a certain homomorphism. This motivates the consideration of an analogous ...
Added: July 12, 2026
Rybakov M., Annals of Pure and Applied Logic 2026 Vol. 177 Article 103811
The paper presents a solution to the question about the decidability of the two-variable fragment of the superintuitionistic predicate logic QLC defined by the class of linear Kripke frames, which is also the ‘superintuitionistic’ fragment of the modal predicate logic QS4.3, under the Gödel translation. We prove that the fragment is undecidable. The result remains true for the ...
Added: July 11, 2026
A. V. Pereskokov, Journal of Mathematical Sciences 2025 Vol. 291 No. 4 P. 544–553
We study the eigenvalue problem for a perturbed resonance oscillator. We find asymptotic
expansions of coherent states of the algebra su(2). We show that the asymptotic
eigenfunctions are localized near a circle and construct an expansion of asymptotic eigenfunctions
near this circle. ...
Added: December 7, 2025
Vybornyi E., Rumiantseva S., Математические заметки 2024 Т. 116 № 6 С. 862–880
In this paper, we consider the problem of constructing a semiclassical asymptotic estimate of the splitting between a pair of close lower-lying energy levels for a quadratic operator defined on the irreducible representation of the Lie algebra su(1,1). In Darboux coordinates on the hyperboloid the Hamiltonian defines the landscape of a symmetric double well. It ...
Added: November 12, 2024
Danilov V., Rakhel M., ZAMM Zeitschrift für Angewandte Mathematik und Mechanik 2024 Vol. 104 No. 3 Article e202300491
In this paper, we propose a scheme for constructing the asymptotics of the fundamental solution of a linear degenerate parabolic equation with a small parameter. The asymptotics is constructed using the operator representation of the Dirac delta function and the non-oscillating Wentzel-Kramers-Brillouin (WKB) method. In addition, a method to justify the obtained asymptotics by proving the ...
Added: December 22, 2023
Lomonosov T., / Series arXiv "math". 2023. No. 2307.07182.
In this paper, an algebraic modification of the method of undetermined coefficients for solving nonhomogeneous linear stationary difference equations for quasipolynomial right-hand sides is proposed. Although the classical method of undetermined coefficients is well-known in both differential equations and difference equations case, its application in the difference equations case is severely limited. For example, it ...
Added: October 30, 2023