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We give global structure of spectra of periodic non-Hermitian Jacobi operators by the discriminant and its stationary points. We also give necessary and sufficient conditions for real spectra and single interval spectra. [ABSTRACT FROM AUTHOR]

In this paper, the authors consider the spectra of second-order left-definite difference operator with linear spectral parameters in two boundary conditions. First, they obtain the exact number of this kind of eigenvalue problem, and prove these eigenvalues are all real and simple. In details, they get that the number of the positive (negative) eigenvalues is related to not only the number of positive (negative) elements in the weight function, but also the parameters in the boundary conditions. Second, they obtain the interlacing properties of these eigenvalues and the sign-changing properties of the corresponding eigenfunctions according to the relations of the parameters in the boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, a generalized diffusion–advection–reaction equation (GDARE) is considered and by using the solitary wave solution two new class of exact finite difference (EFD) schemes are proposed under certain functional relationship between the temporal and space step sizes. Suitable denominator functions are identified from these EFD schemes and a new non-standard finite difference (NSFD) scheme is developed. This NSFD scheme preserves positivity and boundedness properties of the solution of GDARE. The $ L_2 $ L 2 , $ L_\infty $ L ∞ errors and CPU times are obtained for the proposed NSFD and EFD schemes to show the proficiency and accuracy of the schemes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new d-orthogonal polynomial sequence by resolving a characterization problem involving a generalized Dunkl operator. We give some properties of this family, namely, a generating function, the fold symmetry property, a higher-order recurrence relation and differential–difference equation, the inversion formula, the explicit expression of the moments and their integral representations associated with the canonical d-dimensional functional vector ensuring the d-orthogonality as well as the components with respect to a cyclic group of finite order. We recover, as particular cases, some new and known results. [ABSTRACT FROM AUTHOR]

We construct a difference analog of the Lamé operator. Namely, we present a second-order difference operator whose coefficients depend on a small parameter which commutes with a difference operator of order . When the small parameter tends to zero, the difference operator transforms into the Lamé operator. [ABSTRACT FROM AUTHOR]

We establish a fractional extension of the discrete Karamata integration theorem for two types of fractional sum operators. This result, along with other properties of regularly varying sequences and the tools such as comparison theorems and a fixed point theorem in FK-space, is then used to study asymptotic properties of solutions to a nonlinear fractional difference equation. We also establish and utilize a fractional extension of the Stolz–Cesáro theorem, i.e., of the discrete l'Hospital rule. Both, the discrete fractional Karamata theorem and the discrete fractional l'Hospital rule, are believed to find applications in a broader context within the discrete fractional calculus. [ABSTRACT FROM AUTHOR]

AbstractLet the function β be strictly increasing and continuous on an interval I ⊂ ℝ. The β-difference operator is defined by Dβ f (t) = (f(β(t)) − f(t)/(β(t) – t), where t ≠ β(t), and Dβ f (t) = f′ t(t) when t = s0 is a fixed point of the function β. This quantum operator is a generalization of q-Jackson, Hahn, power and other quantum operators. As a convenience of the β-function: β(t) turns into the probability distribution function with the probability measure 1, and the sample space ℝ, in the case of its conditions are relaxed to be increasing and continuous from the right, that is, limt→∞ β(t) = 1 and limt→∞ β(t) = 0, and also by using the Lebesgue-Stieltjes measure of the interval [a, b] to be β(b) − β(a). In this paper, we investigate a β-Taylor’s formula associated with the operator Dβ when the function β has a unique fixed point s0 ∈ I, which may allow for more flexible and accurate approximations of functions. An estimation of its remainder is given. Additionally, the β-power series is defined. Furthermore, as application, the β-expansion form of some fundamental functions is introduced. Finally, we find the unique solution of the β-shifting problem. [ABSTRACT FROM AUTHOR]

In this paper, inverse problems are studied for Sturm–Liouville difference operators having $ \frac {p^{2}-1/4}{m^{2}} $ p 2 − 1 / 4 m 2 type singularity. First, we define the Green's function and the corresponding Weyl function, some of their properties are investigated, and we give a method to find the potential function. Then we give the solution of the inverse nodal problem and present a constructive procedure for the solution of the inverse problem by using nodes. [ABSTRACT FROM AUTHOR]

In this paper, using operational rules, we derive some explicit expansion formulas associated with Brenke type polynomials. In particular, the corresponding connection and linearization coefficients are expressed either in terms of double sums or integral representations or sometimes in closed hypergeometric terms. The derived results are applied to Generalized Gould–Hopper and q-Laguerre polynomials. Some well-known duplication and convolution formulas are deduced as particular cases. [ABSTRACT FROM AUTHOR]

In this paper, we introduce generalized difference weak sequence space classes by utilizing the difference operator Δ ı ȷ \Delta^{\jmath}_{\imath} and the de la Vallée–Poussin mean, denoted as [ ( V , λ ) w , Δ ı ȷ ] m [(\mathscr{V},\lambda)_{w},\Delta^{\jmath}_{\imath}]_{m} for m = 0 m=0 , 1, and ∞. Further, we explore some algebraic and topological properties of these spaces, including their nature as linear, normed, Banach, and BK spaces. Additionally, we examine properties such as solidity, symmetry, and monotonicity. Finally, we define and establish some inclusion relations among generalized difference weak statistical convergence, generalized difference weak 휆-statistical convergence, and generalized difference weak [ V , λ ] [\mathscr{V},\lambda] -convergence. [ABSTRACT FROM AUTHOR]

In order to accurately capture non-local properties and long-term memory effects, this study combines the tempered fractional-order operator with delayed neural networks to investigate its stability, leveraging the introduced decay term of the tempered fractional-order operator. Firstly, the discrete-time tempered fractional-order neural networks model (DTFNNs) is presented. Furthermore, in an effort to better understand the dynamic behavior of complex systems, solutions to discrete-time tempered fractional non-homogeneous equations are obtained. The stability conditions for systems are subsequently established, contributing novel insights to the field. To validate the robustness of these conditions, numerical experiments are conducted, underscoring the practical relevance of the proposed model. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of periodic Schrödinger lattice systems with spectrum 0 and saturable nonlinearities. The existence of ground state solitons of the systems under weak assumptions is obtained. The main novelties are as follows. (1) Some new sufficient conditions for the existence of ground state solitons under the “spectral endpoint” assumption are constructed. (2) Our “non-monotonic” conditions make the proofs of the boundedness of the (PS) sequences to be easier. (3) Our result extends and improves the related results in the literature. Besides, some examples are given to illuminate our result. [ABSTRACT FROM AUTHOR]

We present a general integrodifference pioneer-climax competition system and study the invasion dynamics of the system in both monotone and nonmonotone cases. For the parameter regions in which the system is monotone, we establish the existence of the spreading speed and travelling waves and show that the minimum wave speed is linearly selected if the overall dispersal of the invasive species is larger than that of the native one. For the parameter regions in which the system is nonmonotone, we estimate the spreading speed by constructing suitable auxiliary systems and prove the existence of travelling waves by the fixed point theorem and upper-lower solution method. We apply our theoretical results to an integrodiffference pioneer-climax system which is a spatial version of the model recently studied by Gilbertson and Kot [Theor. Ecol., 14(2021), 501-523]. [ABSTRACT FROM AUTHOR]

In this paper, the problem of solvability and stability for switched discrete-time linear singular (SDLS) systems under Lipschitz perturbations is studied. By definition of SDLS systems of index-1, we first prove the unique existence of solution of SDLS systems under Lipschitz perturbations with different switching rules on two sides. A variation of constants formula of solution is given and the solution manifold is also described. Secondly, we derive some conditions for stability, asymptotical stability and exponential stability of these systems. Finally, some examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

We study the existence of ground state solutions for a class of discrete nonlinear Schrödinger equation with a sign-changing potential which is periodic or asymptotically periodic. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite, the second is that, due to the asymptotically periodic assumption, the associated functional loses the ℤ-translation invariance, and many effective methods for periodic problems cannot be applied to asymptotically periodic ones. These enables us to develop a direct approach to find ground state solutions with asymptotically periodic potential. Two types of ground state solutions are obtained with some new super-quadratic conditions on nonlinearity which are weaker that some well-known ones. Moreover, our conditions can also be used to significantly improve the well-known results of the corresponding continuous nonlinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following discrete nonlinear Schrödinger equation - Δ u n + V n u n - ω u n = f n (u n) , n ∈ Z , lim | n | → ∞ u n = 0 , with non-periodic potentials. We obtain the existence of nontrivial solutions of this equation with various nonlinearities by using Morse theory. In this way, we need to compute the critical groups of the corresponding action functional of the equation. To the best of our knowledge, there is no result focusing on this issue in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we have considered an inverse scattering problem for discrete Dirac operator L with eigenparameter dependent boundary condition. At the outset, the Jost solution and the Jost function of L are given. Then the zeros of the Jost function and some properties of the scattering function are examined. After finding the scattering dataset and the main equation for the operator L, the uniqueness of the kernel, continuity of the scattering function and the appropriate Levinson type formula are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study a two-point boundary value problem involving p-Laplacian and its relationship with the associated discrete ones. Specifically, by using discrete variational methods, we first obtain a sequence of positive solutions for the associated discrete two-point boundary value problems corresponding to different step lengths. Then, utilizing the sequence of positive solutions, we construct a sequence of continuous functions which can be shown to be precompact. Finally, we prove that the limit function of the convergent subsequence is the desired positive solution. In particular, our result covers the cases when the nonlinear function in the equation is (p - 1) -superlinear and asymptotically (p - 1) -linear. [ABSTRACT FROM AUTHOR]

The modification of the coefficients of formal power series is analyzed in order that such variation preserves q-Gevrey asymptotic properties, in particular q-Gevrey asymptotic expansions. A characterization of such sequences is determined, providing a handy tool in practice. The sequence of q-factorials is proved to preserve q-Gevrey asymptotic expansions. [ABSTRACT FROM AUTHOR]

The aim of this research is to present a new generalization of d-orthogonal $ (d\geq 2) $ (d ≥ 2) polynomials of Laguerre type by utilizing a suitable generating function from Sheffer class and employing operational rules associated with the lowering and raising operators that satisfy d-orthogonality. We derive several properties of these polynomials and establish the recurrence relation. Moreover, we provide explicit, connection and inversion formulas, the $ (d+1) $ (d + 1) -order differential equation, and the canonical d-dimensional functional vector. [ABSTRACT FROM AUTHOR]

Like qq-calculus, Hahn calculus (or q,ωq,\omega -calculus) is constructed by defining a difference derivative operator and an integral operator. The q,ωq,\omega -analogs of the integral representations of the Laplace transform and related special functions, such as gamma and beta, are proposed in this article. Then, some basic properties similar to classical and qq-analogs are investigated. Finally, a few examples are given to solve q,ωq,\omega -initial value problems via the newly introduced q,ωq,\omega -Laplace transform.

In the present paper, we are interested in the existence of least energy sign-changing solutions for a class of discrete Schrödinger equations involving asymptotically linear behavior. When the linear potentials are assumed to be unbounded at the infinities and the nonlinearities satisfy certain strict conditions, a least energy sign-changing solution has been obtained via constraint variational method, degree theory and deformation lemma. Moreover, the solution changes sign exactly once and has a fast exponential decay with any order, which is firstly obtained by a comparison principle. [ABSTRACT FROM AUTHOR]

Fractional Hahn boundary value problems are significant tools to describe mathematical and physical phenomena depending on non-differentiable functions. In this work, we develop certain aspects of the theory of fractional Hahn boundary value problems involving fractional Hahn derivatives of the Caputo type. First, we construct the Green function for an αth\alpha {\rm{th}}-order fractional boundary value problem, with 1

Several efforts have been recently devoted to the studies on epidemic mathematical models based on fractional-order operators, by virtue of their capability to take into account memory effects and nonlocal features. The aim of this paper is to make a contribution to the topic by introducing a novel Covid-19 model described by non-integer-order difference equations. By conducting a stability analysis, the paper shows that the conceived system has two fixed points at most, i.e. a disease-free fixed point and an endemic fixed point. In particular, a theorem is proved, which assures the global stability of the disease-free fixed point, indicating that the pandemic will disappear when a simple condition on the system parameters is satisfied. Finally, simulation results are carried out with the aim to highlight the capability of the conceived approach. [ABSTRACT FROM AUTHOR]

In this paper, by using linking methods, we obtain the existence of the nontrivial standing wave solutions for the discrete nonlinear Schrödinger equations with resonance and unbounded potentials. In order to prove the existence of standing wave solutions, we give resonant condition to find a bounded critical sequence, and we show that such a sequence guarantees the existence of one nontrivial standing wave solution in l 2 when the nonlinearity is resonant and the potential is unbounded. To the best of the our knowledge, there is no existence results for the discrete nonlinear Schrödinger equations with resonance in the literature. [ABSTRACT FROM AUTHOR]

On R N equipped with a root system R and a multiplicity function k > 0 , we study the generalized (Dunkl) translations τ x g (- y) of not necessarily radial kernels g. Under certain regularity assumptions on g, we derive bounds for τ x g (- y) by means the Euclidean distance ‖ x - y ‖ and the distance d (x , y) = min σ ∈ G ‖ x - σ (y) ‖ , where G is the reflection group associated with R. Moreover, we prove that τ does not preserve positivity, that is, there is a non-negative Schwartz class function φ , such that τ x φ (- y) < 0 for some points x , y ∈ R N . [ABSTRACT FROM AUTHOR]

In the present paper, we consider the existence of solutions with a prescribed l 2 -norm for the following discrete Schrödinger equations, { − Δ 2 u k − 1 − f (u k) = λ u k k ∈ Z , ∑ k ∈ Z | u k | 2 = α 2 , where Δ 2 u k − 1 = u k + 1 + u k − 1 − 2 u k , f ∈ C (R) , α is a fixed constant, and λ ∈ R arises as a Lagrange multiplier. To get the solutions, we investigate the corresponding minimizing problem with the l 2 -norm constraint: E α = inf { 1 2 ∑ | Δ u k − 1 | 2 − ∑ F (u k) : ∑ | u k | 2 = α 2 }. An elaborative analysis on a minimizing sequence with respect to E α is obtained. We prove that there is a constant α 0 ≥ 0 such that there exists a global minimizer if α > α 0 , and there exists no global minimizer if α < α 0 . It seems that it is the first time to consider the solution with a prescribed l 2 -norm of the discrete Schrödinger equations. [ABSTRACT FROM AUTHOR]

In this article, by using critical point theory, we prove the existence of multiple TT-periodic solutions for difference equations with the mean curvature operator: −Δ(ϕc(Δu(t−1)))+q(t)u(t)=λf(t,u(t)),t∈Z,-\Delta ({\phi }_{c}\left(\Delta u\left(t-1)))+q\left(t)u\left(t)=\lambda f\left(t,u\left(t)),\hspace{1em}t\in {\mathbb{Z}}, where Z{\mathbb{Z}} is the set of integers. As a TT-periodic problem, it does not require the nonlinear term is unbounded or bounded, and thus, our results are supplements to some well-known periodic problems. Finally, we give one example to illustrate our main results.

This paper has been devoted to study the relation between an entire function of hyper order < 1 and its linear c-shift operator sharing a set of small functions. Our first theorem improves a very recent result of the present authors [Meromorphic function of restricted hyper-order sharing one or two sets with its linear C-shift operator. Anal Math. 2021;47(4):747–779.], while the corollary deducted from the same completely provides an answer to an open question posed by Liu [Meromorphic functions sharing a set with applications to difference equations. J Math Anal Appl. 2009;359:384–393]. Most importantly, our another theorem improves and extends a result due to Li [Entire functions sharing a finite set with their difference operators. Comput Methods Funct Theory. 2012;12(1):307–328] which in turn rectifies an erroneous result of Ma-Qi-Zhang [Further results about a special Fermat-type difference equation. J Funct Spaces. 2020. Article ID 3205357]. We have provided a handful number of examples in support of our certain claims. In the last section, as an application of our main result, we extend that of Liu-Ma-Zhai [The generalized Fermat type difference equations. Bull Korean Math Soc. 2018;55(6):1845–1858]. [ABSTRACT FROM AUTHOR]

Discrete tempered fractional calculus, as a fresh generalization of discrete fractional calculus, takes both the advantages of discrete fractional calculus and its tempered fractional counterpart, which has great potential value in the interpretation of special physical phenomena. However, its essential characteristics are far from clarifying. Meanwhile, the primary goal of this study is to enrich the vital characteristics of discrete tempered fractional calculus in the presence of required domains for diverse scenarios. First, the reciprocal properties of the tempered fractional summation and difference operators, as well as the discrete version of Taylor's formula, are established. The formulae of the discrete Laplace transform for discrete tempered fractional calculus are also determined with the aid of discrete convolution. Then, the well-posedness of tempered fractional difference equation is investigated, along with the exploration on the corresponding Volterra summation equation, and the existence, uniqueness and continuous dependence as well. In the aspect of application, the criterion of Ulam–Hyers stability of tempered fractional difference equation on finite time interval is obtained in light of a novel discrete Gronwall inequality. Besides, all examples are showed to verify the effectiveness of the main results. [ABSTRACT FROM AUTHOR]

Caputo fractional (with power-law kernels) and fractional (delta) difference maps belong to a more widely defined class of generalized fractional maps, which are discrete convolutions with some power-law-like functions. The conditions of the asymptotic stability of the fixed points for maps of the orders 0 < α < 1 that are derived in this paper are narrower than the conditions of stability for the discrete convolution equations in general and wider than the well-known conditions of stability for the fractional difference maps. The derived stability conditions for the fractional standard and logistic maps coincide with the results previously observed in numerical simulations. In nonlinear maps, one of the derived limits of the fixed-point stability coincides with the fixed-point—asymptotically period-two bifurcation point. [ABSTRACT FROM AUTHOR]

The main purpose of this study is to examine the properties of scattering solutions and scattering function of a difference equation subject to eigenparameter-dependent boundary and transmission conditions. In general, we are interested in a boundary value problem (BVP) generated by this difference equation, eigenparameter-dependent boundary and transmission conditions. Introducing the Jost solution and scattering solutions of BVP, we obtain the scattering function of this problem. Also, we find the resolvent operator, eigenvalues and continuous spectrum of the operator generated by this BVP. At the end, we present an example to demonstrate the effectiveness of our main results. [ABSTRACT FROM AUTHOR]

We consider a functional equation which is a generalization of Whitehead's functional equation on groups. We present its general solution on free groups. Solutions on other selected groups are also given. [ABSTRACT FROM AUTHOR]

This paper concerns the controllability of autonomous and nonautonomous nonlinear discrete systems, in which linear parts might admit certain degeneracy. By introducing Fredholm operators and coincidence degree theory, sufficient conditions for nonlinear discrete systems to be controllable are presented. In addition, applications are given to illustrate main results. [ABSTRACT FROM AUTHOR]

By applying the combination of discrete variational method and approximation, developed in a previous study [J. Kuang, W. Chen, and Z. Guo, Periodic solutions with prescribed minimal period for second-order even Hamiltonian systems, Commun. Pure Appl. Anal. 21 (2022), no. 1, 47–59], we overcome some difficulties in the absence of Ambrosetti-Rabinowitz condition and obtain new sufficient conditions for the existence of periodic solutions with prescribed minimal period for second-order Hamiltonian systems with asymptotically linear nonlinearities.

In this paper, we consider discrete Caputo delta fractional economic cobweb models. To be more specific, we first discuss some properties of discrete Mittag–Leffler functions, particularly the asymptotic behavior. Some sufficient conditions for the stability of solutions of specific fractional difference equations are presented. In the end, we conclude with a numerical example. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following discrete nonlinear Schrödinger equation - Δ u n + ϵ n u n - ω u n = f n (u n) , n ∈ Z , lim | n | → ∞ u n = 0 , with periodic potentials and ω belongs to a spectral gap of the operator L = - Δ + ϵ defined by L u n = - Δ u n + ϵ n u n . In order to investigate the existence of ground state solutions or infinitely many geometrically distinct solutions, it is commonly assumed that lim | s | → ∞ ∫ 0 s f n (t) d t s 2 = ∞ uniformly for all n ∈ Z in the existing literature. The purpose of this paper is to prove the existence of ground state solutions and infinitely many geometrically distinct solutions under the weaker super-quadratic condition lim | s | → ∞ ∫ 0 s f n (t) d t s 2 = ∞ for n ∈ G just for some set G ⊂ Z . Our result sharply extends and improves some existing ones in the literature. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to introduce and investigate a new d-orthogonal family generalizing the Laguerre polynomial set. We use operational rules associated with corresponding lowering, raising and transfer operators to derive many related properties, namely, recurrence relation, d-orthogonality, functional vector of d-orthogonality, inversion and connection relations, differential equation, among others. [ABSTRACT FROM AUTHOR]

We study the functional equation f 1 (x y z) = f 2 (x y) + f 3 (y z) + f 4 (x z) - f 5 (x) - f 6 (y) - f 7 (z) for all x , y , z ∈ G , f i : G → H , i = 1 , 2 , ... , 7 , where G is a group, H is an abelian group, and we present its general solution when G is a free group. Solutions on other selected groups are also given. [ABSTRACT FROM AUTHOR]

Taking into account of all the previous results, in this paper, we first resolve the "2 CM+1 IM" small functions sharing issue of a meromorphic function of restricted hyper order and its linear shift delay differential operator. We answer two open questions addressed by Qi-Yang (Analysis Math 46(4):843–865, 2020) in a convenient manner. Our second result, related to "2 IM" small function sharing problem of the same system, improves a result of (Analysis Math, 46(4):843–865, 2020). Moreover, concerning shift operator, we have pointed out a shortcoming in the sharing conditions in an example exhibited by Charak-Korhonen-Kumar (J Math Anal Appl 435(2):1241–1248, 2016). In an attempt to resolve the issue, we have solved the "2 IM+1 CM" small functions sharing problem which in turn extend a result of Li-Yi (Bull Korean Math Soc 53(4):1213–1235, 2016). [ABSTRACT FROM AUTHOR]

In the present paper, we study a class of discrete Schrödinger equations with concave–convex nonlinear terms via the variational method. Under certain conditions on the nonlinearities, two nontrivial solutions of the equations have been obtained by finding the minimizers on Nehari manifolds, and one of them is the ground state solution with negative energy. As we all know, there are few results on discrete systems involving concave–convex nonlinearities. [ABSTRACT FROM AUTHOR]

On R N equipped with a normalized root system R, a multiplicity function k (α) > 0 , and the associated measure d w (x) = ∏ α ∈ R | ⟨ x , α ⟩ | k (α) d x , let h t (x , y) denote the heat kernel of the semigroup generated by the Dunkl Laplace operator Δ k . Let d (x , y) = min g ∈ G ‖ x - g (y) ‖ , where G is the reflection group associated with R. We derive the following upper and lower bounds for h t (x , y) : for all c l > 1 / 4 and 0 < c u < 1 / 4 there are constants C l , C u > 0 such that C l w (B (x , t)) - 1 e - c l d (x , y) 2 t Λ (x , y , t) ≤ h t (x , y) ≤ C u w (B (x , t)) - 1 e - c u d (x , y) 2 t Λ (x , y , t) , where Λ (x , y , t) can be expressed by means of some rational functions of ‖ x - g (y) ‖ / t . An exact formula for Λ (x , y , t) is provided. [ABSTRACT FROM AUTHOR]

In the present paper, we consider a p-adic Ising model on a Cayley tree of order three. The existence of G k (2) -periodic non-translation-invariant p-adic generalized Gibbs measures of this model is investigated. Moreover, the boundedness of such kinds of measures is established, which yields the occurrence of a phase transition. [ABSTRACT FROM AUTHOR]

We wish to investigate the continuous dependence on data for a nonhomogeneous second-order difference inclusion corresponding to a general second-order evolution equation of accretive type in a Banach space. Our conclusions are new even for Hilbert spaces and significantly extend some previously known results to the nonhomogeneous case by assuming much weaker conditions on the zeroes of operators. Applicability of our results is demonstrated through a partial-difference equation example. [ABSTRACT FROM AUTHOR]

The renormalization group method has been developed to investigate p-adic q-state Potts models on the Cayley tree of order k. This method is closely related to the examination of dynamical behavior of the p-adic Potts–Bethe mapping which depends on the parameters q, k. In Mukhamedov and Khakimov [Chaotic behavior of the p-adic Potts–Behte mapping. Discrete Contin. Dyn. Syst.38 (2018), 231–245], we have considered the case when q is not divisible by p and, under some conditions, it was established that the mapping is conjugate to the full shift on $\kappa _p$ symbols (here $\kappa _p$ is the greatest common factor of k and $p-1$). The present paper is a continuation of the forementioned paper, but here we investigate the case when q is divisible by p and k is arbitrary. We are able to fully describe the dynamical behavior of the p-adic Potts–Bethe mapping by means of a Markov partition. Moreover, the existence of a Julia set is established, over which the mapping exhibits a chaotic behavior. We point out that a similar result is not known in the case of real numbers (with rigorous proofs). [ABSTRACT FROM AUTHOR]

We discuss the spectral decomposition of the hypergeometric differential operators on the line R e z = 1 / 2 , such operators arise in the problem of decomposition of tensor products of unitary representations of the universal covering of the group S L (2 , R). Our main purpose is a search of natural bases in generalized eigenspaces and variants of the inversion formula. [ABSTRACT FROM AUTHOR]