Recurrence Relations Research Articles

Operational rules for a new family of d-orthogonal polynomials of Laguerre type

The aim of this research is to present a new generalization of d-orthogonal 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 -order differential equation, and the canonical d-dimensional functional vector.

A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation

We are concerned with the Umemura polynomials associated with rational solutions of the third Painlevé equation. We extend Taneda's method, which was developed for the Yablonskii-Vorob'ev polynomials associated with the second Painlevé equation, to give an algebraic proof that the rational functions generated by the nonlinear recurrence relation which determines the Umemura polynomials are indeed polynomials. Our proof is constructive and gives information about the roots of the Umemura polynomials.

Quantum covariant derivative: a tool for deriving adiabatic perturbation theory to all orders

The covariant derivative suitable for differentiating and parallel transporting tangent vectors and other geometric objects induced by a parameter-dependent adiabatic quantum eigenstate is introduced. It is proved to be covariant under gauge and coordinate transformations and compatible with the quantum geometric tensor. For a quantum system driven by a Hamiltonian depending on slowly-varying parameters , , the quantum covariant derivative is used to derive a recurrence relation that determines an asymptotic series for the wave function to all orders in . This adiabatic perturbation theory provides an efficient tool for calculating nonlinear response properties.

A bivariate Q-polynomial structure for the non-binary Johnson scheme

The notion of multivariate P- and Q-polynomial association scheme has been introduced recently, generalizing the well-known univariate case. Numerous examples of such association schemes have already been exhibited. In particular, it has been demonstrated that the non-binary Johnson scheme is a bivariate P-polynomial association scheme. We show here that it is also a bivariate Q-polynomial association scheme for some parameters. This provides, with the P-polynomial structure, the bispectral property (i.e. the recurrence and difference relations) of a family of bivariate orthogonal polynomials made out of univariate Krawtchouk and dual Hahn polynomials. The algebra based on the bispectral operators is also studied together with the subconstituent algebra of this association scheme.

Moment Properties of Record Values from Rayleigh Lomax Distribution and Characterization

In this paper we have obtained explicit expressions for single and product moments of upper record values from Rayleigh Lomax distribution. Also, recursive relationships between moments of upper records are developed. The characterization theorems based on recurrence relations and conditional expectation are also presented.

Counting Polygons and Fishes in Uniform Tessellations of the Hyperbolic Plane

Summary In this article we consider uniform tessellations of the hyperbolic plane. Two counts are performed. The first, considers the number of polygons in layers of the hyptagrid and is shown to be related to Fibonacci numbers. The second, considers the number of fishes in layers of a superimposed octagrid on Escher’s circle limit III. To excute these counts we solve two linear recurrence relations, homogeneous and non-homogeneous. The initial conditions are set up by performing tessellations using a software in hyperbolic geometry.

Non-variant phenomena in heterogeneous systems. New type of solubility diagrams points

The article gives a general classification of non-invariant points in phase equilibrium diagrams of all possible types. The complete topological isomorphism of the diagrams of fusibility, solubility, and liquid-vapor equilibria in various sets of variables is demonstrated. The stability of mono-variant equilibria near the non-variant points is investigated. Recurrent formulas for calculating the number of topological elements of phase diagrams are given. A previously undescribed type of non-invariant points and phase processes in the solubility diagrams is described and characterized. The last ones have no topological analogs in other types of diagrams. Thus, we have carried out, as far as is available to the authors, a complete classification of invariant points and invariant processes in phase equilibrium diagrams of an arbitrary type and with an arbitrary number of components.

Recurrence Relations for Moment Generating Function Based on Progressive First Failure Censoring from Generalized Pareto Distribution and Characterization

In this article, we establish recurrence relations (RR) for single and product moment generating function (MGF) based on based on progressive first failure censoring (PFFC) for generalized Pareto distribution (GPD). Characterization for GPD using RR of single and product MGF of PFFC are also obtained. Further, the results are specialized to the progressively type-II right censored (PTIIRCOS).

Seismotectonic mapping and energy release analysis for Ongole city of Andhra Pradesh, India

An updated homogenised earthquake catalogue comprising 753 events from 1800 AD to 2021 AD for Ongole city andhra Pradesh, was compiled for a circular buffer zone of radius 500km with collector office (Lat 15.49880 N and Long 80.04970 E) as the centre. Dependent events were declustered using the Uhrhammer algorithm and the degree of completeness was computed using CUVI and Stepp’s methods. The analysis shows that around 11% of events were foreshocks and aftershocks. The degree of completeness for the magnitude range of 3.0 ≤ Mw < 3.5, 3.5 ≤ Mw < 4.0, 4.0 ≤ Mw < 4.5, 4.5 ≤ Mw < 5.0, 5.0 ≤ Mw < 5.5 and Mw ≥ 5.5 was found to be 23, 56, 56, 63, 105 and 178 years based on CUVI method and 30, 30, 60, 60, 110 and 160 years as per Stepp’s method respectively. Values of the seismicity parameters a and b based on Gutenberg-Ritcher’s recurrence relationship for the study area were 4.02 and 0.83 respectively. Based on the current study, the maximum earthquake magnitude (Mmax) computed for the Ongole city is 7±0.27 on the moment magnitude scale and cumulative seismic energy release is 2.73x1016J. Finally, a seismotectonic map has been developed in ArcGIS 10.5.1 software comprising past seismicity, regional geology and faults details.

Computing indefinite integrals by difference equations

In teaching mathematics to first-year undergraduates, and thus in the appropriate calculus textbooks, the task of calculating an integral that satisfies a specific first-order or second-order recurrence relation often appears. These relations are obtained mainly by applying the method of integration by parts. Calculating such integrals is usually tedious, especially for an integer n > 2, time-consuming, and presents the possibility of making a large number of errors when computing involves multiple iterative steps. In [1], it is shown that in two cases (Theorems 2.1. and 2.3), the process of calculating integrals satisfying first-order recurrence relations can be performed quickly using easily memorised closed-form formulas for corresponding primitive functions. The question can rightly be asked whether there is a faster way to calculate other integrals of this type. In this paper, our goal is to give an affirmative answer to such a question, though without convering all situations. Since each recurrence relation is equivalent to a difference equation of the same order, the calculation of integrals mentioned above can be reduced to solving the corresponding difference equations. Since every first-order or second-order linear difference equation is solvable, it follows that for every integral which can be reduced to a first- order or second-order recurrence formula, it is possible to find corresponding primitive functions directly. Sometimes such a procedure is much faster than iterative solving of the integral. Closed-form formulas for the integrals discussed in the following sections are not unknown (see [2]). However, here our goal is to present the idea of computing indefinite integrals using difference equations. We will discuss it in more detail in Section 2. In Section 3, we discuss the application of the results obtained to calculate several improper integrals and the application of some of them in different sciences. An exciting example of such an application is the integral , which in the case n = 1 is used in the kinetic theory of gases, particularly in the Maxwell-Boltzmann distribution of gas molecules by energies (see Remark 4). Also, we compare the formulas obtained by the method of difference equations with the formulas obtained using Wolfram Alpha software (see Remark 5).

Linear Localization of Zero Modes in Weakly Coupled Non‐Hermitian Reservoirs

AbstractTopological and symmetry‐protected non‐Hermitian zero modes have attracted considerable interest in the past few years. Here, it is revealed that they can exhibit an unusual behavior when transitioning between the extended and localized regimes: When weakly coupled to a non‐Hermitian reservoir, such a zero mode displays a linearly decreasing amplitude as a function of space, which is not caused by an EP of a Hamiltonian, either of the entire system or the reservoir itself. Instead, this phenomenon is due to the non‐Bloch solution of a linear homogeneous recurrence relation, together with the underlying non‐Hermitian particle‐hole symmetry and the zeroness of its energy.

Homeier-like methods for regularization of nonlinear ill-posed equations in Hilbert space

Homeier-like methods have been extensively studied in the literature for solving non-linear operator equation of the form F(x)=y in the Banach or Hilbert space setting. Motivated by the work of George et al. (2023), we extend some class of Homeier-like methods to approximate ill-posed operator equation in Hilbert space. Two Homeier-like iterative schemes are proposed for approximating the nonlinear ill-posed operator equation F(x)=y in Hilbert space, where F is monotone. Lavrentiev regularization method and recurrence relation are employed for convergence analysis. The regularization parameter is chosen by apriori parameter choice strategy with new source condition as studied by George et al. (2023). We also supply a numerical example for comparison study of the proposed method with some of the Lavrentiev regularization methods in the literature.

On information theoretical modelling of the communications between neurons and the corresponding information rate

AbstractIn a nervous system propagation of spikes among neurons can be interpreted as an information transmission process, where, a neural response carries information about the stimulus. Quantification of this information might be of theoretical and practical importance. In this paper, (i) The interaction between two neurons with an electrically coupled transport of ions is modelled mathematically, where the communication through the junction gap between very close neurons occurs with a specific time delay; (ii) then, the connection structure of neurons is proposed based on a particular Rulkov network model. The corresponding process is modelled as a non‐linear binary channel with memory; and (iii) by assuming a recurrence relation for number of available membrane potential changing due to the arrival of the spike to the axonal terminal, the mutual information between channel input and output is obtained and the input range maximizing mutual information, and hence, the information rate of channel is derived.

Open networks of infinite server queues with non-homogeneous multivariate batch Poisson arrivals

In this paper, we consider the occupancy distribution for an open network of infinite server queues with multivariate batch arrivals following a non-homogeneous Poisson process, and general service time distributions. We derive a probability generating function for the transient occupancy distribution of the network and prove that it is necessary and sufficient for ergodicity that the expected occupancy time for each batch be finite. Further, we recover recurrence relations for the transient probability mass function formulated in terms of a distribution obtained by compounding the batch size with a multinomial distribution.

Matrix Jacobi biorthogonal polynomials via Riemann–Hilbert problem

We consider matrix orthogonal polynomials related to Jacobi type matrices of weights that can be defined in terms of a given matrix Pearson equation. Stating a Riemann–Hilbert problem we can derive first and second order differential-difference relations that these matrix orthogonal polynomials and the second kind functions associated to them verify. For the corresponding matrix recurrence coefficients, non-Abelian extensions of a family of discrete Painlevé d-P I V _{IV} equations are obtained for the three term recurrence relation coefficients.

Moments of inverse Weibull-geometric distribution based on progressive type-II right censored order statistics

In this article, we derive explicit expressions and some recurrence relations for the single and product moments of progressively Type-II right censored order statistics from the inverse Weibull-geometric distribution, and then we use these results to calculate the means and variances of progressively Type-II right censored order statistics. Also, we discuss the best linear unbiased estimators for the location and scale parameters as well as the best linear unbiased predictors of the censored failure times. Finally, a real data application is presented.

The Number of Fillings a 2×2×n prism with 1×1×2 prisms

This paper is inspired by very interesting YouTube video of Burkard Polster, professor of mathematics at Monash University in Melbourne, Australia, which, among other things, concerned the number of ways to fill a part of the plane with dominoes, i.e. 1×2 rectangles. First we deal with the numbers of fillings the 2×2×n prism with elementary 1×1×2 prisms for n=1,2,3,4,5. Special symbolism and figures showing the filling of the prism are used as well as the concept of matching from graph theory and the corresponding graph diagrams. Then we generalize these specific considerations and derive a general recurrence formula for any n≥3, which expresses the number of fillings of the 2×2×n prism with 1×1×2 elementary prisms, which in a way can be considered as spatial domino cubes, if we do not consider their marking with pairs of numbers from 0 to 6.

Approximation of the power distribution from multiple sound sources in the atmosphere using sums of gamma random variables

A sum of gamma random variables is a mathematically tractable approach to approximate multiple sources when the power (amplitude squared) of a single source is nearly gamma distributed because the sum can be expressed analytically for a wide variety of cases. Previous work indicates that the gamma distribution is a good two-parameter empirical approximation of received power from a single, elevated sound source in a turbulent atmosphere. The gamma distribution can be conceptualized as a sum of k independent and identically distributed exponential random variables, each with mean m. Here, k and m are called the shape and scale parameters, respectively. Thus, the summation of multiple gamma random variables with the same scale parameter still yields a gamma distribution. When multiple sources can be approximated well using a gamma distribution, then those sources could be grouped together for simplicity and conceived as a single source in a turbulent atmosphere. In addition, multiple authors have derived analytic expressions for the sum of N independent gamma random variables with distinct parameters in terms of the confluent hypergeometric functions. If the gamma random variables are correlated, then the result is still analytic using recurrence relations.

Matching anti-forcing polynomials of catacondensed hexagonal systems

Lei, Yeh and Zhang put forward the anti-forcing number af(G,M) for a perfect matching M in a graph G, which is the minimum number of edges of G not in M whose deletion results in a subgraph with a unique perfect matching M. The anti-forcing numbers of all perfect matchings form the anti-forcing spectrum of G. The anti-forcing polynomial Af(G,x) of G is a counting polynomial for classifying perfect matchings possessing the same anti-forcing number in G. In this paper, we deduce recurrence formula of the anti-forcing polynomial and continuity of the anti-forcing spectrum for catacondensed hexagonal systems.

Run-and-tumble oscillator: Moment analysis of stationary distributions

When it comes to active particles, even an ideal gas model in a harmonic potential poses a mathematical challenge. An exception is a run-and-tumble particles (RTP) model in one dimension for which a stationary distribution is known exactly. The case of two dimensions is more complex, but the solution is possible. Incidentally, in both dimensions the stationary distributions correspond to a beta function. In three dimensions, a stationary distribution is not known but simulations indicate that it does not have a beta function form. The current work focuses on the three-dimensional RTP model in a harmonic trap. The main result of this study is the derivation of the recurrence relation for generating moments of a stationary distribution. These moments are then used to recover a stationary distribution using the Fourier–Lagrange expansion.