Recurrence Relations Research Articles

Recurrence Relations and General Solution of the Exceptional Hermite Equation

Recurrence Relations and General Solution of the Exceptional Hermite Equation

Sequences of twice-iterated Δw-Gould–Hopper Appell polynomials

ABSTRACTIn this paper, we introduce general sequence of twice-iterated Δw-(degenerate) Gould–Hopper Appell polynomials (TI-DGHAP) via discrete Δw-Gould–Hopper Appell convolution. We obtain some of their characteristic properties such as explicit representation, determinantal representation, recurrence relation, lowering operator (LO), raising operator (RO), difference equation (DE), integro-partial lowering operator (IPLO), integro-partial raising operator (IPRO) and integro-partial difference equation (IPDE). As special cases of these general polynomials, we present TI degenerate Gould–Hopper Bernoulli polynomials, TI degenerate Gould–Hopper Poisson–Charlier polynomials, TI degenerate Gould–Hopper Boole polynomials and TI degenerate Gould–Hopper Poisson–Charlier–Boole polynomials. We also state their corresponding characteristic properties.

On the Jameson-Schneider and Apostol Theorems

We show that the Jameson-Schneider’s theorem for arithmetic functions implies the Apostol theorem, which allows to obtain recurrence relations for several functions of interest in number theory.

Enumeration of spanning trees in prisms of some graphs

In graph theory, a prism over a graph G is the cartesian product of the graph G with P₂. The purpose of this work is to investigate the complexity of the prisms of some path and cycle-related graphs. In particular, we obtain simpler and more explicit formulas for the complexity of a special class of prisms of path-related graphs: fan graph, ladder graph, the composition Pn[P₂] graph, and book graph. Moreover, we obtain straightforward formulas for the complexity of a special class of prisms of cycle-related graphs: wheel graph, gear graph, prism graph, n−crossed prism graph, mirror graph M(Cn) of even cycle Cn, twisted prism, total graph T(Cn) of the cycle Cn, the friendship graph, the flower graph, and planner sunflower graph. These closed formulas are deduced using some basic properties of block matrix, recurrence relation, eigenvalues of circulant matrices, and orthogonal polynomials.

The method of constant terms and k-colored generalized Frobenius partitions

In his 1984 AMS memoir, Andrews introduced the family of k-colored generalized Frobenius partition functions. For any positive integer k, let cϕk(n) denote the number of k-colored generalized Frobenius partitions of n. Among many other things, Andrews proved that for any n≥0, cϕ2(5n+3)≡0(mod5). Since then, many scholars subsequently considered congruence properties of various k-colored generalized Frobenius partition functions, typically with a small number of colors.In 2019, Chan, Wang and Yang systematically studied arithmetic properties of CΦk(q) with 2≤k≤17 by employing the theory of modular forms, where CΦk(q) denotes the generating function of cϕk(n). We notice that many coefficients in the expressions of CΦk(q) are not integers. In this paper, we first observe that CΦk(q) is related to the constant term of a family of bivariable functions, then establish a general symmetric and recurrence relation on the coefficients of these bivariable functions. Based on this relation, we next derive many bivariable identities. By extracting and computing the constant terms of these bivariable identities, we establish the expressions of CΦk(q) with integral coefficients. As an immediate consequence, we prove some infinite families of congruences satisfied by cϕk(n), where k is allowed to grow arbitrary large.

Abacus Algorithms: A Pure Mathematical Approach to Ancient Calculation Tools

The abacus is one of the oldest calculating tools still in use today. Despite its simplicity, the bead-based interface allows users to conduct complex mathematical operations through a system of sliding beads along wires or rods. While the physical abacus itself provides an intuitive and visual approach to calculation, the underlying operations rely on fundamental mathematical principles. This paper provides a comprehensive mathematical framework that formally describes the algorithms behind abacus calculations. Beginning with basic abacus configuration, we define key components like rods, beads, and bead values required to model abacus states. We then characterize the core abacus algorithms for addition, subtraction, multiplication, and division through set notation, recurrence relations, and state transition diagrams. Our formalized abacus algorithms leverage concepts from number theory, modular arithmetic, combinatorics, and algebra. In addition to offering new mathematical insights into ancient technologies, our work helps bridge connections between the tangible abacus interface and the abstract algorithms powering it. Through examples and proofs, we show how bead manipulations precisely correspond to mathematical transformations. This level of formalization not only helps explain the effectiveness of the abacus, but also illustrates how even rudimentary calculation tools utilize profound mathematical ideas. Our mathematical abacus framework lays the foundation for further analysis as well as modifications and extensions of the classic abacus approach.

On Horadam finite operator hybrid numbers

In this paper, we introduce and study a new hybrid number sequence with Horadam numbers and a finite operator, called Horadam finite operator hybrid numbers. We derive recurrence relation, Binet-like formula, ordinary generating function, exponential generating function, Poisson generating function, and summation formula for Horadam finite operator hybrid numbers. Moreover, we give matrix representation and Cassini's identities for these numbers.

Magnetization properties, super-stable points, and cycles of antiferromagnetic spin-1 diamond chains with nodal-nodal interactions

This study is devoted to the discovery of super-stable points and cycles in antiferromagnetic Ising and Ising-Heisenberg models with spin 1 on diamond chains with nodal-nodal interactions. These phenomena are important for understanding the complex behavior of magnetic systems. We specifically investigate their connection with magnetization plateaus, which serve as critical indicators of the model’s characteristics. Employing the recurrence relations technique, we derive multidimensional rational mappings that give insights about the statistical properties of the models. Carefully examining the stability properties of these mappings, in particular, by analyzing the maximum Lyapunov exponent, we have revealed the complex relationship between the magnetization plateau and dynamic properties. Throughout our extensive research, we have comprehensively studied the existence and behavior of super-stable points and cycles for various parameter configurations in spin-1 models on the diamond chains. By highlighting the basic properties of dynamics and stability, our research advances a fundamental understanding of complex magnetic systems and their fascinating properties.

Conformal Mapping of a Z-Shaped Domain

For the problem of conformal mapping of a half-plane onto a Z-shaped domain with arbitrary geometry, an efficient method is developed for finding parameters of the Schwarz–Christoffel integral, i.e., the preimages of the vertices (prevertices) and the pre-integral multiplier. Special attention is given to the situation of crowding prevertices, in which case conventional integration methods face significant difficulties. For this purpose, the concept of a cluster is introduced, its center is determined, and all integrand binomials with prevertices from this cluster are expanded into a fast-convergent series by applying a unified scheme. Next, the arising integrals are reduced to single or double series in terms of Gauss hypergeometric functions F(a, b, c, q). The fast convergence of the resulting expansions is ensured by applying formulas for analytic continuation of F(a, b, c, q) to a neighborhood of the point q = 1 and using numerically stable recurrence relations. The constructed expansions are also fairly efficient for choosing initial approximations for prevertices in Newton’s iteration method. By using the leading terms of these expansions, the approximations for the prevertices are expressed in explicit form in terms of elementary functions, and the subsequent iterations ensure the fast convergence of the algorithm. After finding the parameters in the integral, the desired mapping is constructed as a combination of power series expansions at prevertices, regular expansions at the preimage of the center of symmetry, a Laurent series in a semi-annulus, and special series near the preimages of the vertical segments. Numerical results demonstrate the high efficiency of the developed method, especially in the case of strong crowding of prevertices.

Discrete-Time Quantum Walk on Multilayer Networks.

A Multilayer network is a potent platform that paves the way for the study of the interactions among entities in various networks with multiple types of relationships. This study explores the dynamics of discrete-time quantum walks on a multilayer network. We derive a recurrence formula for the coefficients of the wave function of a quantum walker on an undirected graph with a finite number of nodes. By extending this formula to include extra layers, we develop a simulation model to describe the time evolution of the quantum walker on a multilayer network. The time-averaged probability and the return probability of the quantum walker are studied with Fourier, and Grover walks on multilayer networks. Furthermore, we analyze the impact of decoherence on quantum transport, shedding light on how environmental interactions may impact the behavior of quantum walkers on multilayer network structures.

ON HOMOGENEOUS LINEAR RECURRENCE RELATIONS

We use the Z-transform to motivate the Baldoni et al algorithm to solve homogeneous linear recurrence relations, with applications to Fibonacci numbers and Chebyshev polynomials.

Evaluation of the Gaunt Coefficients by Using Recurrence Relations for Spherical Harmonics

The Gaunt coefficient is one of the important coefficients to be known for calculating molecular integrals in quantum theory of coupling of three angular momenta. Generally, these coefficients are calculated analytically by using the properties of the associated Legendre polynomials. In this study, Gaunt coefficients were calculated algebraically by using the recurrence relations and orthogonality conditions of spherical harmonics and different mathematical expressions were obtained from known analytical expressions for Gaunt coefficients in terms of factorial functions or binomial coefficients. By using the program written in the Mathematica programming language, both the analytical expressions and the algebraic expressions were calculated, and the numerical results obtained were compared. Numerical results are in quite agreement with the literature and each other.

Efficient Analysis of Large-Size Bio-Signals Based on Orthogonal Generalized Laguerre Moments of Fractional Orders and Schwarz–Rutishauser Algorithm

Orthogonal generalized Laguerre moments of fractional orders (FrGLMs) are signal and image descriptors. The utilization of the FrGLMs in the analysis of big-size signals encounters three challenges. First, calculating the high-order moments is a time-consuming process. Second, accumulating numerical errors leads to numerical instability and degrades the reconstructed signals’ quality. Third, the QR decomposition technique is needed to preserve the orthogonality of the higher-order moments. In this paper, the authors derived a new recurrence formula for calculating the FrGLMs, significantly reducing the computational CPU times. We used the Schwarz–Rutishauser algorithm as an alternative to the QR decomposition technique. The proposed method for computing FrGLMs for big-size signals is accurate, simple, and fast. The proposed algorithm has been tested using the MIT-BIH arrhythmia benchmark dataset. The results show the proposed method’s superiority over existing methods in terms of processing time and reconstruction capability. Concerning the reconstructed capability, it has achieved superiority with average values of 25.3233 and 15.6507 with the two metrics PSNR and MSE, respectively. Concerning the elapsed reconstruction time, it also achieved high superiority with an efficiency gain of 0.8. The proposed method is suitable for utilization in the Internet of Healthcare Things.

Some Identities of Fully Degenerate Dowling Polynomials and Numbers

Recently, Kim-Kim introduced the degenerate Whitney numbers of the first and second kind involving the degenerate Dowling polynomials and numbers. In this paper, we introduce the fully degenerate Dowling polynomials and the fully degenerate Bell polynomials and derive some identities involving these polynomials. We also obtain the generating functions, expressions, and recurrence relations for the fully degenerate Dowling polynomials and the fully degenerate Bell polynomials and other special polynomials and numbers.

Alternative to mean and least squares methods used in processing the results of scientific and technical experiments

Increasing the complexity and size of systems of various nature requires constant improvement of modeling and verification of the obtained results by experiment. It is possible to clearly conduct each experiment, objectively evaluate the summaries of the researched process, and spread the material obtained in one study to a series of other studies only if they are correctly set up and processed. On the basis of experimental data, algebraic expressions are selected, which are called empirical formulas, which are used if the analytical expression of some function is complex or does not exist at this stage of the description of the object, system or phenomenon. When selecting empirical formulas, polynomials of the form: у = А0 + А1х+ А2х2+ А3х3+…+ Аnхn are widely used, which can be used to approximate any measurement results if they are expressed as continuous functions. It is especially valuable that even if the exact expression of the solution (polynomial) is unknown, it is possible to determine the value of the coefficients An using the methods of mean and least squares. But in the method of least squares, there is a shift in estimates when the noise in the data is increased, as it is affected by the noise of the previous stages of information processing. Therefore, for real-time information processing procedures, a pseudo-reverse operation is proposed, which is performed using recurrent formulas. This procedure is a procedure of successive updating (with a shift) along the columns of the matrix of given sizes and pseudo-reversal at each step of information change. This approach is straightforward and takes advantage of the bounding method. With pseudo-inversion, it is possible to control the correctness of calculations at each step, using Penrose conditions. The need for pseudo-inversion may arise during optimization, forecasting of certain parameters and characteristics of systems of various purposes, in various problems of linear algebra, statistics, presentation of the structure of the obtained solutions, to understand the content of the incorrectness of the resulting solution, in the sense of Adomar-Tikhonov, and to see the ways of regularization of such solutions.

On Hybrid Hyper k-Pell, k-Pell–Lucas, and Modified k-Pell Numbers

Many different number systems have been the topic of research. One of the recently studied number systems is that of hybrid numbers, which are generalizations of other number systems. In this work, we introduce and study the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers. In order to study these new sequences, we established new properties, generating functions, and the Binet formula of the hyper k-Pell, hyper k-Pell–Lucas, and hyper Modified k-Pell sequences. Thus, we present some algebraic properties, recurrence relations, generating functions, the Binet formulas, and some identities for the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers.

Para-Bannai-Ito Polynomials

New bispectral polynomials orthogonal on a Bannai-Ito bi-lattice (uniform quadri-lattice) are obtained from an unconventional truncation of the untruncated Bannai-Ito and complementary Bannai-Ito polynomials. A complete characterization of the resulting para-Bannai-Ito polynomials is provided, including a three term recurrence relation, a Dunkl-difference equation, an explicit expression in terms of hypergeometric series and an orthogonality relation. They are also derived as a $q\to -1$ limit of the $q$-para-Racah polynomials. A connection to the dual $-1$ Hahn polynomials is also established.

Fractional Zernike functions

We consider and provide an accurate study for the fractional Zernike functions on the punctured unit disc, generalizing the classical Zernike polynomials and their associated β-restricted Zernike functions. Mainly, we give the spectral realization of the latter ones and show that they are orthogonal L2-eigenfunctions for certain perturbed magnetic (hyperbolic) Laplacian. The algebraic and analytic properties for the fractional Zernike functions to be established include the connection to special functions, their zeros, their orthogonality property, as well as the differential equations, recurrence and operational formulas they satisfy. Integral representations are also obtained. Their regularity as poly-meromorphic functions is discussed and their generating functions including a bilinear one of “Hardy–Hille type” are derived. Moreover, we prove that a truncated subclass defines a complete orthogonal system in the underlying Hilbert space giving rise to a specific Hilbertian orthogonal decomposition in terms of a class of generalized Bergman spaces.

A New Generalization of the Alternating Harmonic Series

Kilmer and Zheng (2021) recently introduced a generalized version of the alternating harmonic series. In this paper, we introduce a new generalization of the alternating harmonic series. A special case of our generalization converges to the Kilmer-Zheng series. Then we investigate several interesting and useful properties of this generalized, such as a summation formula related to the Hurwitz -Lerch Zeta function, a duplication formula, an integral representation, derivatives, and the recurrence relationship. Some important special cases of the main results are also discussed.

Off-Diagonally Symmetric Domino Tilings of the Aztec Diamond

We introduce a new symmetry class of domino tilings of the Aztec diamond, called the off-diagonal symmetry class, which is motivated by the off-diagonally symmetric alternating sign matrices introduced by Kuperberg in 2002. We use the method of non-intersecting lattice paths and a modification of Stembridge's Pfaffian formula for families of non-intersecting lattice paths to enumerate our new symmetry class. The number of off-diagonally symmetric domino tilings of the Aztec diamond can be expressed as a Pfaffian of a matrix whose entries satisfy a nice and simple recurrence relation.