Recurrence Relations Research Articles

On Some Formulas for the Lauricella Function

Lauricella, G. in 1893 defined four multidimensional hypergeometric functions FA, FB, FC and FD. These functions depended on three variables but were later generalized to many variables. Lauricella’s functions are infinite sums of products of variables and corresponding parameters, each of them has its own parameters. In the present work for Lauricella’s function FA(n), the limit formulas are established, some expansion formulas are obtained that are used to write recurrence relations, and new integral representations and a number of differentiation formulas are obtained that are used to obtain the finite and infinite sums. In the presentation and proof of the obtained formulas, already known expansions and integral representations of the considered FA(n) function, definitions of gamma and beta functions, and the Gaussian hypergeometric function of one variable are used.

On the eigenvalue-separation properties of real tridiagonal matrices

In this paper, we give a simple sufficient condition for the eigenvalue-separation properties of real tridiagonal matrices T. This result is much more than the statement that the pertinent eigenvalues are distinct. Its derivation is based on recurrence formulae satisfied by the polynomials made up by the minors of the characteristic polynomial det(xE-T) that are proven to form a Sturm sequence. This is a new result, and it proves the simple spectrum property of a symmetric tridiagonal matrix studied in Grünbaum's paper. Two numerical examples underpin the theoretical findings. The style of the paper is expository in order to address a large readership.

Analytical Solution of Mixed Problems for the One-Dimensional Ionization Equations in the Case of Constant Velocities of Atoms and Ions

We consider the main initial–boundary value (mixed) problems for the nonlinear system of one-dimensional gas ionization equations in the case of constant velocities of gas atoms and ions resulting from ionization. The atom and ion concentrations are the unknowns in this system. We find a general formula for a sufficiently smooth solution of the system. It is shown that mixed problems for the system of one-dimensional ionization equations admit integration in closed-form analytical expressions. In the case of a mixed problem for a finite interval, the analytical solution is constructed using recurrence formulas each of which is defined in a triangle belonging to some triangulation, specified in the paper, of the domain where the unknown functions are defined.

Existence of sequences satisfying bilinear type recurrence relations

We study sequences $\left\{A_n\right\}_{n=-\infty}^{+\infty}$ of elements of an arbitrary field $\mathbb{F}$ that satisfy decompositions of the form $A_{m+n} A_{m-n}=a_1(m) b_1(n)+a_2(m) b_2(n)$, $A_{m+n+1} A_{m-n}=\tilde a_1(m) \tilde b_1(n)+\tilde a_2(m) \tilde b_2(n)$, where $a_1,a_2,b_1,b_2\colon \mathbb{Z}\to\mathbb{F}$. We prove some results concerning the existence and uniqueness of such sequences. The results are used to construct analogs of the Diffie-Hellman and ElGamal cryptographic algorithms. The discrete logarithm problem is considered in the group $(S,+)$, where the set $S$ consists of quadruples $S(n)=(A_{n-1},A_n, A_{n+1}, A_{n+2})$, $n\in\mathbb{Z}$, and $S(n)+S(m)=S(n+m)$.

Construction of Polynomial Eigenfunctions of a Second-Order Linear Differential Equation

A system of third-order recurrence relations for the coefficients of polynomial eigenfunctions (PEFs) of a differential equation is solved. A recurrence relation for three consecutive PEFs and a formula for differentiating PEFs are obtained. We consider differential equations one of which generalizes the Hermite and Laguerre differential equations and the other is a generalization of the Jacobi differential equation. For these equations, we construct functions bringing them to self-adjoint form and find conditions under which these functions become weight functions. Examples are given where the PEFs for nonweight functions do not have real zeros.

Series of formulas for bhattacharyya parameters in the theory of polar codes

Bhattacharyya parameters are used in the theory of polar codes to determine positions of frozen and information bits. These parameters characterize rate of polarization of channels WN(i), 1 ≤ i ≤ N, which are constructed in a special way from the original channel W, where N = 2n is the channel length, n = 1, 2, .... In the case where W is a binary symmetric memoryless channel, we present two series of formulas for the parameters Z(WN(i)): for i = N - 2k + 1, 0 ≤ k ≤ n, and for i = N/2 - 2k + 1, 1 ≤ k ≤ n - 2. The formulas require of the order of $\binom{2^{n-k}+2^k-1}{2^k} 2^{2^k}$ addition operations for the first series and of the order of $\binom{2^{n-k-1}+2^k-1}{2^k} 2^{2^k}$ for the second. In the cases i = 1, N/4 + 1, N/2 + 1, N, the obtained expressions for the parameters have been simplified by computing the sums in them. We show potential generalizations for the values of i in the interval (N/4, N). We also study combinatorial properties of the polarizing matrix GN of a polar code with Arıkan’s kernel. In particular, we establish simple recurrence relations between rows of the matrices GN and GN/2.

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.