Recurrence Relations Research Articles

Existence of Sequences Satisfying Bilinear Type Recurrence Relations

Existence of Sequences Satisfying Bilinear Type Recurrence Relations

Hyper-Fibonacci and Hyper-Lucas Polynomials

In this paper, hyper-Fibonacci and hyper-Lucas polynomials are defined and some of their algebraic and combinatorial properties such as the recurrence relations, summation formulas, and generating functions are presented. In addition, some relationships between the hyper-Fibonacci and hyper-Lucas polynomials are given.

Recurrence and Eigenfunction Methods for Non-Trivial Models of Discrete Binary Choice.

Understanding how systems relax to equilibrium is a core theme of statistical physics, especially in economics, where systems are known to be subject to extrinsic noise not included in simple agent-based models. In models of binary choice-ones not much more complicated than Kirman's model of ant recruitment-such relaxation dynamics become difficult to determine analytically and require solving a three-term recurrence relation in the eigendecomposition of the stochastic process. In this paper, we derive a concise closed-form solution to this linear three-term recurrence relation. Its solution has traditionally relied on cumbersome continued fractions, and we instead employ a linear algebraic approach that leverages the properties of lower-triangular and tridiagonal matrices to express the terms in the recurrence relation using a finite set of orthogonal polynomials. We pay special attention to the power series coefficients of Heun functions, which are also important in fields such as quantum mechanics and general relativity, as well as the binary choice models studied here. We then apply the solution to find equations describing the relaxation to steady-state behavior in social choice models through eigendecomposition. This application showcases the potential of our solution as an off-the-shelf solution to the recurrence that has not previously been reported, allowing for the easy identification of the eigenspectra of one-dimensional, one-step, continuous-time Markov processes.

Discrete harmonic analysis associated with Jacobi expansions III: The Littlewood–Paley–Stein [formula omitted]-functions and the Laplace type multipliers

The research about harmonic analysis associated with Jacobi expansions carried out in Arenas et al. (2020) and Arenas et al. (2022) is continued in this paper. Given the operator J(α,β)=J(α,β)−I, where J(α,β) is the three-term recurrence relation for the normalized Jacobi polynomials and I is the identity operator, we define the corresponding Littlewood–Paley–Stein gk(α,β)-functions associated with it and we prove an equivalence of norms with weights for them. As a consequence, we deduce a result for Laplace type multipliers.

Multiple orthogonal polynomials associated with the exponential integral

AbstractWe introduce a new family of multiple orthogonal polynomials satisfying orthogonality conditions with respect to two weights on the positive real line, with the gamma density and a density related to the exponential integral . We give explicit formulas for the type I functions and type II polynomials, their Mellin transform, Rodrigues formulas, hypergeometric series, and recurrence relations. We determine the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials and make a connection to random matrix theory. Finally, we also consider two related families of mixed‐type multiple orthogonal polynomials.

Using Interval Recurrence Formula to Solve Two Improper Fractional Integrals

Using Interval Recurrence Formula to Solve Two Improper Fractional Integrals

A new perspective on bicomplex numbers with Leonardo number components

In the present paper, the bicomplex Leonardo numbers will be introduced with the use of Leonardo numbers and some important algebraic properties including recurrence relation, generating function, Catalan’s and Cassini’s identities, Binet’s formula, sum formulas will also be obtained.

Metamaterial-Based Analog Recurrent Neural Network Toward Machine Intelligence

Wave-based analog computation platforms hold the promise of realizing intelligent information processing with high speed and low computational cost, but current design principles face intractable problems in processing low-frequency wave information end to end at the physical layer. Here we theoretically demonstrate a concept of a metamaterial-based analog recurrent neural network (Meta-RNN) for intelligent classification of physical information carried by mechanical vibrations. With the spatiotemporal recurrence relation induced by coupled local resonators of the simulated metamaterial model, the Meta-RNN is capable of memorizing the internal hidden state in the dynamic displacement field. The trained coupling and synergy of local resonators lead to desirable vibration energy localization with selective frequency extraction capability, which can distinguish the intrinsic characteristics of vibration information. The proposed Meta-RNN provides a foundation for implementing a promising mechanical analog processor for intelligent vibration information processing and classification, and paves the way to efficient machine learning platforms for machine intelligence.

Remarks on higher Schwarzians

The Schwarzian derivative has recently received renewed attention in connection with the study of the Sachdev–Ye–Kitaev model. In mathematics literature, various higher order generalizations of the Schwarzian derivative are known due to Aharonov, Bertilsson, and Schippers. Physical applications of the higher Schwarzian derivatives have not yet been discussed in any detail. In this work, we link Bertilsson's variant to the ℓ–conformal Galilei group, as well as discuss some of its interesting peculiarities. These include a recurrence relation, which allows one to construct the higher Schwarzians iteratively, a composition law, and symmetry transformations.

Modulated Bi-Orthogonal Polynomials on the Unit Circle: The 2j-k and j-2k Systems

We construct the systems of bi-orthogonal polynomials on the unit circle where the Toeplitz structure of the moment determinants is replaced by det (w_{2j-k})_{0le j,k le N-1} and the corresponding Vandermonde modulus squared is replaced by prod _{1 le j < k le N}(zeta _k - zeta _j)(zeta ^{-2}_k - zeta ^{-2}_j) . This is the simplest case of a general system of pj-qk with p, q co-prime integers. We derive analogues of the structures well known in the Toeplitz case: third order recurrence relations, determinantal and multiple-integral representations, their reproducing kernel and Christoffel–Darboux sum, and associated (Carathéodory) functions. We close by giving full explicit details for the system defined by the simple weight w(zeta )=e^{zeta }, which is a specialisation of a weight arising from averages of moments of derivatives of characteristic polynomials over textrm{USp}(2N), textrm{SO}(2N) and textrm{O}^-(2N).

Matrix Approaches for Gould–Hopper–Laguerre–Sheffer Matrix Polynomial Identities

The Gould–Hopper–Laguerre–Sheffer matrix polynomials were initially studied using operational methods, but in this paper, we investigate them using matrix techniques. By leveraging properties of Pascal functionals and Wronskian matrices, we derive several identities for these polynomials, including recurrence relations. It is highlighted that these identities, acquired via matrix techniques, are distinct from the ones obtained when using operational methods.

Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations

In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power q=1m,m∈Z+, where the fractional derivative of order α=qγ, for which γ∈Z+. This assumption will help us to write the integral, the kernel, and the nonhomogeneous terms as a LPS with the same power. The recurrence relations for finding the series coefficients can be constructed using this form. To demonstrate the algorithm’s accuracy, the residual error is defined and calculated for several values of the fractional derivative. Two strongly nonlinear examples are discussed to provide the efficiency of the algorithm. The algorithm gains powerful results for this kind of fractional problem. Under Caputo meaning of the symmetry order, the obtained results are illustrated numerically and graphically. Geometrically, the behavior of the obtained solutions declares that the changing of the fractional derivative parameter values in their domain alters the style of these solutions in a symmetric meaning, as well as indicates harmony and symmetry, which leads them to fully coincide at the value of the ordinary derivative. From these simulations, the results report that the recommended novel algorithm is a straightforward, accurate, and superb tool to generate analytic-approximate solutions for integral and integro-differential equations of fractional order.

Holographic torus correlators of stress tensor in AdS3/CFT2

In this work, we investigate holographic correlators of the stress tensor of a conformal field theory (CFT) on a torus within the context of AdS3/CFT2. To compute the correlators of the stress tensor, we employ the Einstein-Hilbert theory of gravity and perturbatively solve Einstein’s equation in the bulk. In addition, we present an explicit prescription for developing a recurrence relation that simplifies the computation of higher point correlators. Our results show that the correlators and recurrence relation are consistent with known results in CFTs. Additionally, in line with the proposed cutoff-AdS/ Toverline{T} -CFT holography, we extend our computation program to investigate holographic torus correlators at a finite cutoff in the AdS3 and derive a parallel recurrence relation associated with higher point correlators.

On the Uniqueness of the Bounded Solution for the Fractional Nonlinear Partial Integro-Differential Equation with Approximations

This paper studies the uniqueness of the bounded solution to a new Cauchy problem of the fractional nonlinear partial integro-differential equation based on the multivariate Mittag–Leffler function as well as Banach’s contractive principle in a complete function space. Applying Babenko’s approach, we convert the fractional nonlinear equation with variable coefficients to an implicit integral equation, which is a powerful method of studying the uniqueness of solutions. Furthermore, we construct algorithms for finding analytic and approximate solutions using Adomian’s decomposition method and recurrence relation with the order convergence analysis. Finally, an illustrative example is presented to demonstrate constructions for the numerical solution using MATHEMATICA.

On polynomials generated by a four-term recurrence relation and linear coefficients

On polynomials generated by a four-term recurrence relation and linear coefficients

Scrutiny of a More Flexible Counterpart of Huang–Kotz FGM’s Distributions in the Perspective of Some Information Measures

In this work, we reveal some distributional traits of concomitants of order statistics (COSs) arising from the extended Farlie–Gumbel–Morgenstern (FGM) bivariate distribution, which was developed and studied in recent work. The joint distribution and product moments of COSs for this family are discussed. Moreover, some useful recurrence relations between single and product moments of concomitants are obtained. In addition, the asymptotic behavior of the concomitant’s rank for order statistics (OSs) is studied. The information measures, differential entropy, Kullback–Leibler (KL) distance, Fisher information number (FIN), and cumulative past inaccuracy (CPI) are theoretically and numerically studied.

The Asymptotic Number of Score Sequences

A tournament on a graph is an orientation of its edges. The score sequence lists the in-degrees in non-decreasing order. Works by Winston and Kleitman (J Comb Theory Ser A 35(2):208–230, 1983) and Kim and Pittel (J Comb Theory Ser A 92(2):197–206, 2000) showed that the number S_n of score sequences on the complete graph K_n satisfies S_n=Theta (4^n/n^{5/2}). By combining a recent recurrence relation for S_n in terms of the Erdős–Ginzburg–Ziv numbers N_n with the limit theory for discrete infinitely divisible distributions, we observe that n^{5/2}S_n/4^nrightarrow e^lambda /2sqrt{pi }, where lambda =sum _{k=1}^infty N_k/k4^k. This limit agrees numerically with the asymptotics of S_n conjectured by Takács (J Stat Plan Inference 14(1):123–142, 1986). We also identify the asymptotic number of strong score sequences, and show that the number of irreducible subscores in a random score sequence converges in distribution to a shifted negative binomial with parameters r=2 and p=e^{-lambda }.

Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials

The aim of this paper is to construct general forms of ordinary generating functions for special numbers and polynomials involving Fibonacci type numbers and polynomials, Lucas numbers and polynomials, Chebyshev polynomials, Sextet polynomials, Humbert-type numbers and polynomials, chain and anti-chain polynomials, rank polynomials of the lattices, length of any alphabet of words, partitions, and other graph polynomials. By applying the Euler transform and the Lambert series to these generating functions, many new identities and relations are derived. By using differential equations of these generating functions, some new recurrence relations for these polynomials are found. Moreover, general Binet's type formulas for these polynomials are given. Finally, some new classes of polynomials and their corresponding certain family of special numbers are investigated with the help of these generating functions.

Complete Bell Polynomials and Recurrence Relations for Arithmetic Functions

We use the Z-transform to solve a type of recurrence relation satisfied by the number of representations of an integer n as the sum of squares and as the sum of triangular numbers, and also by the color partitions of n; the corresponding solution is in terms of the complete Bell polynomials.

Casimir effect for a stack of graphene sheets

We consider a stack of parallel sheets composed of conducting planes with tensorial conductivities. Using the scattering matrix approach, we derive explicit formulas for the Casimir energy of two, three, and four planes as well as a recurrence relation for arbitrary planes. Specifically, for a stack of graphene, we solve the recurrence relations and obtain formulas for the Casimir energy and force acting on the planes within the stack. Moreover, we calculate the binding energy in the graphene stack with graphite interplane separation, which amounts to ${E}_{ib}=9.9$ meV/atom. Notably, the Casimir force on graphene sheets decreases rapidly for planes beyond the first one. For the second graphene layer in the stack, the force is 35 times smaller than that experienced by the first layer.