Recurrence Relations Research Articles

Some infinite series summations involving linear recurrence relations of order 2 and 3

Some infinite series summations involving linear recurrence relations of order 2 and 3

Generalised (k, t)-Narayana sequence

Generalised (k, t)-Narayana sequence

Deflection of Light by a Reissner-Nordström Black Hole and Painlevé VI equation

Abstract We consider the bending angle of the trajectory of a photon incident from and deflected to infinity around a Reissner-Nordström black hole. We treat the bending angle as a function of the squared reciprocal of the impact parameter and the squared electric charge of the background normalized by the mass of the black hole. It is shown that the bending angle satisfies a system of two inhomogeneous linear partial differential equations with polynomial coefficients. This system can be understood as an isomonodromic deformation of the inhomogeneous Picard-Fuchs equation satisfied by the bending angle in the Schwarzschild spacetime, where the deformation parameter is identified as the background electric charge. Furthermore, the integrability condition for these equations is found to be a specific type of the Painlevé VI equation that allows an algebraic solution. We solve the differential equations both at the weak and strong deflection limits. In the weak deflection limit, the bending angle is expressed as a power series expansion in terms of the squared reciprocal of the impact parameter and we obtain the explicit full-order expression for the coefficients. In the strong deflection limit, we obtain the asymptotic form of the bending angle that consists of the divergent logarithmic term and the finite O(1) term supplemented by linear recurrence relations which enable us to straightforwardly derive higher order coefficients. In deriving these results, the isomonodromic property of the differential equations plays an important role. Lastly, we briefly discuss the applicability of our method to other types of spacetimes such as a spinning black hole.

Confidence intervals for mean response times of a open queueing network: calibration approach

Response time is a key factor in the analysis of the different queueing network model properties. This study computes a series of response times using a data-based recurrence relation. The true average response time is determined using the sample averages from those response times. Several confidence intervals are created for the open queueing net- work model’s average response times using the calibration approach. Using a numerical simulation analysis, the accuracy of the various confidence intervals is evaluated. Journal of Statistical Research 2023, Vol 57, No.1-2, pp.69-79

An orthogonal class of $p$-Legendre polynomials on variable interval

The work incorporates a generalization of the Legendre polynomial by introducing a parameter $p>0$ in its generating function. The coefficients thus generated, constitute a class of the polynomials which are termed as the $p$-Legendre polynomials. It is shown that this class turns out to be orthogonal with respect to the weight function: $(1-\sqrt{p}\ x)^{\frac{p+1}{2p}-1}(1+\sqrt{p}\ x)^{\frac{p+1}{2p}-1}$ over the interval $(-\frac{1}{\sqrt{p}}, \frac{1}{\sqrt{p}}).$ Among the other properties derived, include the Rodrigues formula, normalization, recurrence relation and zeros. A graphic depiction for $p=0.5, 1, 2,$ and $3$ is shown. The $p$-Legendre polynomials are used to estimate a function using the least squares approach. The approximations are graphically depicted for $p=0.7, 1, 2.$

Certain properties of [formula omitted]-analogue of M-function

The intent of this work is to create and study certain fundamental characteristics of new generalizations of the M-function using q-calculus. We establish its characteristics, like the convergence condition, recurrence relation, integral representations, q-derivative formulae, q-Laplace and q-Sumudu transformations, and image formulas for the Caputo fractional q-derivative and the Hilfer fractional q-derivative. Furthermore, we analyze some specific situations to demonstrate of our key findings.

On the computation of moments in the Super-Transition-Arrays model for radiative opacity calculations

In the Super-Transition-Array statistical method for the computation of radiative opacity of hot dense matter, the moments of the absorption or emission features involve partition functions with reduced degeneracies, occurring through the calculation of averages of products of subshell populations. In the present work, we discuss several aspects of the computation of such peculiar partition functions, insisting on the precautions that must be taken in order to avoid numerical difficulties. In a previous work, we derived a formula for supershell partition functions, which takes the form of a functional of the distribution of energies within the supershell and allows for fast and accurate computations, truncating the number of terms in the expansion. The latter involves coefficients for which we obtained a recursion relation and an explicit formula. We show that such an expansion can be combined with the recurrence relation for shifted partition functions. We also propose, neglecting the effect of fine structure as a first step, a positive-definite formula for the Super-Transition-Array moments of any order, providing an insight into the asymmetry and sharpness of the latter. The corresponding formulas are free of alternating sums. Several ways to speed up the calculations are also presented.

Words over a finite alphabet avoiding 1243

. In this paper, we establish a system of recurrence relations and an algorithm for finding the generating function Ak (x) for the number of words over alphabet [k] of length n that avoid 1243, for arbitrary k. In particular, we present an explicit formula for the generating function Ak (x) for 1 ≤ k ≤ 10.

The zero locus and some combinatorial properties of certain exponential Sheffer sequences

We present combinatorial and analytical results concerning a Sheffer sequence with an exponential generating function of the form G(s,z)=eczs+αz2+βz4, where α,β,c∈R with β<0 and c≠0. We demonstrate that the zeros of all polynomials in such a Sheffer sequence are either real, or purely imaginary. Additionally, using the properties of Riordan matrices we show that our Sheffer sequence satisfies a four-term recurrence relation of order 4, and we also exhibit a connection between the coefficients of these Sheffer polynomials and the number of nodes with a given label in certain marked generating trees.

On Some Relations Involving the Ramanujan’s Tau Function

It is known a recurrence relation for the Ramanujan’s tau-function involving the sum of divisors function𝝈(𝒏), whose solution gives a closed formula for 𝝉(𝒏) in terms of complete Bell polynomials, and a determinantal expression for 𝝈(𝒎) where participate the values 𝝉(𝒌).

On the spectrum of tridiagonal operators in the context of orthogonal polynomials

AbstractThe basis for our studies is a large class of orthogonal polynomial sequences $$(P_n)_{n\in {{\mathbb {N}}}_0}$$ ( P n ) n ∈ N 0 , which is normalized by $$P_n(x_0)=1$$ P n ( x 0 ) = 1 for all $$n\in {\mathbb {N}}_0$$ n ∈ N 0 where the coefficients in the three-term recurrence relation are bounded. The goal is to check if $$x_0 \in {\mathbb {R}}$$ x 0 ∈ R is in the support of the orthogonalization measure $$\mu $$ μ . For this purpose, we use, among other things, a result of G. H. Hardy concerning Cesàro operators on weighted $$l^2$$ l 2 -spaces. These investigations generalize ideas from Lasser et al. (Arch Math 100:289–299, 2013).

Explicit and approximate series solutions for nonlinear fractional wave-like differential equations with variable coefficients

This work investigates the exact and accurate approximate series solutions of the strongly nonlinear wave-like differential equations of fractional order with variable coefficients utilizing the Laplace residual power series technique in the 2D, and 3D-space. The posed model is a notable model in mathematical physics, engineering physics, and biophysics, to characterize the variations in laser light intensity, and fluid particle velocity distributions in turbulent flows. In the meaning of Caputo fractional derivative, the posed models are solved analytically via simulation of the generalized Taylor formula in the Laplace space. Different from the other analytic methods, an easy recurrence formula for finding the Laplace power series coefficients is derived. To validate the proposed approach, it is applied to three interesting problems. The approximate analytical solutions of studied fractional problems are expressed in fast-converging Caputo-fractional Maclaurin formulas. The results are numerically and graphically analyzed to demonstrate the efficiency and applicability of the technique, as well as to examine the effects of partial arrangement on the behavior of the solutions. Ultimately, the results and numerical simulations indicate that the used technique is an effective approach and accurate tool for solving fractional evolution models and other fractional models arising in nonlinear phenomena.

$SO(3)$ квазі-мономіальні сім'ї многочленів

Let $H$ be a subgroup of the affine space group ${\rm Aff} 3)$, considered with its natural action on the vector space of three-variable polynomials. The polynomial family $\{ B_{m,n,k}(x,y,z) \}$ is called quasi-monomial with respect to $H$ if the group operators in two different bases $\{ x^m y^n z^k \}$ and $\{ B_{m,n,k}(x,y,z) \}$ have identical atrices. We derive a criterion for quasi-monomiality when the group $H$ is the special orthogonal group $SO(3)$. This criterion is expressed through the exponential generating function of the polynomial family $\{B_{m,n,k}(x,y,z)\}$. It has been proven that Appel's biorthogonal polynomials are quasi-monomials with respect to $SO(3)$ and recurrence relations have been found for them.

Some Identities on Generalized Arithmetic Triangles

Summary We generalize the arithmetic triangle of Blaise Pascal, Yang Hui, and others, by maintaining its recurrence relation, but replacing the traditional 1s on the boundary with arbitrary sequences. Within these structures, we examine some identities commonly studied in Pascal’s version, including those related to sums and alternating sums of entries in rows as well as the so-called hockey stick identities. We use recurrence relations and elementary generating functions to derive general results and see how these results can be used in some interesting special cases.

Second-order concomitants of order statistics from bivariate Cambanis family: some information measures–estimation

In this paper, the marginal distribution of second-order concomitants of order statistics (SCOSs) from the Cambanis family is determined. An advantageous recurrence relation between single moments of SCOSs and concomitants of order statistics (COSs) is established based on the Cambanis family. We also use this family to derive an elegant beneficial recurrence relation between single moments of the SCOSs. Some information measures, such as Shannon differential entropy, differential extropy, and negative cumulative extropy (NCEX) of SCOSs based on this family, are investigated. Furthermore, we present some numerical examples to back up the theoretical findings. In addition, a set of bivariate data from the real world was examined. Finally, an estimate of a marginal variable’s scale parameter in the copula Cambanis family is obtained.

On Higher-Order Generalized Fibonacci Hybrinomials: New Properties, Recurrence Relations and Matrix Representations

This paper presents a comprehensive survey of the generalization of hybrid numbers and hybrid polynomials, particularly in the fields of mathematics and physics. In this paper, by using higher-order generalized Fibonacci polynomials, we introduce higher-order generalized Fibonacci hybrid polynomials called higher-order generalized Fibonacci hybrinomials. We obtain some special cases and algebraic properties of the higher-order generalized Fibonacci hybrinomials, such as the recurrence relation, generating function, exponential generating function, Binet formula, Vajda’s identity, Catalan’s identity, Cassini’s identity and d’Ocagne’s identity. We also present three different matrices whose components are higher-order generalized Fibonacci hybrinomials, higher-order generalized Fibonacci polynomials and Lucas polynomials. By using these matrices, we obtain some identities related to these newly established hybrinomials.

Positivity-hardness results on Markov decision processes

This paper investigates a series of optimization problems for one-counter Markov decision processes (MDPs) and integer-weighted MDPs with finite state space. Specifically, it considers problems addressing termination probabilities and expected termination times for one-counter MDPs, as well as satisfaction probabilities of energy objectives, conditional and partial expectations, satisfaction probabilities of constraints on the total accumulated weight, the computation of quantiles for the accumulated weight, and the conditional value-at-risk for accumulated weights for integer-weighted MDPs. Although algorithmic results are available for some special instances, the decidability status of the decision versions of these problems is unknown in general. The paper demonstrates that these optimization problems are inherently mathematically difficult by providing polynomial-time reductions from the Positivity problem for linear recurrence sequences. This problem is a well-known number-theoretic problem whose decidability status has been open for decades and it is known that decidability of the Positivity problem would have far-reaching consequences in analytic number theory. So, the reductions presented in the paper show that an algorithmic solution to any of the investigated problems is not possible without a major breakthrough in analytic number theory. The reductions rely on the construction of MDP-gadgets that encode the initial values and linear recurrence relations of linear recurrence sequences. These gadgets can flexibly be adjusted to prove the various Positivity-hardness results.

Recognition of 3D Images by Fusing Fractional-Order Chebyshev Moments and Deep Neural Networks.

In order to achieve efficient recognition of 3D images and reduce the complexity of network parameters, we proposed a novel 3D image recognition method combining deep neural networks with fractional-order Chebyshev moments. Firstly, the fractional-order Chebyshev moment (FrCM) unit, consisting of Chebyshev moments and the three-term recurrence relation method, is calculated separately using successive integrals. Next, moment invariants based on fractional order and Chebyshev moments are utilized to achieve invariants for image scaling, rotation, and translation. This design aims to enhance computational efficiency. Finally, the fused network embedding the FrCM unit (FrCMs-DNNs) extracts depth features to analyze the effectiveness from the aspects of parameter quantity, computing resources, and identification capability. Meanwhile, the Princeton Shape Benchmark dataset and medical images dataset are used for experimental validation. Compared with other deep neural networks, FrCMs-DNNs has the highest accuracy in image recognition and classification. We used two evaluation indices, mean square error (MSE) and peak signal-to-noise ratio (PSNR), to measure the reconstruction quality of FrCMs after 3D image reconstruction. The accuracy of the FrCMs-DNNs model in 3D object recognition was assessed through an ablation experiment, considering the four evaluation indices of accuracy, precision, recall rate, and F1-score.

Rotation numbers and bounded deviations for quasi-periodic monotone recurrence relations

It is known that monotone recurrence relations are defined by finding equilibria of generalized Frenkel-Kontorova models. In this paper, we intend to study quasi-periodic monotone recurrence relations. By introducing a countable set consisting of integers with bounded distances and improving a method of Angenent used for studying periodic monotone recurrence relations, we derive a criterion for the existence of solutions with bounded deviations from the rigid rotation. With the help of the criterion, a range of parameters is provided to guarantee the quasi-periodic standard map has orbits with bounded deviations from the rigid rotation corresponding to each given rotation number.

Intrinsic Solutions to the Biquadratic Diophantine Equations

Abstract: The Biquadratic Diophantine equation with five unknowns has non-zero unique intrinsic integral solutions obtained. The recurrence relation between the solutions is also obtained.