Recurrence Relations Research Articles

Recurrence Relations for Moment Generating Function of Progressive First Failure Censoring and Characterizations of Right Truncated Exponential Distribution

Recurrence Relations for Moment Generating Function of Progressive First Failure Censoring and Characterizations of Right Truncated Exponential Distribution

Skew-Orthogonal Polynomials and Pfaff Lattice Hierarchy Associated With an Elliptic Curve

Abstract Starting with a skew-symmetric inner product over an elliptic curve, we propose the concept of elliptic skew-orthogonal polynomials. Inspired by the Landau–Lifshitz hierarchy and its corresponding time evolutions, we obtain the recurrence relation and the $\tau $-function representation for such a novel class of skew-orthogonal polynomials. Furthermore, a bilinear integral identity is derived through the so-called Cauchy–Stieljes transformation, from which we successfully establish the connection between the elliptic skew-orthogonal polynomials and an elliptic extension of the Pfaff lattice hierarchy.

The generalized K-function and its recurrence relations

The generalized K-function and its recurrence relations

Dimension reduction based on approximate gramians via Laguerre polynomials for coupled systems

Abstract In this paper, we focus on the topic of model order reduction (MOR) for coupled systems. At first, an approximation via Laguerre polynomials expansions to controllability and observability gramians for such systems are presented, which provides a low-rank decomposition form whose factors are constructed from a recurrence formula instead of Lyapunov equations. Then, in combination of balanced truncation and dominant subspace projection method, a series of MOR algorithms are proposed that preserve the coupled structures. What’s more, some main properties of reduced-order models, such as stability preservation, are well discussed. Finally, three numerical simulations are provided to illustrate the effectiveness of our algorithms.

Central factorial numbers are Pólya frequency sequences

The central factorial numbers are closely related to the central difference operator, and have received increasing attention from combinatorialists recently. Known properties of the central factorial numbers include their generating functions, recurrence relation and combinatorial interpretations. The purpose of this work is to adjoin to the list the analytical property that the generating functions of central factorial numbers of odd, and even, indices, are simply real-rooted, thus establishing the numbers being Pólya frequency sequences.

Record-setters in the Stern sequence

Stern's diatomic series, denoted by (a(n))n≥0, is defined by the recurrence relations a(2n)=a(n) and a(2n+1)=a(n)+a(n+1) for n≥1, with initial values a(0)=0 and a(1)=1. A record-setter for a sequence (s(n))n≥0 is an index v such that s(i)<s(v) holds for all i<v. In this paper, we provide a complete description of the record-setters for the Stern sequence. As a consequence, we prove that the running max sequence of the Stern sequence is not 2-regular.

Using easy coefficients conjecture for rotation symmetric Boolean functions

A Boolean function in n variables is rotation symmetric (RS) if it is invariant under powers of ρ(x1,…,xn)=(x2,…,xn,x1). An RS function is called monomial rotation symmetric (MRS) if it is generated by applying powers of ρ to a single monomial. Completing earlier research on special cases, the author showed in 2018 that for any RS function fn in n variables, the sequence of Hamming weights wt(fn) for all values of n satisfies a linear recurrence relation. It was also proved that the associated recursion polynomial could be explicitly calculated as the minimal polynomial of a rules matrix and an algorithm for computing the rules matrix was explained. Examples showed that the usual formula which gives the values of wt(fn)−2n−1 as a linear combination of powers of the irrational roots of the minimal polynomial has simple coefficients which are all 1/2 if the multiset made up of the roots of the characteristic polynomial of the rules matrix is used instead, no matter what the degree of the Boolean function is. The conjecture that this is always true is called the Easy Coefficients Conjecture (ECC). The special case of MRS quadratics was proved in an earlier paper. The present paper gives some ECC applications valid for any function for which the ECC is true, even though there is a proof only for the quadratic MRS case so far.

Earthquakes on the Once-Punctured Torus

We study earthquake deformations on Teichmüller space associated with simple closed curves of the once-punctured torus. We describe two methods to get an explicit form of the earthquake deformation for any simple closed curve. The first method is rooted in linear recurrence relations, the second in hyperbolic geometry. The two methods align, providing both an algebraic and geometric interpretation of the earthquake deformations. We convert the expressions to other coordinate systems for Teichmüller space to examine earthquake deformations further. Two families of curves are used as examples. Examining the limiting behavior of each gives insight into earthquakes about measured geodesic laminations, of which simple closed curves are a special case.

A note on the post quantum-Sheffer polynomial sequences

Abstract In this article, the post quantum analogue of Sheffer polynomial sequences is introduced using concepts of post quantum calculus. The series representation, recurrence relations, determinant expression and certain other properties of this class are established. Further, the 2D-post quantum-Sheffer polynomials are introduced via generating function and their properties are established. Certain identities and integral representations for the 2D-post quantum-Hermite polynomials, 2D-post quantum-Laguerre polynomials, and 2D-post quantum-Bessel polynomials are also considered.

Active oscillator: Recurrence relation approach

The present work analyzes stationary distributions of active Brownian particles in a harmonic trap. Generally, obtaining stationary distributions for this system is non-trivial, and to date, no exact expressions are available. In this work, we develop and explore a method based on a transformation of the Fokker–Planck equation into a recurrence relation for generating moments of a distribution. The method, therefore, offers an analytically tractable approach, an alternative to numerical simulations, in a situation where more direct analytical approaches fail. Although the current work focuses on the active Brownian particle model, the method is general and valid for any type of active dynamics and any system dimension.

Weighted extropy measures in general Morgenstern family under k-record values with application to medical data

In this paper, we study the marginal distribution of concomitants of k-record (KR) values from generalized Farlie–Gumbel–Morgenstern (GFGM) of bivariate distributions. In addition, the joint distribution of concomitants of KR for this family is obtained. Furthermore, some useful recurrence relations between moments of concomitants are derived. In addition, the hazard rate, the reversed hazard rate, and mean residual life functions of concomitants for this family are obtained. Some recent new measures of information, such as weighted extropy, weighted cumulative past extropy, and weighted cumulative residual extropy, are investigated for the concomitant of KR under the GFGM family. A non-parametric estimator of the proposed measure is provided by combining the empirical method with the concurrent use of KR in the GFGM family. Finally, we analyzed real-world data to examine our findings.

Curve fitting for seismic waves of earthquake with Hermite polynomials

We investigate and study on mathematical structures involving mathematical models and others associated with seismic waves in an earthquake. Our first aim is to give some novel formulas and certain finite sums including the Bernoulli numbers and the Hermite polynomials with the aid of generating functions, the Riemann integral, and the Volkenborn integral. The second aim is to examine the seismic wave propagation in different geological units with the help of special polynomials containing the Hermite polynomials and their graph fitting of functions. To evaluate the shape of the seismic waves propagating within the ground (rock and/or soil), we use comparing method with the graph of the Hermite polynomials and functions and the polynomial Rocking Bearings. Furthermore, we also define generating function for the polynomial type Rocking Bearings. We give open problems on this generating function and earthquake facts. By applying partial derivative operator to the generating function for the m-parametric Hermite type polynomials, we give a novel recurrence relation and derivative formulas for these polynomials. We also give a new general formula for monomials in terms of these polynomials. Moreover, for the purpose of visualizing curve fitting approach to the seismic waves, we draw many plots of the Hermite functions with Mathematica (Version 12.0.0) with their codes. Finally, with the aid of these graphs, we give useful evaluation on the shapes of the seismic waves propagated in the ground (rock and/or soil).

Advancements in $ q $-Hermite-Appell polynomials: a three-dimensional exploration

&lt;p&gt;In this research, we leverage various $ q $-calculus identities to introduce the notion of $ q $-Hermite-Appell polynomials involving three variables, elucidating their formalism. We delve into numerous properties and unveil novel findings regarding these $ q $-Hermite-Appell polynomials, encompassing their generating function, series representation, summation equations, recurrence relations, $ q $-differential formula, and operational principles. Our investigation sheds light on the intricate nature of these polynomials, elucidating their behavior and facilitating deeper understanding within the realm of $ q $-calculus.&lt;/p&gt;

The altered Hermite matrix: implications and ramifications

&lt;p&gt;Matrix theory is essential for addressing practical problems and executing computational tasks. Matrices related to Hermite polynomials are essential due to their applications in quantum mechanics, numerical analysis, probability, and signal processing. Their orthogonality, recurrence relations, and spectral properties make them a valuable tool for both theoretical research and practical applications. From a different perspective, we introduced a variant of the Hermite matrix that incorporates triple factorials and demonstrated that this matrix satisfies various properties. By utilizing effective matrix algebra techniques, various algebraic properties of this matrix have been determined, including the product formula, inverse matrix and eigenvalues. Additionally, we extended this matrix to a more generalized form and derived several identities.&lt;/p&gt;

Integral Transforms and Extended Hermite-Apostol Type Frobenius-Genocchi Polynomials

The schemata for applications of the integral transforms of mathematical physics to recurrence relations, differential, integral, integro-differential equations and in the theory of special functions has been developed. The article aims to introduce and present operational representations for a new class of extended Hermite-Apostol type Frobenius-Genocchi polynomials via integral transforms. The recurrence relations and some identities involving these polynomials are established. The article concludes by establishing a determinant form and quasi-monomial properties for the Hermite-Apostol type Frobenius-Genocchi polynomials and for their extended forms.

On the study of the recurrence relations and characterizations based on progressive first-failure censoring

&lt;abstract&gt; &lt;p&gt;In this research, the progressive first-failure censored data (PFFC) from the Kumaraswamy modified inverse-Weibull distribution (KMIWD) were used to obtain the recurrence relations and characterizations for single and product moments. The recurrence relationships allow for a rapid and efficient assessment of the means, variances and covariances for any sample size. Additionally, the paper outcomes can be boiled down to the traditional progressive type-II censoring. Also, some special cases are limited to some lifetime distributions as the exponentiated modified inverse Weibull and Kumaraswamy inverse exponential.&lt;/p&gt; &lt;/abstract&gt;

On Stirling and bell numbers of order 1/2

The Stirling numbers of order 1/2 (of the second kind) introduced by Katugampola are discussed and it is shown that they are given by a scaled subfamily of the generalized Stirling numbers introduced by Hsu and Shiue. This allows to deduce in a straightforward fashion many properties of the Stirling and Bell numbers of order 1/2, for example, recurrence relations, generating functions, Dobi?ski formula, and Spivey formula. The even Bell polynomials of order 1/2 are shown to be closely related to generalized Laguerre polynomials of order ?1/2. Generalized Stirling numbers of order 1/2 of the first kind are defined and studied. An analog of the Weyl algebra is introduced and proposed as a natural algebraic setting where the Stirling numbers of order 1/2 of both kinds appear as ordering coefficients. This algebra contains the Weyl algebra as a subalgebra.

Representations of quadratic Heisenberg-Weyl algebras and polynomials in the fourth Painlevé transcendent

&lt;p&gt;We provide new insights into the solvability property of a Hamiltonian involving the fourth Painlevé transcendent and its derivatives. This Hamiltonian is third-order shape invariant and can also be interpreted within the context of second supersymmetric quantum mechanics. In addition, this Hamiltonian admits third-order lowering and raising operators. We have considered the case when this Hamiltonian is irreducible, i.e., when no special solutions exist for given parameters $ \alpha $ and $ \beta $ of the fourth Painlevé transcendent $ P_{IV}(x, \alpha, \beta) $. This means that the Hamiltonian does not admit a potential in terms of rational functions (or the hypergeometric type of special functions) for those parameters. In such irreducible cases, the ladder operators are as well involving the fourth Painlevé transcendent and its derivative. An important case for which this occurs is when the second parameter (i.e., $ \beta $) of the fourth Painlevé transcendent $ P_{IV}(x, \alpha, \beta) $ is strictly positive, $ \beta &amp;gt; 0 $. This Hamiltonian was studied for all hierarchies of rational solutions that come in three families connected to the generalized Hermite and Okamoto polynomials. The explicit form of ladder, the associated wavefunctions involving exceptional orthogonal polynomials, and recurrence relations were also completed described. Much less is known for the irreducible case, in particular the excited states. Here, we developed a description of the induced representations based on various commutator identities for the highest and lowest weight type representations. We also provided for such representations a new formula concerning the explicit form of the related excited states from the point of view of the Schrödinger equation as two-variables polynomials that involve the fourth Painlevé transcendent and its derivative.&lt;/p&gt;

On Leonardo Pisano dual quaternions

In this work, firstly we introduce the Leonardo Pisano dual quaternions combining Leonardo Pisano numbers and dual quaternions. Then we examine some fundamental properties and identities of the Leonardo Pisano dual quaternions, such as recurrence relations, generating function, summing formulas, Binet’s formula, Cassini and Catalan’s identities.

Prediction of Rainfall Trends Using Forecasting Approaches Based on Singular Spectrum Analysis

Advanced technologies such as the Internet of Things provide an integrated platform for weather focusing, including rainfall and flood prediction. Large rainfall data frequently contain noise, which can be difficult to analyze using a standard time series model due to violated assumptions. Singular spectrum analysis (SSA) is a model-free time series analysis method that is widely used. This study aims to predict the rainfall trends in the Special Region of Yogyakarta, Indonesia, using the Recurrent SSA (SSA-R) and Vector SSA (SSA-V). The SSA-R forecasts using the recurrent continuation directly with the linear recurrent formula, while the SSA-V is a modified recurrent method. This study used 50 years of monthly rainfall data (1970-2019) from 25 stations in the special region of Yogyakarta, Indonesia. The SSA steps for forecasting rainfall data include decomposition (embedding and singular value decomposition), reconstruction (grouping and diagonal averaging), and evaluating the SSA model using w-correlation (if w-correlation is close to zero, returning to the decomposition stage; otherwise, continue the process), forecasting, evaluating the forecast results using root mean square error (RMSE), mean absolute error, r, and mean forecast error, and finally selecting the best model (either the SSA-R or SSA-V model). The results showed that the SSA-R performed better than SSA-V due to the smallest RMSE in the dry, rainy, and inter-monsoon seasons. The SSA-R model’s forecast results revealed faint, constant patterns for the dry, and rainy seasons and an increasing pattern for the inter-monsoon season. The novelty of this study is to compare the performance of the SSA-R and SSA-V models in the large rainfall data in the special region of Yogyakarta, Indonesia.