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The most general description of quantum evolution up to a time τ is a completely positive tracing preserving map known as a dynamical map Λ ̂ (τ). Here, we consider Λ ̂ (τ) arising from suddenly coupling a system to one or more thermal baths with a strength that is neither weak nor strong. Given no clear separation of characteristic system/bath time scales, Λ ̂ (τ) is generically expected to be non-Markovian; however, we do assume the ensuing dynamics has a unique steady state, implying the baths possess a finite memory time τm. By combining several techniques within a tensor network framework, we directly and accurately extract Λ ̂ (τ) for a small number of interacting fermionic modes coupled to infinite non-interacting Fermi baths. First, we use an orthogonal polynomial mapping and thermofield doubling to arrive at a purified chain representation of the baths whose length directly equates to a time over which the dynamics of the infinite baths is faithfully captured. Second, we employ the Choi–Jamiolkowski isomorphism so that Λ ̂ (τ) can be fully reconstructed from a single pure state calculation of the unitary dynamics of the system, bath and their replica auxiliary modes up to time τ. From Λ ̂ (τ) , we also compute the time local propagator L ̂ (τ). By examining the convergence with τ of the instantaneous fixed points of these objects, we establish their respective memory times τ m Λ and τ m L . Beyond these times, the propagator L ̂ (τ) and dynamical map Λ ̂ (τ) accurately describe all the subsequent long-time relaxation dynamics up to stationarity. These timescales form a hierarchy τ m L ≤ τ m Λ ≤ τ re , where τre is a characteristic relaxation time of the dynamics. Our numerical examples of interacting spinless Fermi chains and the single impurity Anderson model demonstrate regimes where τre ≫ τm, where our approach can offer a significant speedup in determining the stationary state compared to directly simulating the long-time limit. Our results also show that having access to Λ ̂ (τ) affords a number of insightful analyses of the open system thus far not commonly exploited. [ABSTRACT FROM AUTHOR]

In non-perturbative non-Markovian open quantum systems, reaching either low temperatures with the hierarchical equations of motion (HEOM) or high temperatures with the Thermalized Time Evolving Density Operator with Orthogonal Polynomials Algorithm (T-TEDOPA) formalism in Hilbert space remains challenging. We compare different ways of modeling the environment. Sampling the Fourier transform of the bath correlation function, also called temperature dependent spectral density, proves to be very effective. T-TEDOPA [Tamascelli et al., Phys. Rev. Lett. 123, 090402 (2019)] uses a linear chain of oscillators with positive and negative frequencies, while HEOM is based on the complex poles of an optimized rational decomposition of the temperature dependent spectral density [Xu et al., Phys. Rev. Lett. 129, 230601 (2022)]. Resorting to the poles of the temperature independent spectral density and of the Bose function separately is an alternative when the problem due to the huge number of Bose poles at low temperatures is circumvented. Two examples illustrate the effectiveness of the HEOM and T-TEDOPA approaches: a benchmark pure dephasing case and a two-bath model simulating the dynamics of excited electronic states coupled through a conical intersection. We show the efficiency of T-TEDOPA to simulate dynamics at a finite temperature by using either continuous spectral densities or only all the intramolecular oscillators of a linear vibronic model calibrated from ab initio data of a phenylene ethynylene dimer. [ABSTRACT FROM AUTHOR]

In this work, the concept of quasi‐type Kernel polynomials with respect to a moment functional is introduced. Difference equation satisfied by these polynomials along with the criterion for orthogonality conditions are discussed. The process of recovering orthogonality for the linear combination of a quasi‐type kernel polynomial with another orthogonal polynomial, which is identified by involving linear spectral transformation, is provided. This process involves an expression of ratio of iterated kernel polynomials. This leads to considering the limiting case of ratio of kernel polynomials involving continued fractions. Special cases of such ratios in terms of certain continued fractions are exhibited. Moreover, examples are discussed for obtaining the orthogonality of quasi‐type kernel polynomials of order 1. [ABSTRACT FROM AUTHOR]

The use of orthonormal polynomial bases has been found to be efficient in preventing ill‐conditioning of the system matrix in the primal formulation of Virtual Element Methods (VEM) for high values of polynomial degree and in presence of badly‐shaped polygons. However, we show that using the natural extension of a orthogonal polynomial basis built for the primal formulation is not sufficient to cure ill‐conditioning in the mixed case. Thus, in the present work, we introduce an orthogonal vector‐polynomial basis which is built ad hoc for being used in the mixed formulation of VEM and which leads to very high‐quality solution in each tested case. Furthermore, a numerical experiment related to simulations in Discrete Fracture Networks (DFN), which are often characterised by very badly‐shaped elements, is proposed to validate our procedures. [ABSTRACT FROM AUTHOR]

The study of P -polynomial association schemes (distance-regular graphs) and Q -polynomial association schemes, and in particular P - and Q -polynomial association schemes, has been a central theme not only in the theory of association schemes but also in the whole study of algebraic combinatorics in general. Leonard's theorem (1982) says that the spherical functions (or the character tables) of P - and Q -polynomial association schemes are described by Askey-Wilson orthogonal polynomials or their relatives. These polynomials are one-variable orthogonal polynomials. It seems that the new attempt to define and study higher rank P - and Q -polynomial association schemes had been hoped for, but had gotten only limited success. The first very successful attempt was initiated recently by Bernard-Crampé-d'Andecy-Vinet-Zaimi, and then followed by Bannai-Kurihara-Zhao-Zhu. The general theory and some explicit examples of families of higher rank (multivariate) P - and/or Q -polynomial association schemes have been obtained there. The main purpose of the present paper is to prove that some important families of association schemes are shown to be bivariate Q -polynomial. Namely, we show that all the nonbinary Johnson association schemes and all the association schemes obtained from attenuated spaces are bivariate Q -polynomial. It should be noted that the parameter restrictions needed in the previous papers are completely lifted in this paper. Our proofs are done by explicitly calculating the Krein parameters of these association schemes. At the end, we mention some speculations and indications of what we can expect in the future study. [ABSTRACT FROM AUTHOR]

We study the approximation and quadrature error of points that satisfy Marcinkiewicz-Zygmund inequalities. First, we investigate the use of Marcinkiewicz-Zygmund inequalities in an abstract Hilbert space for an abstract approximation and quadrature rule. The setting is then specified to Sobolev spaces induced by Freud weights e^{-2\sigma |x|^\alpha } with \alpha >1 and \sigma >0, and we derive specific bounds for the approximation and quadrature error. For the Gaussian weight e^{-2\pi x^2}, we verify that the Sobolev spaces essentially coincide with a specific class of modulation spaces that are well known in (time-frequency) analysis. [ABSTRACT FROM AUTHOR]

ABSTRACT In this paper, we consider an appropriate ordering of the Laurent monomials xiyj$x^{i}y^{j}$, i,j∈Z$i,j \in \mathbb {Z}$ that allows us to study sequences of orthogonal Laurent polynomials of the real variables x$x$ and y$y$ with respect to a positive Borel measure μ$\mu$ defined on R2$\mathbb {R}^2$ such that ({x=0}∪{y=0})∩supp(μ)=∅$(\lbrace x=0 \rbrace \cup \lbrace y=0 \rbrace) \cap \textrm {supp}(\mu)= \emptyset$. This ordering is suitable for considering the multiplication plus inverse multiplication operator on each variable (x+1x$(x+\frac{1}{x}$ and y+1y)$ y+\frac{1}{y})$, and as a result we obtain five‐term recurrence relations, Christoffel–Darboux and confluent formulas for the reproducing kernel and a related Favard's theorem. A connection with the one‐variable case is also presented, along with some applications for future research. [ABSTRACT FROM AUTHOR]

Additive perturbations, specifically matrix Uvarov transformations for matrix orthogonal polynomials, are under consideration. Christoffel–Uvarov formulas are deduced for the perturbed biorthogonal families, along with their matrix norms. These formulations are expressed in terms of the spectral jets of the Christoffel–Darboux kernels. [ABSTRACT FROM AUTHOR]

The stretching of a composite specimen taking into account temperature influence is modelled in terms of tensile stress applied at the opposite boundaries of the elastic rectangular domain. Two other boundaries are supposed to be fixed, it allows to formulate the stated problem as a boundary value problem. The analytical solution is realized with the help of the integral transform method. According to it, the problem is reduced to the one‐dimensional non‐homogeneous ordinary differential equations with corresponding boundary conditions in the transform's domain. The apparatus of Green's matrix‐function was applied to solve it. It let to the system of two singular integral equations, which was solved with the help of orthogonal polynomials method. The final formulae for displacements and stress were used to analyze the stress state of the rectangular elastic specimen. Obtained numerical results were compared with some experimental data for the stretching rectangular ceramic specimen. [ABSTRACT FROM AUTHOR]

This article aims to significantly advance geometric function theory by providing a valuable contribution to analytic and multivalent functions. It focuses on differential subordination and superordination, which characterize the interactions between analytic functions. To achieve our goal, we employ a method that relies on the characteristics of differential subordination and superordination. As one of the latest advancements in this field, this technique is able to derive several results about differential subordination and superordination for multivalent functions defined by the new operator M λ , p m v , ρ ; η F ξ within the open unit disk A . Additionally, by employing the technique, the differential sandwich outcome is achieved. Therefore, this work presents crucial exceptional instances that follow the results. The findings of this paper can be applied to a wide range of mathematical and engineering problems, including system identification, orthogonal polynomials, fluid dynamics, signal processing, antenna technology, and approximation theory. Furthermore, this work significantly advances the knowledge and understanding of the analytical functions of the unit and its interactive higher relations. The characteristics and consequences of differential subordination theory are symmetric to those of differential superordination theory. By combining them, sandwich-type theorems can be derived. [ABSTRACT FROM AUTHOR]

The study of −1$-1$ orthogonal polynomials viewed as q→−1$q\rightarrow -1$ limits of the q$q$‐orthogonal polynomials is pursued. This paper presents the continuous polynomials part of the −1$-1$ analog of the q$q$‐Askey scheme. A compendium of the properties of all the continuous −1$-1$ hypergeometric polynomials and their connections is provided. [ABSTRACT FROM AUTHOR]

We integrate in the framework of the semilocal trace formula two recent discoveries on the spectral realization of the zeros of the Riemann zeta function by introducing a semilocal analogue of the prolate wave operator. The latter plays a key role both in the spectral realization of the low lying zeros of zeta—using the positive part of its spectrum—and of their ultraviolet behavior—using the Sonin space which corresponds to the negative part of the spectrum. In the archimedean case the prolate operator is the sum of the square of the scaling operator with the grading of orthogonal polynomials, and we show that this formulation extends to the semilocal case. We prove the stability of the semilocal Sonin space under the increase of the finite set of places which govern the semilocal framework and describe their relation with Hilbert spaces of entire functions. Finally, we relate the prolate operator to the metaplectic representation of the double cover of SL (2 , R) with the goal of obtaining (in a forthcoming paper) a second candidate for the semilocal prolate operator. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate Sobolev orthogonality and families of orthogonal polynomials on the triangle as a generalization of the results in Xu, Y. Constr. Approx. 46, 349-434 (2017). We define inner products or pseudo-inner products that contain derivatives up to third-order in the Sobolev space on a triangle and study associated orthogonal polynomials that are analogs of the Jacobi polynomials J k , n α , β , γ but with α , β , γ being - 1 , - 2 , or - 3 , and two other families derived under simultaneous permutations of x , y , 1 - x - y and α , β , γ . [ABSTRACT FROM AUTHOR]

Stochastic models can effectively describe the operating characteristics of power electronic converters with stochastic parameters. However, it is difficult to implement the models in field programmable gate array– (FPGA) based real-time simulation, because their high order leads to a large calculation. This article proposes an optimized real-time stochastic modeling method for power electronic converters based on generalized polynomial chaos. First, an orthogonal polynomials construction method is used based on Schmidt orthogonalization to describe stochastic variables with atypical probability distributions and provide conditions for simplifying the system model. Second, the method of probability space transformation is adopted to divide the system model into multiple sub-models to suppress the exponential growth of the model order while maintaining the statistical properties. This method has performed over the traditional stochastic modeling method. The proposed model is built on FPGA-based hardware-in-the-loop experiment platform with 1us simulation step. The optimized model uses approximately 37% fewer resources than the traditional stochastic model while maintaining the same level of accuracy. [ABSTRACT FROM AUTHOR]

The problems of constructing an exact or approximate solution of system of singular integro‐differential equations related to the problems of adhesive interaction between elastic thin finite or infinite nonhomogeneous patch and elastic plate are investigated. For the patch loaded with horizontal and vertical forces, the usual model of beam bending in combination with the uniaxial stress state model is valid. Using the methods of theory of analytic functions, integral transformation, or orthogonal polynomials, the singular integro‐differential equations reduced to the different boundary value problems (Karleman type problem with displacements and Riemann problem) of the theory of analytic functions or to the infinite system of linear algebraic equations. The asymptotic analysis of problem is carried out. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of 2D version special functions generalizing the famous Slepian functions. As a consequence, many desirable spectral properties of the corresponding weighted finite Fourier transform are deduced from the rich literature. In particular, a similar aspect related to Slepian's seminal papers is the investigation of differential operators that commute with appropriate integral operators. Finally, we provide some analytic expressions for the finite Fourier transforms of the disk polynomials and the two variables Gegenbauer polynomials. [ABSTRACT FROM AUTHOR]

We describe the set of all possible masses at a fixed point of the real axis in the context of the most general case of a truncated matricial (possibly degenerate) Hamburger moment problem. This set is a matricial interval, which we explicitly construct from the given data in the non-degenerate and degenerate cases simultaneously. In particular, we determine the maximum possible mass. [ABSTRACT FROM AUTHOR]

In this contribution we deal with sequences of polynomials orthogonal with respect to a Sobolev type inner product. A banded symmetric operator is associated with such a sequence of polynomials according to the higher order difference equation they satisfy. Taking into account the Darboux transformation of the corresponding matrix we deduce the connection with a sequence of orthogonal polynomials associated with a Christoffel perturbation of the measure involved in the standard part of the Sobolev inner product. A connection with matrix orthogonal polynomials is stated. The Laguerre-Sobolev type case is studied as an illustrative example. Finally, the bispectrality of such matrix orthogonal polynomials is pointed out. [ABSTRACT FROM AUTHOR]

In this paper, a generalized version of dynamic asymmetric simple exclusion process (ASEP) is introduced, and it is shown that the process has a Markov duality property with the same process on the reversed lattice. The duality functions are multivariate q -Racah polynomials, and the corresponding orthogonality measure is the reversible measure of the process. By taking limits in the generator of dynamic ASEP, its reversible measure, and the duality functions, we obtain orthogonal and triangular dualities for several other interacting particle systems. In this sense, the duality of dynamic ASEP sits on top of a hierarchy of many dualities. For the construction of the process, we rely on representation theory of the quantum algebra U q (sl 2) . In the standard representation, the generator of generalized ASEP can be constructed from the coproduct of the Casimir. After a suitable change of representation, we obtain the generator of dynamic ASEP. The corresponding intertwiner is constructed from q -Krawtchouk polynomials, which arise as eigenfunctions of twisted primitive elements. This gives a duality between dynamic ASEP and generalized ASEP with q -Krawtchouk polynomials as duality functions. Using this duality, we show the (almost) self-duality of dynamic ASEP. [ABSTRACT FROM AUTHOR]

From the literature, it is known that the choice of basis functions in hp-FEM heavily influences the computational cost in order to obtain an approximate solution. Depending on the choice of the reference element, suitable tensor product like basis functions of Jacobi polynomials with different weights lead to optimal properties due to condition number and sparsity. This paper presents biorthogonal basis functions to the primal basis functions mentioned above. The authors investigate hypercubes and simplices as reference elements, as well as the cases of H1 and H(Curl). The functions can be expressed as sums of tensor products of Jacobi polynomials with maximal two summands. [ABSTRACT FROM AUTHOR]

We investigate the properties of matrices that emerge from the application of Fourier series to Jacobi matrices. Specifically, we focus on functions defined by the coefficients of a Fourier series expressed in orthogonal polynomials. In the operational formulation of integro-differential problems, these infinite matrices play a fundamental role. We have derived precise calculation formulas for their elements, enabling exact computation of these operational matrices. Numerical results illustrate the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

Definition: This survey highlights the significant role of exponential operators and the monomiality principle in the theory of special polynomials. Using operational calculus formalism, we revisited classical and current results corresponding to a broad class of special polynomials. For instance, we explore the 2D Hermite polynomials and their generalizations. We also present an integral representation of Gegenbauer polynomials in terms of Gould–Hopper polynomials, establishing connections with a simple case of Gegenbauer–Sobolev orthogonality. The monomiality principle is examined, emphasizing its utility in simplifying the algebraic and differential properties of several special polynomial families. This principle provides a powerful tool for deriving properties and applications of such polynomials. Additionally, we review advancements over the past 25 years, showcasing the evolution and extensive applicability of this operational formalism in understanding and manipulating special polynomial families. [ABSTRACT FROM AUTHOR]

In the present work, we investigate certain algebraic and differential properties of the orthogonal polynomials with respect to a discrete–continuous Sobolev-type inner product defined in terms of the Jacobi measure. [ABSTRACT FROM AUTHOR]

This paper will be devoted to the study of weighted (deformed) free Poisson random variables from the viewpoint of orthogonal polynomials and statistics of non-crossing partitions. A family of weighted (deformed) free Poisson random variables will be defined in a sense by the sum of weighted (deformed) free creation, annihilation, scalar, and intermediate operators with certain parameters on a weighted (deformed) free Fock space together with the vacuum expectation. We shall provide a combinatorial moment formula of non-commutative Poisson random variables. This formula gives us a very nice combinatorial interpretation to two parameters of weights. One can see that the deformation treated in this paper interpolates free and boolean Poisson random variables, their distributions and moments, and yields some conditionally free Poisson distribution by taking limit of the parameter. [ABSTRACT FROM AUTHOR]

Split-step quantum walk operators can be expressed as a generalized version of CMV operators with complex transmission coefficients, which we call rotated CMV operators. Following the idea of Cantero, Moral and Velazquez's original construction of the original CMV operators from the theory of orthogonal polynomials on the unit circle (OPUC), we show that rotated CMV operators can be constructed similarly via a rotated version of OPUCs with respect to the same measure, and admit an analogous $ \mathcal {LM} $ LM -factorization as the original CMV operators. We also develop the rotated second kind polynomials corresponding to the rotated OPUCs. We then use the $ \mathcal {LM} $ LM -factorization of rotated alternate CMV operators to compute the Gesztesy–Zinchenko transfer matrices for rotated CMV operators. [ABSTRACT FROM AUTHOR]

In this paper, we present four characterizations of q-Dunkl-classical orthogonal polynomials. The first one is a $ T_{\theta,q} $ T θ , q -distributional equation of Pearson type fulfilled by its associated form, where $ T_{\theta,q} $ T θ , q is the q-Dunkl operator. The second is a second order linear $ T_{\theta,q} $ T θ , q -difference equation. The third is a first order linear $ T_{\theta,q} $ T θ , q -difference equation with polynomial coefficients satisfied by the corresponding Stieltjes function. The fourth is the so-called structure relation fulfilled by q-Dunkl-classical polynomials. Moreover, we provide an example of a non-symmetric sequence of $ T_{\theta,q} $ T θ , q -classical orthogonal polynomials. [ABSTRACT FROM AUTHOR]

Recently, the author determined the higher-order differential operator having the Jacobi-Sobolev polynomials as its eigenfunctions for certain eigenvalues. These polynomials form an orthogonal system with respect to an inner product equipped with the Jacobi measure on the interval $ [-1,1] $ [ − 1 , 1 ] with parameters $ \alpha \in \mathbb {N}_{0},\beta \gt -1 $ α ∈ N 0 , β > − 1 and two point masses N, S>0 at the right end point of the interval involving functions and their first derivatives. The first purpose of the present paper is to reveal how the Jacobi-Sobolev equation reduces to the differential equation satisfied by the Laguerre-Sobolev polynomials on the positive half line via a confluent limiting process as $ \beta \rightarrow \infty $ β → ∞. Secondly, we explicitly establish the differential equation for the symmetric Gegenbauer-Sobolev polynomials by employing a quadratic transformation of the argument. Each of the three differential operators involved is of order $ 4\alpha +10 $ 4 α + 10 and symmetric with respect to the corresponding Sobolev inner product. [ABSTRACT FROM AUTHOR]

We study the monic orthogonal polynomials with respect to a singularly perturbed Airy weight. By using Chen and Ismail’s ladder operator approach, we derive a discrete system satisfied by the recurrence coefficients for the orthogonal polynomials. We find that the orthogonal polynomials satisfy a second-order linear ordinary differential equation, whose coefficients are all expressed in terms of the recurrence coefficients. By considering the time evolution, we obtain a system of differential-difference equations satisfied by the recurrence coefficients. Finally, we study the asymptotics of the recurrence coefficients when the degrees of the orthogonal polynomials tend to infinity. [ABSTRACT FROM AUTHOR]

Trajectory planning in cable-driven robots is more challenging than rigidlink ones. To maintain the robot control, the cable tensions must be positive during motion. This paper presents a direct collocation approach to solve the optimal trajectory planning based on the minimization of a robot's tension and tension-rate objective functions. Besides, during robot motion, the cables must be tensile. The configuration of a cable parallel robot composed of a 3-cable and a prismatic actuator neutralizes the moving platform's weight while improving tensionability. To generate smooth trajectories, the proposed method is compared with two standard approaches: GPOPS-II software package which uses Legendre-Gauss-Radu quadrature orthogonal collocation polynomials and direct collocation by using Bspline interpolation curves. Despite the efficiency of using B-spline functions in trajectory planning, numerical simulations demonstrate that the Hermite-Simpson direct collocation approach has a substantial benefit in the computation cost and accuracy for trajectory planning of a cable-driven parallel robot. Also, by choosing appropriate constraints and cost functions, the cable forces in the parallel robot can be well managed. [ABSTRACT FROM AUTHOR]

As more and more numerical and analytical solutions to the linear neutron transport equation become available, verification of the numerical results becomes increasingly important. This presentation concerns the development of another benchmark for the linear neutron transport equation in a benchmark series, each employing a different method of solution. In 1D, there are numerous ways of analytically solving the monoenergetic transport equation, such as the Wiener–Hopf method, based on the analyticity of the solution, the method of singular eigenfunctions, inversion of the Laplace and Fourier transform solutions, and analytical discrete ordinates in the limit, which is arguably one of the most straightforward, to name a few. Another potential method is the PN (Legendre polynomial order N) method, where one expands the solution in terms of full-range orthogonal Legendre polynomials, and with orthogonality and series truncation, the moments form an open set of first-order ODEs. Because of the half-range boundary conditions for incoming particles, however, full-range Legendre expansions are inaccurate near material discontinuities. For this reason, a double PN (DPN) expansion in half-range Legendre polynomials is more appropriate, where one separately expands incoming and exiting flux distributions to preserve the discontinuity at material interfaces. Here, we propose and demonstrate a new method of solution for the DPN equations for an isotropically scattering medium. In comparison to a well-established fully analytical response matrix/discrete ordinate solution (RM/DOM) benchmark using an entirely different method of solution for a non-absorbing 1 mfp thick slab with both isotropic and beam sources, the DPN algorithm achieves nearly 8- and 7-place precision, respectively. [ABSTRACT FROM AUTHOR]

This paper addresses the challenge of solving Fokker–Planck equations, which are prevalent mathematical models across a myriad of scientific fields. Due to factors like fractional-order derivatives and non-linearities, obtaining exact solutions to this problem can be complex. To overcome these challenges, our framework first discretizes the given equation using the Crank-Nicolson finite difference method, transforming it into a system of ordinary differential equations. Here, the approximation of time dynamics is done using forward difference or an L1 discretization technique for integer or fractional-order derivatives, respectively. Subsequently, these ordinary differential equations are solved using a novel strategy based on a kernel-based machine learning algorithm, named collocation least-squares support vector regression. The effectiveness of the proposed approach is demonstrated through multiple numerical experiments, highlighting its accuracy and efficiency. This performance establishes its potential as a valuable tool for tackling Fokker–Planck equations in diverse applications. [ABSTRACT FROM AUTHOR]

This paper presents a new approach for the unified Chebyshev polynomials (UCPs). It is first necessary to introduce the three basic formulas of these polynomials, namely analytic form, moments, and inversion formulas, which will later be utilized to derive further formulas of the UCPs. We will prove the basic formula that shows that these polynomials can be expressed as a combination of three consecutive terms of Chebyshev polynomials (CPs) of the second kind. New derivatives and connection formulas between two different classes of the UCPs are established. Some other expressions of the derivatives of UCPs are given in terms of other orthogonal and non-orthogonal polynomials. The UCPs are also the basis for additional derivative expressions of well-known polynomials. A new linearization formula (LF) of the UCPs that generalizes some well-known formulas is given in a simplified form where no hypergeometric forms are present. Other product formulas of the UCPs with various polynomials are also given. As an application to some of the derived formulas, some definite and weighted definite integrals are computed in closed forms.

In this paper, the vibration characteristics (free vibration and forced vibration) of fiber-reinforced thin-walled conical–cylindrical composite shells (FTCCS) are analyzed by combining theory and experiment. Based on the classical shell theory and Kirchhoff–Love assumption, the overall structure of the FTCCS is theoretically modeled. The artificial spring technology is used to simulate the arbitrary boundary conditions at the joint, which is divided into a main spring and a secondary spring. At the same time, the displacement functions of the two sub-structure shells of the FTCCS are established by the orthogonal polynomial method, and the energy function of the FTCCS is constructed according to the continuity condition of the bolt joint. Then, the vibration characteristics of the FTCCS are solved using the Rayleigh–Ritz method and the modal superposition method. Finally, the TC300/epoxy resin-based fiber-reinforced thin-walled conical–cylindrical composite shells are selected as the research object, and the rationality of the mathematical model is verified by pulse, frequency sweeping and resonance excitation tests. The experimental results show that the errors between the natural frequency and the resonance displacement calculated by the theory and the experimental results are 1.75–4.76% and 3.79–9.28%, which confirms the rationality of the proposed model. By changing the length of the shell, the lamination angle and the semi-cone angle of the conical shell, the vibration characteristics parameters under different conditions are calculated to evaluate the influence of three physical parameters on the vibration characteristics of the FTCCS. [ABSTRACT FROM AUTHOR]

In this article, we use the notion of generalized orthogonality for a sequence of polynomials introduced by Bryc, Fakhfakh, and Mlotkowski (2019) to extend the characterizations of the Feinsilver, Meixner, and Shanbhag based on orthogonal polynomials. These new versions subsume the real natural exponential families (NEFs) having polynomial variance function in the mean of arbitrary degree. We also relate generalized orthogonality to Sheffer systems. We show that the generalized orthogonality of Sheffer systems occurs if and only if the associated classical additive convolution semigroup of probability measures generates NEFs with polynomial variance function. In addition, we use the raising and lowering operators for quasi-monomial polynomials associated with NEFs to give a characterization of NEFs with polynomial variance function of arbitrary degree. [ABSTRACT FROM AUTHOR]

Boron (B) is an essential micro nutrient that is needed by oil palms, especially to control productivity. The aim of this research was to determine the optimal dose of B for mature oil palms. The research was conducted at a smallholder oil palm plantation located in Katingan Region, Central Kalimantan Province from January to December 2022. The field experiment was a single factor arranged in a Randomized Complete Block Design (RCBD) with three blocks as replications. The factor tested was the dose of B fertilization, consisting of five doses, namely 0 g. trunk-1. semester-1 (control); 25 g. trunk-1. semester-1; 50 g. trunk-1. semester-1; 75 g. trunk-1. semester-1; and 100 g. trunk-1. semester-1. Observations were done on several variables of micro-weather characteristics at the research site, nutrient and physiological characteristics of leaf, morphological characters and plant growth, and yield and yield components of oil palms. The data obtained were then analyzed with analysis of variance (ANOVA) a=5 %, and data showing significant differences between treatments were tested with an orthogonal polynomial. The results showed that B had positive effects on leaf B, P and K concentration and absorption, leaf chlorophyll content, number of midribs, plant height, leaf area per trunk, leaf area index, crop dry weight, pollen fertility, fruit set, pollen viability, fresh fruit bunch weight (FFB) and FFB productivity. The optimal dose of B to optimize oil palms productivity was 60.24 g. trunk-1. semester-1 with a maximum value of FFB productivity of 6.94 tons. trunk-1. semester? [ABSTRACT FROM AUTHOR]

We call a translation function class a set of zonal translation networks if it is produced by the translations of a convolutional structure, which is a composite function reunited with an even function defined on $ [-1, 1] $ and a geodesic distance. In the present paper, we consider the density of a class of zonal translation networks produced by a convolution structure on the unit ball from the view of Fourier-Laplace series and give a sufficient condition to ensure the zonal translation class is density in $ L^p $ spaces defined on the unit ball. In particular, we construct with De La Vallée Poussin operators a sequence of zonal translation networks and show the approximation order. [ABSTRACT FROM AUTHOR]

We study orthogonal polynomials (OPs) for a weight function defined over a domain of revolution, where the domain is formed from rotating a two‐dimensional region and goes beyond the quadratic domains. Explicit constructions of orthogonal bases are provided for weight functions on a number of domains. Particular attention is paid to the setting when an orthogonal basis can be constructed explicitly in terms of known polynomials of either one or two variables. Several new families of OPs are derived, including a few families that are eigenfunctions of a spectral operator and their reproducing kernels satisfy an addition formula. [ABSTRACT FROM AUTHOR]

Exceptional orthogonal Hermite polynomials have been linked to the k-step extension of the harmonic oscillator. The exceptional polynomials allow the existence of different supercharges in the Darboux–Crum and Krein–Adler constructions. They also allow the existence of different types of ladder relations and their associated recurrence relations. The existence of such relations is a unique property of these polynomials. Those relations have been used to construct 2D models which are superintegrable and display an interesting spectrum, degeneracies and finite-dimensional unitary representations. In previous works, only the physical or polynomial part of the spectrum was discussed. It is known that the general solutions are associated with other types of recurrence/ladder relations. We discuss in detail the case of the exceptional Hermite polynomials X 2 (1) and explicitly present new chains obtained by acting with different types of ladder operators. We exploit a recent result by one of the authors (Chalifour and Grundland in Ann Henri Poincaré 21:3341, 2020), where the general analytic solution was constructed and connected with the non-degenerate confluent Heun equation. The analogue Rodrigues formulas for the general solution are constructed. The set of finite states from which the other states can be obtained algebraically is not unique, but the vanishing arrow and diagonal arrow from the diagram of the 2-chain representations can be used to obtain minimal sets. These Rodrigues formulas are then exploited, not only to construct the states, polynomial and non-polynomial, in a purely algebraic way, but also to obtain coefficients from the action of ladder operators in an algebraic manner based on further commutation relations between monomials in the generators. [ABSTRACT FROM AUTHOR]

In this article, a new operational matrix of fractional integration of Hermite polynomials is derived to solve multi-order linear fractional differential equations (FDEs) with spectral tau approach. We firstly convert the FDEs into an integrated-form through multiple fractional integration in association with the Riemann-Liouville sense. This integral equation is then formulated as an algebraic equation system with Hermite polynomials. Finally, linear multi-order FDEs with initial conditions are solved with this method. We present exact and approximated solutions for a number of representative examples. Numerical results indicate that the proposed method provides a high degree of accuracy to solve the linear multi-order FDEs. [ABSTRACT FROM AUTHOR]

An extended two‐dimensional (2D) elasticity theory based on the weighted residual concept is developed in an attempt to capture thickness effects in vibrating thick plates. Plate theories can only provide approximate solutions due to certain kinematics assumptions. The weighting functions are expressed in terms of Legendre orthogonal polynomials. A formulation in rectangular Cartesian coordinates serves as the basis for an implementation of the finite element method. The applicability of the extended 2D elasticity theory is demonstrated through examples of rectangular, thick plates with different aspect ratios, thickness ratios, and boundary conditions. The accuracy of the extended 2D elasticity theory is assessed through a three‐dimensional finite element analysis. The frequencies and modal stresses show excellent agreement with three‐dimensional finite element calculations. [ABSTRACT FROM AUTHOR]

Fish feed production is fraught with high costs due to the inclusion of expensive fishmeal and animal proteins that can be sourced from aquatic macrophytes. Limited attempts have been made on use of the latter for fish feed production. Further, the quality of feed given to fish is known to affect its growth and reproductive performance. Role of feeds containing duckweed (Lemna minor) at 0%‐control feed, 10%, 15%, 20% and 25% inclusion levels on growth and reproductive performance of Oreochromis niloticus of size 18 ± 1 g were evaluated for 12 weeks. The fish were fed twice daily at 10% body weight at 9.00 a.m. and 4.00 p.m. Length–weight measurements were done fortnightly using a measuring board and a weighing balance, respectively. Female mouth‐brooding fish were used to evaluate reproductive performance indicators, namely %: fertilization, hatchability and survivability. Data were subjected to one‐way analysis of variance followed by post hoc and polynomial orthogonal analysis to identify L. minor diets with significant differences (p < 0.05). Fish fed on a diet containing 10% L. minor inclusion showed significantly better growth performance and feed conversion ratio than those fed on the control diet. All L. minor diets gave good fish condition factors above 1.0. Fish fed on a diet containing 10% L. minor and those fed on the control gave reasonably high survival rates of 85.55% and 83.33%, respectively, whereas those fed on 20% L. minor produced the same growth performance as control. Orthogonal polynomial analysis for the final weights across the L. minor diets – 0%–25% – showed a cubic polynomial model (p = 0.000), whereas final lengths portrayed a linear inverse significant relationship (p < 0.05). Inclusion of L. minor in the diets resulted to slightly better fertilization, hatchability and survivability rates at 10%, 20% and 15%, respectively. In conclusion, inclusion of L. minor from 10% to 20% in fish feeds is recommended for enhancing growth and reproductive performance of O. niloticus. [ABSTRACT FROM AUTHOR]

In this article, new results are investigated in the context of the recently introduced Abu-Shady–Kaabar fractional derivative. First, we solve the generalized Legendre fractional differential equation. As in the classical case, the generalized Legendre polynomials constitute notable solutions to the aforementioned fractional differential equation. In the sense of the fractional derivative of Abu-Shady–Kaabar, we establish important properties of the generalized Legendre polynomials such as Rodrigues formula and recurrence relations. Special attention is also devoted to another very important property of Legendre polynomials and their orthogonal character. Finally, the representation of a function f ∈ L α 2 ([ − 1 , 1 ]) in a series of generalized Legendre polynomials is addressed. [ABSTRACT FROM AUTHOR]

This paper presents a new approach for the unified Chebyshev polynomials (UCPs). It is first necessary to introduce the three basic formulas of these polynomials, namely analytic form, moments, and inversion formulas, which will later be utilized to derive further formulas of the UCPs. We will prove the basic formula that shows that these polynomials can be expressed as a combination of three consecutive terms of Chebyshev polynomials (CPs) of the second kind. New derivatives and connection formulas between two different classes of the UCPs are established. Some other expressions of the derivatives of UCPs are given in terms of other orthogonal and non-orthogonal polynomials. The UCPs are also the basis for additional derivative expressions of well-known polynomials. A new linearization formula (LF) of the UCPs that generalizes some well-known formulas is given in a simplified form where no hypergeometric forms are present. Other product formulas of the UCPs with various polynomials are also given. As an application to some of the derived formulas, some definite and weighted definite integrals are computed in closed forms. [ABSTRACT FROM AUTHOR]

Quantum Markov chains (QMCs) are positive maps on a trace-class space describing open quantum dynamics on graphs. Such objects have a statistical resemblance with classical random walks, while at the same time they allow for internal (quantum) degrees of freedom. In this work we study continuous-time QMCs on the integer line, half-line and finite segments, so that we are able to obtain exact probability calculations in terms of the associated matrix-valued orthogonal polynomials and measures. The methods employed here are applicable to a wide range of settings, but we will restrict ourselves to classes of examples for which the Lindblad generators are induced by a single positive map, and such that the Stieltjes transforms of the measures and their inverses can be calculated explicitly. [ABSTRACT FROM AUTHOR]

The present manuscript proposes different modification procedures for the standard Adomian Decomposition Method (ADM). These procedures are based on the application of orthogonal polynomials that play vital parts in approximation theories. Moreover, the study also scrutinizes four nonlinear inhomogeneous initial-value problems, and distinctively examines their respective absolute error differences. Remarkably, different computational benefits of the proposed modification are noted with regard high-level of accuracy and fewer computational steps. [ABSTRACT FROM AUTHOR]

By learning the mappings between infinite function spaces using carefully designed neural networks, the operator learning methodology has exhibited significantly more efficiency than traditional methods in solving differential equations, but faces concerns about their accuracy and reliability. To overcome these limitations through robustly enforcing boundary conditions (BCs), a general neural architecture named spectral operator learning is introduced by combining with the structures of the spectral numerical method. One variant called the orthogonal polynomial neural operator (OPNO) is proposed later, aiming at PDEs with Dirichlet, Neumann, and Robin BCs. The strict BC satisfaction properties and the universal approximation capacity of the OPNO are theoretically proven. A variety of numerical experiments with physical backgrounds demonstrate that the OPNO outperforms other existing deep learning methodologies, showcasing potential of comparable accuracy with the traditional second-order finite difference method that employs a considerably fine mesh (with relative errors on the order of 10–6 ), and is up to almost 5 magnitudes faster over the traditional method. [ABSTRACT FROM AUTHOR]

This research focuses on the development of a streamlined numerical technique founded on the hybridization of two shifted polynomial basis functions to address a specific category of nonlinear Partial Differential Equations. Within this approach, a solution based on power series is employed, utilizing Chebyshev and Legendre shifted polynomials to meet the specific conditions of the Partial Differential Equation. Plugging the candidate solution series into the provided Partial Differential Equation, and employing suitable points of collocation, a linear system of algebraic equations with unspecified hybridization coefficients was obtained and numerically solved by Gaussian elimination. Furthermore, different discretization patterns were examined to comprehend how the outcomes vary with alterations in the placement of the collocation points within the domain. Two instances were examined using the numerical method to determine the method's efficiency in terms of its reliability, effectiveness, and accuracy. The results obtained were benchmarked and validated with existing results in the literature. However, the combination of the two shifted orthogonal polynomials (Chebyshev and Legendre) greatly improved performance past that in prior literature. [ABSTRACT FROM AUTHOR]

The aim of this contribution is to study orthogonal polynomials via cubic decomposition in the framework of the Laguerre-Hahn class. We consider two monic orthogonal polynomial sequences {Wn}n≥0 and {Pn}n≥0, and we let w and u be, respectively, the corresponding regular linear functionals such that W3n(x) = Pn(x³), n ≥ 0. We prove that if either w or u is a Laguerre-Hahn linear functional, then so is the other one. Based on this result, we deduce a complete analysis of the class s of the Laguerre-Hahn linear functional w. More precisely, we show that 3s' s 3s' + 6, where s' is the class of u. An illustrative example of class 1 is analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce multiple skew-orthogonal polynomials and investigate their connections with classical integrable systems. By using Pfaffian techniques, we show that multiple skew-orthogonal polynomials can be expressed by multi-component Pfaffian tau-functions upon appropriate deformations. Moreover, a two-component Pfaff lattice hierarchy, which is equivalent to the Pfaff–Toda hierarchy studied by Takasaki, is obtained by considering the recurrence relations and Cauchy transforms of multiple skew-orthogonal polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we study the two-parameter spectral element method based on weighted shifted orthogonal polynomials for solving singularly perturbed diffusion equation on an interval [0, 1] which are modeled with singular parameters. We continue our study to estimate the lower bound of the weighted orthogonal polynomial coefficient and the upper bound of a posteriori error estimates of the method through different weighted norms to minimize the computational cost. Numerical examples are implemented to study the applicability and efficiency of the technique. The obtained error bounds for the coefficient of orthogonal polynomials and the posteriori estimates fall within the bounds derived in the theoretical section. It is also observed that the two weighted norms decreases when the values of N 1 and N 2 increases for the three choices of and for different values of x and y. The quality and accuracy of the solution can be realized through figures and tables. [ABSTRACT FROM AUTHOR]