Showing papers on "Discrete orthogonal polynomials published in 2022"
TL;DR: Krylov complexity measures operator growth with respect to a basis, which is adapted to the Heisenberg time evolution as discussed by the authors , and it relies on the Lanczos algorithm, also known as the recursion method.
24 citations
8 citations
8 citations
TL;DR: In this article , the algebraic structure of the rank two Racah algebra is studied in detail, and an automorphism group of this algebra is provided, which is isomorphic to the permutation group of five elements.
7 citations
TL;DR: The symmetric additive and multiplicative convolutions of polynomials were introduced by Walsh and Szegö in different contexts, and have been studied for a century as mentioned in this paper .
Abstract: We study three convolutions of polynomials in the context of free probability theory. We prove that these convolutions can be written as the expected characteristic polynomials of sums and products of unitarily invariant random matrices. The symmetric additive and multiplicative convolutions were introduced by Walsh and Szegö in different contexts, and have been studied for a century. The asymmetric additive convolution, and the connection of all of them with random matrices, is new. By developing the analogy with free probability, we prove that these convolutions produce real rooted polynomials and provide strong bounds on the locations of the roots of these polynomials.
6 citations
TL;DR: In this paper , a unified class of Apostol-Genocchi polynomials based on poly-Daehee polynomial variations and their extensions have been explored extensively and found applications in a variety of research fields.
Abstract: Numerous polynomial variations and their extensions have been explored extensively and found applications in a variety of research fields. The purpose of this research is to establish a unified class of Apostol–Genocchi polynomials based on poly-Daehee polynomials and to explore some of their features and identities. We investigate these polynomials via generating functions and deduce various identities, summation formulae, differential and integral formulas, implicit summation formulae, and several characterized generating functions for new numbers and polynomials. Finally, by using an operational version of Apostol–Genocchi polynomials, we derive some results in terms of new special polynomials. Due to the generic nature of the findings described here, they are used to reduce and generate certain known or novel formulae and identities for relatively simple polynomials and numbers.
5 citations
TL;DR: In this article , a unified family of Legendre-based generalized Apostol-Bernoulli, Euler, and Genocchi polynomials with appropriate constraints for the Maclaurin series is presented.
Abstract: Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate constraints for the Maclaurin series. Then we look at the formulae and identities that are involved, including an integral formula, differential formulas, addition formulas, implicit summation formulas, and general symmetry identities. We also provide an explicit representation for these new polynomials. Due to the generality of the findings given here, various formulae and identities for relatively simple polynomials and numbers, such as generalized Bernoulli, Euler, and Genocchi numbers and polynomials, are indicated to be deducible. Furthermore, we employ the umbral calculus theory to offer some additional formulae for these new polynomials.
5 citations
TL;DR: A unified presentation of a class of Humbert-Hermite polynomials in two variables is given in this paper , which generalizes the well known class of Gegenbauer, Humberts, Legendre, Chebycheff, Pincherle, Horadam, Kinney, Horador, Pethe, Horadi, Pithers, and Pathan and Khan.
Abstract: Version: 11.03.2022 Abstract: A unified presentation of a class of Humbert’s polynomials in two variables which generalizes the well known class of Gegenbauer, Humbert, Legendre, Chebycheff, Pincherle, Horadam, Kinney, Horadam–Pethe, Djordjević, Gould, Milovanovi ´ c and Djordjevi ´ c , Pathan and Khan polynomials and many not so called ’named’ polynomials has inspired the present paper. We define here generalized Humbert–Hermite polynomials of two variables. Several expansions of Humbert-Hermite polynomials, Hermite–Gegenbaurer (or ultraspherical) polynomials and Hermite–Chebyshev polynomials
5 citations
TL;DR: In this paper , the authors investigated model order reduction (MOR) methods of discrete-time systems via discrete orthogonal polynomials in the time domain, which can match the first several expansion coefficients of the original output.
Abstract: This paper investigates model order reduction (MOR) methods of discrete-time systems via discrete orthogonal polynomials in the time domain. First, this system is expanded under discrete orthogonal polynomials, and the expansion coefficients are computed from a linear equation. Then the reduced-order system is produced by using the orthogonal projection matrix that is defined in terms of the expansion coefficients, which can match the first several expansion coefficients of the original output, and preserve the asymptotic stability and bounded-input/bounded-output (BIBO) stability. We also study the output error between the original system and its reduced-order system. Besides, the MOR method using discrete Walsh functions is proposed for discrete-time systems with inhomogeneous initial conditions. Finally, three numerical examples are given to illustrate the feasibility of the proposed methods.
4 citations
TL;DR: In this article , a degenerate version of type 2 Bernoulli polynomials and numbers was proposed by modifying a generating function, and explicit expressions and their representations were derived.
Abstract: Research on the degenerate versions of special polynomials provides a new area, introducing the $ \lambda $-analogue of special polynomials and numbers, such as $ \lambda $-Sheffer polynomials. In this paper, we propose a new variant of type 2 Bernoulli polynomials and numbers by modifying a generating function. Then we derive explicit expressions and their representations that provide connections among existing $ \lambda $-Sheffer polynomials. Also, we provide the explicit representations of the proposed polynomials in terms of the degenerate Lah-Bell polynomials and the higher-order degenerate derangement polynomials to confirm the presented identities.
4 citations
TL;DR: In this article , a new type of degenerate Changhee-genocchi numbers and polynomials which are different from those previously introduced by Kim et al. are investigated.
Abstract: A remarkably large number of polynomials and their extensions have been presented and studied. In this paper, we consider a new type of degenerate Changhee–Genocchi numbers and polynomials which are different from those previously introduced by Kim et al. (J. Ineq. Appl. 294, 2017). We investigate some properties of these numbers and polynomials. We also introduce a higher-order new type of degenerate Changhee–Genocchi numbers and polynomials which can be represented in terms of the degenerate logarithm function. Finally, we derive their summation formulae.
TL;DR: In this paper , the authors investigate symmetric q-differential equations of a higher order by applying symmetric properties of q-Euler polynomials and q-Genocchi polynomorphisms.
Abstract: One finds several q-differential equations of a higher order for q-Euler polynomials and q-Genocchi polynomials. Additionally, we have a few q-differential equations of a higher order, which are mixed with q-Euler numbers and q-Genocchi polynomials. Moreover, we investigate some symmetric q-differential equations of a higher order by applying symmetric properties of q-Euler polynomials and q-Genocchi polynomials.
TL;DR: In this article , the Jacobi-Piñeiro multiple orthogonal polynomials with respect to two weights on the step-line are considered, and a connection between different dual spectral matrices, one banded (recursion matrix) and one Hessenberg, respectively, and the Gauss-Borel factorization of the moment matrix is given.
Abstract: Abstract Multiple orthogonal polynomials with respect to two weights on the step-line are considered. A connection between different dual spectral matrices, one banded (recursion matrix) and one Hessenberg, respectively, and the Gauss–Borel factorization of the moment matrix is given. It is shown a hidden freedom exhibited by the spectral system related to the multiple orthogonal polynomials. Pearson equations are discussed, a Laguerre–Freud matrix is considered, and differential equations for type I and II multiple orthogonal polynomials, as well as for the corresponding linear forms are given. The Jacobi–Piñeiro multiple orthogonal polynomials of type I and type II are used as an illustrating case and the corresponding differential relations are presented. A permuting Christoffel transformation is discussed, finding the connection between the different families of multiple orthogonal polynomials. The Jacobi–Piñeiro case provides a convenient illustration of these symmetries, giving linear relations between different polynomials with shifted and permuted parameters. We also present the general theory for the perturbation of each weight by a different polynomial or rational function aka called Christoffel and Geronimus transformations. The connections formulas between the type II multiple orthogonal polynomials, the type I linear forms, as well as the vector Stieltjes–Markov vector functions is also presented. We illustrate these findings by analyzing the special case of modification by an even polynomial.
TL;DR: In this article , the authors give generating functions for parametrically generalized polynomials that are related to the combinatorial numbers, the Bernoulli polynomial, the Euler polynomorphism, the cosine-Bernoulli, the sine-Euler, and the Sine Euler.
Abstract: The aim of this paper is to give generating functions for parametrically generalized polynomials that are related to the combinatorial numbers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the cosine-Bernoulli polynomials, the sine-Bernoulli polynomials, the cosine-Euler polynomials, and the sine-Euler polynomials. We investigate some properties of these generating functions. By applying Euler’s formula to these generating functions, we derive many new and interesting formulas and relations related to these special polynomials and numbers mentioned as above. Some special cases of the results obtained in this article are examined. With this special case, detailed comments and comparisons with previously available results are also provided. Furthermore, we come up with open questions about interpolation functions for these polynomials. The main results of this paper highlight the existing symmetry between numbers and polynomials in a more general framework. These include Bernouilli, Euler, and Catalan polynomials.
TL;DR: In this paper , a generating function for mix type Apostol-genocchi polynomials of order η associated with Bell polynomial is provided. But it is not shown how to generate symmetric identities.
Abstract: In this paper, we provide a generating function for mix type Apostol–Genocchi polynomials of order η associated with Bell polynomials. We also derive certain important identities of Apostol Genocchi polynomials of order η based on Bell polynomials, such as the correlation formula, the implicit summation formula, the derivative formula, some correlation with Stirling numbers and their special instances. Moreover, we discover some symmetric identities and their related known results.
05 Jan 2022
TL;DR: In this paper , a quasisymmetric analogue of the Macdonald polynomials is introduced, which is based on the compact "multiline queue" formula for the modified and integral form.
Abstract: We present several new and compact formulas for the modified and integral form of the Macdonald polynomials, building on the compact “multiline queue” formula for Macdonald polynomials due to Corteel, Mandelshtam, and Williams. We also introduce a new quasisymmetric analogue of Macdonald polynomials. These “quasisymmetric Macdonald polynomials" refine the (symmetric) Macdonald polynomials and specialize to the quasisymmetric Schur polynomials defined by Haglund, Luoto, Mason, and van Willigenburg.
TL;DR: In this article , the authors investigate theoretically a kind of orthogonal polynomials, namely, generalized Jacobi poynomials (GJPs), and derive the moment and high-order derivative formulas of the GJPs.
Abstract: The main goal of this article is to investigate theoretically a kind of orthogonal polynomials, namely, generalized Jacobi polynomials (GJPs). These polynomials can be expressed as certain combinations of Legendre polynomials. Some basic formulas of these polynomials such as the power form representation and inversion formula of these polynomials are first introduced, and after that, some interesting formulas concerned with these polynomials are established. The formula of the derivatives of the moments of these polynomials is developed. As special cases of this formula, the moment and high-order derivative formulas of the GJPs are deduced. New expressions for the high-order derivatives of the GJPs, but in terms of different symmetric and non-symmetric polynomials, are also established. These expressions lead to some interesting connection formulas between the GJPs and some various polynomials.
TL;DR: In this article , the authors define degenerate Genocchi polynomials and numbers of the second kind by using logarithmic functions, and investigate some of their analytical properties and some applications.
Abstract: The main aim of this study is to define degenerate Genocchi polynomials and numbers of the second kind by using logarithmic functions, and to investigate some of their analytical properties and some applications. For this purpose, many formulas and relations for these polynomials, including some implicit summation formulas, differentiation rules and correlations with the earlier polynomials by utilizing some series manipulation methods, are derived. Additionally, as an application, the zero values of degenerate Genocchi polynomials and numbers of the second kind are presented in tables and multifarious graphical representations for these zero values are shown.
TL;DR: In this paper , the authors considered a novel kinds of (p,q)-extensions of geometric polynomials and acquired several properties and identities by making use of some series manipulation methods.
Abstract: Utilizing p,q-numbers and p,q-concepts, in 2016, Duran et al. considered p,q-Genocchi numbers and polynomials, p,q-Bernoulli numbers and polynomials and p,q-Euler polynomials and numbers and provided multifarious formulas and properties for these polynomials. Inspired and motivated by this consideration, many authors have introduced (p,q)-special polynomials and numbers and have described some of their properties and applications. In this paper, using the (p,q)-cosine polynomials and (p,q)-sine polynomials, we consider a novel kinds of (p,q)-extensions of geometric polynomials and acquire several properties and identities by making use of some series manipulation methods. Furthermore, we compute the p,q-integral representations and p,q-derivative operator rules for the new polynomials. Additionally, we determine the movements of the approximate zerosof the two mentioned polynomials in a complex plane, utilizing the Newton method, and we illustrate them using figures.
TL;DR: In this article , the degenerate generalized Apostol-Bernoulli, Euler, and Genocchi polynomials of order α and level α in the variable x were studied.
Abstract: The aim of this paper is to study new classes of degenerated generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order $\alpha$ and level $m$ in the variable $x$. Here the degenerate polynomials are a natural extension of the classic polynomials. In more detail, we derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. Most of the results are proved by using generating function methods.
TL;DR: The main objective of as discussed by the authors is to deduce some interesting algebraic relationships that connect the degenerated generalized generalized Apostol-Bernoulli, Euler and Genocchi polynomials.
Abstract: The main objective of this work is to deduce some interesting algebraic relationships that connect the degenerated generalized Apostol–Bernoulli, Apostol–Euler and Apostol– Genocchi polynomials and other families of polynomials such as the generalized Bernoulli polynomials of level m and the Genocchi polynomials. Futher, find new recurrence formulas for these three families of polynomials to study.
TL;DR: In this paper , a generalized 2D extension of the q-Bessel polynomials is introduced, and the generating equation, series expansion and determinant form for the generalized 2-D q-bessel poynomials are established.
TL;DR: In this paper , the authors show how to construct exceptional orthogonal polynomials (XOP) using isospectral deformations of classical orthogonality.
Abstract: In this paper, we show how to construct exceptional orthogonal polynomials (XOP) using isospectral deformations of classical orthogonal polynomials. The construction is based on confluent Darboux transformations, where repeated factorizations at the same eigenvalue are allowed. These factorizations allow us to construct Sturm–Liouville problems with polynomial eigenfunctions that have an arbitrary number of real-valued parameters. We illustrate this new construction by exhibiting the class of deformed Gegenbauer polynomials, which are XOP families that are isospectral deformations of classical Gegenbauer polynomials.
TL;DR: In this paper , Fourier transform of multivariate orthogonal polynomials on the simplex is presented and several recurrence relations are derived using the Parseval's identity.
Abstract: Abstract In this paper, Fourier transform of multivariate orthogonal polynomials on the simplex is presented. A new family of multivariate orthogonal functions is obtained using the Parseval’s identity and several recurrence relations are derived.
TL;DR: In this article , the derivatives of Jacobi polynomials of the third-kind Chebyshev polynomial are derived and the moments of the moments are derived.
Abstract: This paper investigates certain Jacobi polynomials that involve one parameter and generalize the well-known orthogonal polynomials called Chebyshev polynomials of the third-kind. Some new formulas are developed for these polynomials. We will show that some of the previous results in the literature can be considered special ones of our derived formulas. The derivatives of the moments of these polynomials are derived. Hence, two important formulas that explicitly give the derivatives and the moments of these polynomials in terms of their original ones can be deduced as special cases. Some new expressions for the derivatives of different symmetric and non-symmetric polynomials are expressed as combinations of the generalized third-kind Chebyshev polynomials. Some new linearization formulas are also given using different approaches. Some of the appearing coefficients in derivatives and linearization formulas are given in terms of different hypergeometric functions. Furthermore, in several cases, the existing hypergeometric functions can be summed using some standard formulas in the literature or through the employment of suitable symbolic algebra, in particular, Zeilberger’s algorithm.
TL;DR: In this paper , a generalization of Laurent biorthogonal polynomials is proposed and its recurrence relation and Christoffel transformation are derived, which yields an extension of the fully discrete relativistic Toda lattice, one of which can reduce to the Narita-Itoh-Bogoyavlensky lattice.
Abstract: This paper is concerned about certain generalization of Laurent biorthogonal polynomials together with the corresponding related integrable lattices. On one hand, a generalization for Laurent biorthogonal polynomials is proposed and its recurrence relation and Christoffel transformation are derived. On the other hand, it turns out the compatibility condition between the recurrence relation and the Christoffel transformation for the generalized Laurent biorthogonal polynomials yields an extension of the fully discrete relativistic Toda lattice. And also, it is shown that isospectral deformations of the generalized Laurent biorthogonal polynomials lead to two different generalizations of the continuous-time relativistic Toda lattice, one of which can reduce to the Narita–Itoh–Bogoyavlensky lattice.
TL;DR: An extension of the Laplace transform obtained by using the Laguerre-type exponentials is first shown in this article , and the solution of the Blissard problem by means of the Bell polynomials gives the possibility to associate to any numerical sequence a Laplace-type transform depending on that sequence.
Abstract: An extension of the Laplace transform obtained by using the Laguerre-type exponentials is first shown. Furthermore, the solution of the Blissard problem by means of the Bell polynomials gives the possibility to associate to any numerical sequence a Laplace-type transform depending on that sequence. Computational techniques for the corresponding transform of analytic functions, involving Bell polynomials, are derived.