Editorial

3 pages, 192 KiB

Open AccessEditorial

Polynomial Sequences and Their Applications

by Francesco Aldo Costabile, Maria Italia Gualtieri and Anna Napoli

Mathematics 2022, 10(24), 4804; https://doi.org/10.3390/math10244804 - 16 Dec 2022

Cited by 2 | Viewed by 1399

The purpose of this Special Issue is to present, albeit partially, the state of the art on the theory and application of polynomial sequences [...] Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

Research

Jump to: Editorial, Review

18 pages, 1503 KiB

Open AccessArticle

A Matlab Toolbox for Extended Dynamic Mode Decomposition Based on Orthogonal Polynomials and p-q Quasi-Norm Order Reduction

by Camilo Garcia-Tenorio and Alain Vande Wouwer

Mathematics 2022, 10(20), 3859; https://doi.org/10.3390/math10203859 - 18 Oct 2022

Cited by 1 | Viewed by 3416

Abstract

Extended Dynamic Mode Decomposition (EDMD) allows an approximation of the Koopman operator to be derived in the form of a truncated (finite dimensional) linear operator in a lifted space of (nonlinear) observable functions. EDMD can operate in a purely data-driven way using either [...] Read more.

Extended Dynamic Mode Decomposition (EDMD) allows an approximation of the Koopman operator to be derived in the form of a truncated (finite dimensional) linear operator in a lifted space of (nonlinear) observable functions. EDMD can operate in a purely data-driven way using either data generated by a numerical simulator of arbitrary complexity or actual experimental data. An important question at this stage is the selection of basis functions to construct the observable functions, which in turn is determinant of the sparsity and efficiency of the approximation. In this study, attention is focused on orthogonal polynomial expansions and an order-reduction procedure called p-q quasi-norm reduction. The objective of this article is to present a Matlab library to automate the computation of the EDMD based on the above-mentioned tools and to illustrate the performance of this library with a few representative examples. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

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21 pages, 617 KiB

Open AccessArticle

Rate of Weighted Statistical Convergence for Generalized Blending-Type Bernstein-Kantorovich Operators

by Faruk Özger, Ekrem Aljimi and Merve Temizer Ersoy

Mathematics 2022, 10(12), 2027; https://doi.org/10.3390/math10122027 - 11 Jun 2022

Cited by 16 | Viewed by 1676

Abstract

An alternative approach, known today as the Bernstein polynomials, to the Weierstrass uniform approximation theorem was provided by Bernstein. These basis polynomials have attained increasing momentum, especially in operator theory, integral equations and computer-aided geometric design. Motivated by the improvements of Bernstein polynomials [...] Read more.

An alternative approach, known today as the Bernstein polynomials, to the Weierstrass uniform approximation theorem was provided by Bernstein. These basis polynomials have attained increasing momentum, especially in operator theory, integral equations and computer-aided geometric design. Motivated by the improvements of Bernstein polynomials in computational disciplines, we propose a new generalization of Bernstein–Kantorovich operators involving shape parameters

λ

,

α

and a positive integer as an original extension of Bernstein–Kantorovich operators. The statistical approximation properties and the statistical rate of convergence are also obtained by means of a regular summability matrix. Using the Lipschitz-type maximal function, the modulus of continuity and modulus of smoothness, certain local approximation results are presented. Some approximation results in a weighted space are also studied. Finally, illustrative graphics that demonstrate the approximation behavior and consistency of the proposed operators are provided by a computer program. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

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22 pages, 476 KiB

Open AccessArticle

General Odd and Even Central Factorial Polynomial Sequences

by Francesco Aldo Costabile, Maria Italia Gualtieri and Anna Napoli

Mathematics 2022, 10(6), 978; https://doi.org/10.3390/math10060978 - 18 Mar 2022

Cited by 4 | Viewed by 1354

Abstract

The

δ^{2} (\cdot)

operator, where

δ (\cdot)

is the known central difference operator, is considered. The associated odd and even polynomial sequences are determined and their generalizations studied. Particularly, matrix and determinant forms, recurrence formulas, generating functions and [...] Read more.

The

δ^{2} (\cdot)

operator, where

δ (\cdot)

is the known central difference operator, is considered. The associated odd and even polynomial sequences are determined and their generalizations studied. Particularly, matrix and determinant forms, recurrence formulas, generating functions and an algorithm for effective calculation are provided. An interesting property of biorthogonality is also demonstrated. New examples of odd and even central polynomial sequences are given. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

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8 pages, 258 KiB

Open AccessArticle

Solution of the Problem P = L

by Sergey Goncharov and Andrey Nechesov

Mathematics 2022, 10(1), 113; https://doi.org/10.3390/math10010113 - 31 Dec 2021

Cited by 6 | Viewed by 1645

Abstract

The problems associated with the construction of polynomial complexity computer programs require new techniques and approaches from mathematicians. One of such approaches is representing some class of polynomial algorithms as a certain class of special logical programs. Goncharov and Sviridenko described a logical [...] Read more.

The problems associated with the construction of polynomial complexity computer programs require new techniques and approaches from mathematicians. One of such approaches is representing some class of polynomial algorithms as a certain class of special logical programs. Goncharov and Sviridenko described a logical programming language

L_{0}

, where programs inductively are obtained from the set of

Δ_{0}

-formulas using special terms. In their work, a new idea has been proposed to look at the term as a program. The computational complexity of such programs is polynomial. In the same years, a number of other logical languages with similar properties were created. However, the following question remained: can all polynomial algorithms be described in these languages? It is a long-standing problem, and the method of describing some polynomial algorithm in a not Turing complete logical programming language was not previously clear. In this paper, special types of terms and formulas have been found and added to solve this problem. One of the main contributions is the construction of p-iterative terms that simulate the work of the Turing machine. Using p-iterative terms, the work showed that class P is equal to class L, which extends the programming language

L_{0}

with p-iterative terms. Thus, it is shown that L is quite expressive and has no halting problem, which occurs in high-level programming languages. For these reasons, the logical language L can be used to create fast and reliable programs. The main limitation of the language L is that the implementation of algorithms of complexity is not higher than polynomial. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

10 pages, 9470 KiB

Open AccessArticle

Approximation Properties of Chebyshev Polynomials in the Legendre Norm

by Cuixia Niu, Huiqing Liao, Heping Ma and Hua Wu

Mathematics 2021, 9(24), 3271; https://doi.org/10.3390/math9243271 - 16 Dec 2021

Cited by 2 | Viewed by 2302

Abstract

In this paper, we present some important approximation properties of Chebyshev polynomials in the Legendre norm. We mainly discuss the Chebyshev interpolation operator at the Chebyshev–Gauss–Lobatto points. The cases of single domain and multidomain for both one dimension and multi-dimensions are considered, respectively. [...] Read more.

In this paper, we present some important approximation properties of Chebyshev polynomials in the Legendre norm. We mainly discuss the Chebyshev interpolation operator at the Chebyshev–Gauss–Lobatto points. The cases of single domain and multidomain for both one dimension and multi-dimensions are considered, respectively. The approximation results in Legendre norm rather than in the Chebyshev weighted norm are given, which play a fundamental role in numerical analysis of the Legendre–Chebyshev spectral method. These results are also useful in Clenshaw–Curtis quadrature which is based on sampling the integrand at Chebyshev points. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

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17 pages, 500 KiB

Open AccessArticle

Roots of Characteristic Polynomial Sequences in Iterative Block Cyclic Reductions

by Masato Shinjo, Tan Wang, Masashi Iwasaki and Yoshimasa Nakamura

Mathematics 2021, 9(24), 3213; https://doi.org/10.3390/math9243213 - 12 Dec 2021

Cited by 3 | Viewed by 2294

Abstract

The block cyclic reduction method is a finite-step direct method used for solving linear systems with block tridiagonal coefficient matrices. It iteratively uses transformations to reduce the number of non-zero blocks in coefficient matrices. With repeated block cyclic reductions, non-zero off-diagonal blocks in [...] Read more.

The block cyclic reduction method is a finite-step direct method used for solving linear systems with block tridiagonal coefficient matrices. It iteratively uses transformations to reduce the number of non-zero blocks in coefficient matrices. With repeated block cyclic reductions, non-zero off-diagonal blocks in coefficient matrices incrementally leave the diagonal blocks and eventually vanish after a finite number of block cyclic reductions. In this paper, we focus on the roots of characteristic polynomials of coefficient matrices that are repeatedly transformed by block cyclic reductions. We regard each block cyclic reduction as a composition of two types of matrix transformations, and then attempt to examine changes in the existence range of roots. This is a block extension of the idea presented in our previous papers on simple cyclic reductions. The property that the roots are not very scattered is a key to accurately solve linear systems in floating-point arithmetic. We clarify that block cyclic reductions do not disperse roots, but rather narrow their distribution, if the original coefficient matrix is symmetric positive or negative definite. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

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11 pages, 296 KiB

Open AccessArticle

Polynomial Analogue of Gandy’s Fixed Point Theorem

by Sergey Goncharov and Andrey Nechesov

Mathematics 2021, 9(17), 2102; https://doi.org/10.3390/math9172102 - 31 Aug 2021

Cited by 9 | Viewed by 2232

Abstract

The paper suggests a general method for proving the fact whether a certain set is p-computable or not. The method is based on a polynomial analogue of the classical Gandy’s fixed point theorem. Classical Gandy’s theorem deals with the extension of a predicate [...] Read more.

The paper suggests a general method for proving the fact whether a certain set is p-computable or not. The method is based on a polynomial analogue of the classical Gandy’s fixed point theorem. Classical Gandy’s theorem deals with the extension of a predicate through a special operator

Γ_{Φ (x)}^{Ω^{*}}

and states that the smallest fixed point of this operator is a

Σ

-set. Our work uses a new type of operator which extends predicates so that the smallest fixed point remains a p-computable set. Moreover, if in the classical Gandy’s fixed point theorem, the special

Σ

-formula

Φ (\bar{x})

is used in the construction of the operator, then a new operator uses special generating families of formulas instead of a single formula. This work opens up broad prospects for the application of the polynomial analogue of Gandy’s theorem in the construction of new types of terms and formulas, in the construction of new data types and programs of polynomial computational complexity in Turing complete languages. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

28 pages, 449 KiB

Open AccessArticle

New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas

by Waleed Mohamed Abd-Elhameed and Badah Mohamed Badah

Mathematics 2021, 9(13), 1573; https://doi.org/10.3390/math9131573 - 04 Jul 2021

Cited by 7 | Viewed by 1649

Abstract

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the [...] Read more.

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type

4 F_{3} (1)

, which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

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45 pages, 1281 KiB

Open AccessFeature PaperArticle

The Legacy of Peter Wynn

by Claude Brezinski, F. Alexander Norman and Michela Redivo-Zaglia

Mathematics 2021, 9(11), 1240; https://doi.org/10.3390/math9111240 - 28 May 2021

Cited by 4 | Viewed by 2990

Abstract

After the death of Peter Wynn in December 2017, manuscript documents he left came to our knowledge. They concern continued fractions, rational (Padé) approximation, Thiele interpolation, orthogonal polynomials, moment problems, series, and abstract algebra. The purpose of this paper is to analyze them [...] Read more.

After the death of Peter Wynn in December 2017, manuscript documents he left came to our knowledge. They concern continued fractions, rational (Padé) approximation, Thiele interpolation, orthogonal polynomials, moment problems, series, and abstract algebra. The purpose of this paper is to analyze them and to make them available to the mathematical community. Some of them are in quite good shape, almost finished, and ready to be published by anyone willing to check and complete them. Others are rough notes, and need to be reworked. Anyway, we think that these works are valuable additions to the literature on these topics and that they cannot be left unknown since they contain ideas that were never exploited. They can lead to new research and results. Two unpublished papers are also mentioned here for the first time. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

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29 pages, 1535 KiB

Open AccessArticle

General Bivariate Appell Polynomials via Matrix Calculus and Related Interpolation Hints

by Francesco Aldo Costabile, Maria Italia Gualtieri and Anna Napoli

Mathematics 2021, 9(9), 964; https://doi.org/10.3390/math9090964 - 25 Apr 2021

Cited by 8 | Viewed by 1777

Abstract

An approach to general bivariate Appell polynomials based on matrix calculus is proposed. Known and new basic results are given, such as recurrence relations, determinant forms, differential equations and other properties. Some applications to linear functional and linear interpolation are sketched. New and [...] Read more.

An approach to general bivariate Appell polynomials based on matrix calculus is proposed. Known and new basic results are given, such as recurrence relations, determinant forms, differential equations and other properties. Some applications to linear functional and linear interpolation are sketched. New and known examples of bivariate Appell polynomial sequences are given. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

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28 pages, 479 KiB

Open AccessArticle

On Certain Properties and Applications of the Perturbed Meixner–Pollaczek Weight

by Abey S. Kelil, Alta S. Jooste and Appanah R. Appadu

Mathematics 2021, 9(9), 955; https://doi.org/10.3390/math9090955 - 24 Apr 2021

Cited by 1 | Viewed by 1526

Abstract

This paper deals with monic orthogonal polynomials orthogonal with a perturbation of classical Meixner–Pollaczek measure. These polynomials, called Perturbed Meixner–Pollaczek polynomials, are described by their weight function emanating from an exponential deformation of the classical Meixner–Pollaczek measure. In this contribution, we investigate certain [...] Read more.

This paper deals with monic orthogonal polynomials orthogonal with a perturbation of classical Meixner–Pollaczek measure. These polynomials, called Perturbed Meixner–Pollaczek polynomials, are described by their weight function emanating from an exponential deformation of the classical Meixner–Pollaczek measure. In this contribution, we investigate certain properties such as moments of finite order, some new recursive relations, concise formulations, differential-recurrence relations, integral representation and some properties of the zeros (quasi-orthogonality, monotonicity and convexity of the extreme zeros) of the corresponding perturbed polynomials. Some auxiliary results for Meixner–Pollaczek polynomials are revisited. Some applications such as Fisher’s information, Toda-type relations associated with these polynomials, Gauss–Meixner–Pollaczek quadrature as well as their role in quantum oscillators are also reproduced. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

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17 pages, 285 KiB

Open AccessArticle

Method for Obtaining Coefficients of Powers of Bivariate Generating Functions

by Dmitry Kruchinin, Vladimir Kruchinin and Yuriy Shablya

Mathematics 2021, 9(4), 428; https://doi.org/10.3390/math9040428 - 22 Feb 2021

Cited by 6 | Viewed by 2094

Abstract

In this paper, we study methods for obtaining explicit formulas for the coefficients of generating functions. To solve this problem, we consider the methods that are based on using the powers of generating functions. We propose to generalize the concept of compositae to [...] Read more.

In this paper, we study methods for obtaining explicit formulas for the coefficients of generating functions. To solve this problem, we consider the methods that are based on using the powers of generating functions. We propose to generalize the concept of compositae to the case of generating functions in two variables and define basic operations on such compositae: composition, addition, multiplication, reciprocation and compositional inversion. These operations allow obtaining explicit formulas for compositae and coefficients of bivariate generating functions. In addition, we present several examples of applying the obtained results for getting explicit formulas for the coefficients of bivariate generating functions. The introduced mathematical apparatus can be used for solving different problems that are related to the theory of generating functions. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

13 pages, 282 KiB

Open AccessArticle

Two-Variable Type 2 Poly-Fubini Polynomials

by Ghulam Muhiuddin, Waseem Ahmad Khan and Ugur Duran

Mathematics 2021, 9(3), 281; https://doi.org/10.3390/math9030281 - 31 Jan 2021

Cited by 14 | Viewed by 1770

Abstract

In the present work, a new extension of the two-variable Fubini polynomials is introduced by means of the polyexponential function, which is called the two-variable type 2 poly-Fubini polynomials. Then, some useful relations including the Stirling numbers of the second and the first [...] Read more.

In the present work, a new extension of the two-variable Fubini polynomials is introduced by means of the polyexponential function, which is called the two-variable type 2 poly-Fubini polynomials. Then, some useful relations including the Stirling numbers of the second and the first kinds, the usual Fubini polynomials, and the higher-order Bernoulli polynomials are derived. Also, some summation formulas and an integral representation for type 2 poly-Fubini polynomials are investigated. Moreover, two-variable unipoly-Fubini polynomials are introduced utilizing the unipoly function, and diverse properties involving integral and derivative properties are attained. Furthermore, some relationships covering the two-variable unipoly-Fubini polynomials, the Stirling numbers of the second and the first kinds, and the Daehee polynomials are acquired. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

Review

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28 pages, 424 KiB

Open AccessFeature PaperReview

Towards the Centenary of Sheffer Polynomial Sequences: Old and Recent Results

by Francesco Aldo Costabile, Maria Italia Gualtieri and Anna Napoli

Mathematics 2022, 10(23), 4435; https://doi.org/10.3390/math10234435 - 24 Nov 2022

Cited by 3 | Viewed by 1204

Abstract

Sheffer’s work is about to turn 100 years after its publication. In reporting this important event, we recall some interesting old and recent results, aware of the incompleteness of the wide existing literature. Particularly, we recall Sheffer’s approach, the theory of Rota and [...] Read more.

Sheffer’s work is about to turn 100 years after its publication. In reporting this important event, we recall some interesting old and recent results, aware of the incompleteness of the wide existing literature. Particularly, we recall Sheffer’s approach, the theory of Rota and his collaborators, the isomorphism between the group of Sheffer polynomial sequences and the so-called Riordan matrices group. This inspired the most recent approaches based on elementary matrix calculus. The interesting problem of orthogonality in the context of Sheffer sequences is also reported, recalling the results of Sheffer, Meixner, Shohat, and the very recent one of Galiffa et al., and of Costabile et al. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

17 pages, 371 KiB

Open AccessReview

Multivalue Collocation Methods for Ordinary and Fractional Differential Equations

by Angelamaria Cardone, Dajana Conte, Raffaele D’Ambrosio and Beatrice Paternoster

Mathematics 2022, 10(2), 185; https://doi.org/10.3390/math10020185 - 07 Jan 2022

Cited by 8 | Viewed by 1776

Abstract

The present paper illustrates some classes of multivalue methods for the numerical solution of ordinary and fractional differential equations. In particular, it focuses on two-step and mixed collocation methods, Nordsieck GLM collocation methods for ordinary differential equations, and on two-step spline collocation methods [...] Read more.

The present paper illustrates some classes of multivalue methods for the numerical solution of ordinary and fractional differential equations. In particular, it focuses on two-step and mixed collocation methods, Nordsieck GLM collocation methods for ordinary differential equations, and on two-step spline collocation methods for fractional differential equations. The construction of the methods together with the convergence and stability analysis are reported and some numerical experiments are carried out to show the efficiency of the proposed methods. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

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