Special Polynomials Research Articles

An Improved Method for Order Reduction of High Order Uncertain SISO Dynamic Systems by Affine Arithmetic

This article presents a refined algorithm using Modified Polynomial Differentiation (MPD) method through Affine Arithmetic (AA) to reduce the high order uncertain systems. This new algorithm is applicable for the reduction of Continuous SISO systems. Measurement errors due to variation in ambient conditions may result the system as an uncertain system unlike other methods in literature. This strategy, unlike previous ones found in the Iteration literature, can produce a reduced order model that is stable from a original high order uncertain system that is also stable. Applications of Affine Arithmetic steer clear of many of the drawbacks of Interval Arithmetic, including unbounded solutions in a number of important applications. Utilizing common numerical examples from interval literature, the proposed approach has been validated

A note on infinite series whose terms involve truncated degenerate exponentials

The degenerate exponentials are degenerate versions of the ordinary exponential and the truncated degenerate exponentials are obtained from the Taylor expansions of them by truncating the first finitely many terms. The degenerate exponentials play an important role in recent studies on degenerate versions of many special numbers and polynomials, the degenerate gamma function, the degenerate umbral calculus and the degenerate q-umbral calculus. The aim of this note is to consider infinite series whose terms involve truncated degenerate exponentials together with binomial coefficients, the generalized falling factorials and the Stirling numbers of the second kind, and to find either their values or some other expressions of them as finite sums.

Some identities related to degenerate r-Bell and degenerate Fubini polynomials

Many works have been done in recent years as to explorations for degenerate versions of some special polynomials and numbers, which began with the pioneering work of Carlitz on the degenerate Bernoulli and degenerate Euler polynomials. This paper is focused on the study of some properties, recurrence relations and identities related to the degenerate r-Bell polynomials, the two variable degenerate Fubini polynomials and the degenerate r-Stirling numbers of the second kind. Especially, we express the power series in terms of the degenerate r-Bell polynomials, of the degenerate r-Stirling numbers of the second kind and of the degenerate Fubini polynomials.

An Algebraic Approach to the Δh-Frobenius–Genocchi–Appell Polynomials

In recent years, the generating function of mixed-type special polynomials has received growing interest in several fields of applied sciences and physics. This article intends to study a new class of polynomials, called the Δh-Frobenius–Genocchi–Appell polynomials. The generating function of Δh-Frobenius–Genocchi–Appell polynomials is constructed and some of their fundamental properties are studied. By making use of this generating function, we investigate some novel and interesting results, such as recurrence relations, explicit representations, and implicit formulas for the Δh-Frobenius–Genocchi–Appell polynomials. The quasi-monomiality and determinant form for these polynomials are established. The Δh-Genocchi–Appell polynomials are explored as a special case and several results for Δh-Genocchi–Appell polynomials are also obtained.

On Generalized Bivariate (p,q)-Bernoulli–Fibonacci Polynomials and Generalized Bivariate (p,q)-Bernoulli–Lucas Polynomials

Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this paper, we define the generalized (p,q)-Bernoulli–Fibonacci and generalized (p,q)-Bernoulli–Lucas polynomials and numbers by using the (p,q)-Bernoulli numbers, unified (p,q)-Bernoulli polynomials, h(x)-Fibonacci polynomials, and h(x)-Lucas polynomials. We also introduce the generalized bivariate (p,q)-Bernoulli–Fibonacci and generalized bivariate (p,q)-Bernoulli–Lucas polynomials and numbers. Then, we derive some properties of these newly established polynomials and numbers by using their generating functions with their functional equations. Finally, we provide some families of bilinear and bilateral generating functions for the generalized bivariate (p,q)-Bernoulli–Fibonacci polynomials.

Constructions of cyclic codes and extended primitive cyclic codes with their applications

Linear codes with a few weights have many nice applications including combinatorial design, distributed storage system, secret sharing schemes and so on. In this paper, we construct two families of linear codes with a few weights based on special polynomials over finite fields. The first family of linear codes are extended primitive cyclic codes which are affine-invariant. The second family of linear codes are reducible cyclic codes. The parameters of these codes and their duals are determined. As the first application, we prove that these two families of linear codes hold t-designs, where t=2,3. As the second application, the minimum localities of the codes are also determined and optimal locally recoverable codes are derived.

Combinatorial identities involving degenerate harmonic and hyperharmonic numbers

In recent years, some degenerate versions of quite a few special numbers and polynomials are introduced and investigated by means of various methods. The aim of this paper is to study some results on degenerate harmonic numbers, degenerate hyperharmonic numbers, degenerate Fubi polynomials and degenerate r-Fubini polynomials from a general identity which is valid for any two formal power series and involves the degenerate r-Stirling numbers of the second kind.

Polynomial Differentiation Threshold based Edge Detection of Contrast Enhanced Images

Polynomial Differentiation Threshold based Edge Detection of Contrast Enhanced Images

A Theorem on Separated Transformations of Basis Vectors of Polynomial Space and its Applications in Special Polynomials and Related Sí (2, R) Lie Algebra

A Theorem on Separated Transformations of Basis Vectors of Polynomial Space and its Applications in Special Polynomials and Related Sí (2, R) Lie Algebra

On (p,q)–Fibonacci and (p,q)–Lucas Polynomials Associated with Changhee Numbers and Their Properties

Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties have been studied in the literature with the help of generating functions and their functional equations. In this paper, using the (p,q)–Fibonacci polynomials, (p,q)–Lucas polynomials, and Changhee numbers, we define the (p,q)–Fibonacci–Changhee polynomials and (p,q)–Lucas–Changhee polynomials, respectively. We obtain some important identities and relations of these newly established polynomials by using their generating functions and functional equations. Then, we generalize the (p,q)–Fibonacci–Changhee polynomials and the (p,q)–Lucas–Changhee polynomials called generalized (p,q)–Fibonacci–Lucas–Changhee polynomials. We derive a determinantal representation for the generalized (p,q)–Fibonacci–Lucas–Changhee polynomials in terms of the special Hessenberg determinant. Finally, we give a new recurrent relation of the (p,q)–Fibonacci–Lucas–Changhee polynomials.

Generalized Mixed Type Bernoulli-Gegenbauer Polynomials

The generalized mixed type Bernoulli-Gegenbauer polynomials of order α >−1/2 are special polynomials obtained by use of the generating function method. These polynomials represent an interesting mixture between two classes of special functions, namely generalized Bernoulli polynomials and Gegenbauer polynomials. The main purpose of this paper is to discuss some of their algebraic and analytic properties.

Studies on Special Polynomials Involving Degenerate Appell Polynomials and Fractional Derivative

The focus of the research presented in this paper is on a new generalized family of degenerate three-variable Hermite–Appell polynomials defined here using a fractional derivative. The research was motivated by the investigations on the degenerate three-variable Hermite-based Appell polynomials introduced by R. Alyosuf. We show in the paper that, for certain values, the well-known degenerate Hermite–Appell polynomials, three-variable Hermite–Appell polynomials and Appell polynomials are seen as particular cases for this new family. As new results of the investigation, the operational rule for this new generalized family is introduced and the explicit summation formula is established. Furthermore, using the determinant formulation of the Appell polynomials, the determinant form for the new generalized family is obtained and the recurrence relations are also determined considering the generating expression of the polynomials contained in the new generalized family. Certain applications of the generalized three-variable Hermite–Appell polynomials are also presented showing the connection with the equivalent results for the degenerate Hermite–Bernoulli and Hermite–Euler polynomials with three variables.

Degenerate 2D bivariate Appell polynomials: properties and applications

The development of certain aspects of hybrid special polynomials after incorporating monomiality principle, operational rules, and other properties and their aspects is obvious and indisputable. The study presented in this paper follows this line of research. By using the monomiality principle, new outcomes are produced, thus in line with prior facts, our aim is to introduce the degenerate 2D bivariate Appell polynomials . Further, we obtain some well-known main properties and explicit forms satisfied by these polynomials. Also, generating relations corresponding to the degenerate 2D bivariate Bernoulli, Euler, and Genocchi polynomials are established, they will further be helpful in establishing the corresponding results for these polynomials.

Certain Properties and Applications of Δh Hybrid Special Polynomials Associated with Appell Sequences

The development of certain aspects of special polynomials in line with the monomiality principle, operational rules, and other properties and their aspects is obvious and indisputable. The study presented in this paper follows this line of research. By using the monomiality principle, new outcomes are produced, and their differential equation and series representation is obtained, which are important in several branches of mathematics and physics. Thus, in line with prior facts, our aim is to introduce the Δh hybrid special polynomials associated with Hermite polynomials denoted by ΔhHQm(u,v,w;h). Further, we obtain some well-known main properties and explicit forms satisfied by these polynomials.

Numerical solution of neutral delay differential equations using orthogonal neural network

In this paper, an efficient orthogonal neural network (ONN) approach is introduced to solve the higher-order neutral delay differential equations (NDDEs) with variable coefficients and multiple delays. The method is implemented by replacing the hidden layer of the feed-forward neural network with the orthogonal polynomial-based functional expansion block, and the corresponding weights of the network are obtained using an extreme learning machine(ELM) approach. Starting with simple delay differential equations (DDEs), an interest has been shown in solving NDDEs and system of NDDEs. Interest is given to consistency and convergence analysis, and it is seen that the method can produce a uniform closed-form solution with an error of order 2^{-n}, where n is the number of neurons. The developed neural network method is validated over various types of example problems(DDEs, NDDEs, and system of NDDEs) with four different types of special orthogonal polynomials.

The mixed mock modularity of a new $U$-type function related to the Andrews-Gordon identities

In this paper we resolve a question by Bringmann, Lovejoy, and Rolen on a new vector-valued $U$-type function. We obtain an expression for a corresponding family of Hecke--Appell-type sums in terms of mixed mock modular forms; that is, we express the sum in terms of Appell functions and theta functions. This $U$-type function appears from considering the special polynomials related to generating functions for the partitions occurring in Gordon’s generalization of the Rogers--Ramanujan identities.

Quasi-Monomiality Principle and Certain Properties of Degenerate Hybrid Special Polynomials

This article aims to introduce degenerate hybrid type Appell polynomials HQm(u,v,w;η) and establishes their quasi-monomial characteristics. Additionally, a number of features of these polynomials are established, including symmetric identities, implicit summation formulae, differential equations, series definition and operational formalism.

Recurrences in Terms of Special Polynomials for Exponential Sums of Elementary Symmetric Polynomials over Finite Fields

Recurrences in Terms of Special Polynomials for Exponential Sums of Elementary Symmetric Polynomials over Finite Fields

Multi-lump wave patterns of KPI via integer partitions

The connection between the theory of integer partitions and a class of nonsingular rational solutions of the Kadomtsev–Petviashvili (KP) I equation referred to as multi-lumps is investigated. It is shown that specific partitions of a given positive integer N lead to multi-lump solutions that exhibit distinctive surface wave patterns. The pattern formations are described by the complex roots of special polynomials such as the Yablonskii–Vorob’ev, Okamoto and generalized Okamoto polynomials arising in the study of rational solutions of Painlevé II and IV equations. Stationary KPI multi-lumps which correspond to rational solutions of the Boussinesq equation are also discussed.

On a parametric K-fold series and its connection to Nielsen-Kölbig relations

The goal of this paper is twofold. First, we investigate the generating function of the parametric sum∑n1≥⋯≥nK≥1∏j=1K1nj⋅(nj+x),K∈N, where x is real with |x|<1. With the help of this approach, we generate a class of multiple zeta-star sums SK,λ⋆:=∑ζ⋆(s) with |s|=2K+λ, si≥2, λ∈N0, and evaluate them as special polynomial values at ζ(i). Second, as a somewhat surprising application, we restate our result concerning SK,λ⋆ with the help of the non-starred alternating multiple zeta values of the form ζ(u‾,{1}v) and establish a new system of equations connected with Nielsen-Kölbig relations. This enables us a relatively direct insight into the dependence among such specific values.