New Classes of Degenerate Unified Polynomials (original) (raw)

New degenerated polynomials arising from non-classical Umbral Calculus

arXiv: Number Theory, 2018

We introduce new generalizations of the Bernoulli, Euler, and Genocchi polynomials and numbers based on the Carlitz-Tsallis degenerate exponential function and concepts of the Umbral Calculus associated with it. Also, we present generalizations of some familiar identities and connection between these kinds of Bernoulli, Euler, and Genocchi polynomials. Moreover, we establish a new analogue of the Euler identity for the degenerate Bernoulli numbers.

A unified presentation of certain classical polynomials

Mathematics of Computation, 1972

This paper attempts to present a unified treatment of the classical orthogonal polynomials, viz. Jacobi, Laguerre and Hermite polynomials, and their generalizations introduced from time to time. The results obtained here include a number of linear, bilinear and bilateral generating functions and operational formulas for the polynomials { T n ( α , β ) ( x , a , b , c , d , p , r ) | n = 0 , 1 , 2 , ⋯ } \{ T_n^{(\alpha ,\beta )}(x,a,b,c,d,p,r)|n = 0,1,2, \cdots \} , defined by Eq. (3) below.*

A Note on Type-Two Degenerate Poly-Changhee Polynomials of the Second Kind

Symmetry

In this paper, we first define type-two degenerate poly-Changhee polynomials of the second kind by using modified degenerate polyexponential functions. We derive new identities and relations between type-two degenerate poly-Changhee polynomials of the second kind. Finally, we derive type-two degenerate unipoly-Changhee polynomials of the second kind and discuss some of their identities.

On degenerate Apostol-type polynomials and applications

Boletín de la Sociedad Matemática Mexicana

The main object of the current paper is to introduce and investigate a new unified class of the degenerate Apostol-type polynomials. These polynomials are studied by means of the generating function, series definition and are framed within the context of monomiality principle. Several important recurrence relations and explicit representations for the antecedent class of polynomials are derived. As the special cases, the degenerate Apostol-Bernoulli, Euler and Genocchi polynomials are obtained and corresponding results are also proved. A fascinating example is constructed in terms of truncated-exponential polynomials, which gives the applications of these polynomials to produce their hybridized forms.

A Note on Type 2 Degenerate Poly-Fubini Polynomials and Numbers

2020

Abstract. In this paper, we construct the degenerate poly-Fubini polynomials, called the type 2 degenerate poly-Fubini polynomials, by using the modified degenerate polyexponential function and derive several properties on the degenerate poly-Fubini polynomials and numbers. In the last section, we introduce type 2 degenerate unipolyFubini polynomials attached to an arithmetic function, by using the modified degenerate polyexponential function and investigate some identities for those polynomials. Furthermore, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

On Degenerate Truncated Special Polynomials

The main aim of this paper is to introduce the degenerate truncated forms of multifarious special polynomials and numbers and is to investigate their various properties and relationships by using the series manipulation method and diverse special proof techniques. The degenerate truncated exponential polynomials are first considered and their several properties are given. Then the degenerate truncated Stirling polynomials of the second kind are defined and their elementary properties and relations are proved. Also, the degenerate truncated forms of the bivariate Fubini and Bell polynomials and numbers are introduced and various relations and formulas for these polynomials and numbers, which cover several summation formulas, addition identities, recurrence relationships, derivative property and correlations with the degenerate truncated Stirling polynomials of the second kind, are acquired. Thereafter, the truncated degenerate Bernoulli and Euler polynomials are considered and multifarious correlations and formulas including summation formulas, derivation rules and correlations with the degenerate truncated Stirling numbers of the second are derived. In addition, regarding applications, by introducing the degenerate truncated forms of the classical Bernstein polynomials, we obtain diverse correlations and formulas. Some interesting surface plots of these polynomials in the special cases are provided.

On orthogonal polynomials spanning a non-standard flag

Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics, 2012

We survey some recent developments in the theory of orthogonal polynomials defined by differential equations. The key finding is that there exist orthogonal polynomials defined by 2nd order differential equations that fall outside the classical families of Jacobi, Laguerre, and Hermite polynomials. Unlike the classical families, these new examples, called exceptional orthogonal polynomials, feature non-standard polynomial flags; the lowest degree polynomial has degree m > 0. In this paper we review the classification of codimension m = 1 exceptional polynomials, and give a novel, compact proof of the fundamental classification theorem for codimension 1 polynomial flags. As well, we describe the mechanism or rational factorizations of 2nd order operators as the analogue of the Darboux transformation in this context. We finish with the example of higher codimension generalization of Jacobi polynomials and perform the complete analysis of parameter values for which these families have non-singular weights.

On Apostol-Type Hermite Degenerated Polynomials

2023

This article presents a generalization of new classes of degenerated Apostol–Bernoulli, Apostol–Euler, and Apostol–Genocchi Hermite polynomials of level m. We establish some algebraic and differential properties for generalizations of new classes of degenerated Apostol–Bernoulli polynomials. These results are shown using generating function methods for Apostol–Euler and Apostol–Genocchi Hermite polynomials of level m.

A note on degenerate poly-Genocchi polynomials

International Journal of ADVANCED AND APPLIED SCIENCES

In this article, we introduce degenerate poly-Genocchi polynomials and numbers. We derive summation formulas, recurrence relations, and identities of these polynomials by using summation techniques series. Also, we establish symmetric identities by using power series methods, respectively.

The phase problem of ultraflat unimodular polynomials: The resolution of the conjecture of Saffari

Mathematische Annalen, 2001

Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by ∂D. Let Kn := pn : pn(z) = n k=0 a k z k , a k ∈ C , |a k | = 1. The class Kn is often called the collection of all (complex) unimodular polynomials of degree n. Given a sequence (εn) of positive numbers tending to 0, we say that a sequence (Pn) of unimodular polynomials Pn ∈ Kn is (εn)-ultraflat if (1 − εn) √ n + 1 ≤ |Pn(z)| ≤ (1 + εn) √ n + 1 , z ∈ ∂D , n ∈ N. The existence of ultraflat unimodular polynomials seemed very unlikely, in view of a 1957 conjecture of P. Erdős (Problem 22 in [Er1]) asserting that, for all Pn ∈ Kn with n ≥ 1, max z∈∂D |Pn(z)| ≥ (1 + ε) √ n + 1 , where ε > 0 is an absolute constant (independent of n). Yet, combining some probabilistic lemmas from Körner's paper [Kö] with some constuctive methods (Gauss polynomials, etc.), which were completely unrelated to the deterministic part of Körner's paper, Kahane [Ka] proved that there exists a sequence (Pn) with Pn ∈ Kn which is (εn)-ultraflat, where εn = O n −1/17 √ log n. Thus the Erdős conjecture was disproved for the classes Kn. In this paper we study ultraflat sequences (Pn) of unimodular polynomials Pn ∈ Kn in general, not necessarily those produced by Kahane in his paper [Ka]. We prove a few conjectures of Saffari [Sa] (see also [QS2]). Most importantly the following one. Uniform Distribution Conjecture for the Angular Speed. Let (Pn) be a εnultraflat sequence of unimodular polynomials Pn ∈ Kn. Let Pn(e it) = Rn(t)e iαn (t) , Rn(t) = |Pn(e it)|. In the interval [0, 2π], the distribution of the normalized angular speed α ′ n (t)/n converges to the uniform distribution as n → ∞. More precisely, we have m{t ∈ [0, 2π] : 0 ≤ α ′ n (t) ≤ nx} = 2πx + on(x) for every x ∈ [0, 1], where limn→∞ on(x) = 0 for every x ∈ [0, 1].