Polyexponential Functions Research Articles

Degenerate Apostol-Frobenius Type Poly-Genocchi Polynomials of Higher Order with Parameters a and b

This paper introduces another variation of poly-Genocchi polynomials by mixing the concept of modified degenerate polyexponential function, Apostol-Genocchi polynomials and Frobenius polynomials. These polynomials are called the degenerate Apostol-Frobenius-type poly-Genocchi polynomials with parameters a and b. Several identities and formulas are derived including recurrence relations, explicit formulas and certain differential identity. Moreover, some relations are established connecting these polynomials to degenerate Stirling numbers of the first and second kind, higher order degenerate Bernoulli polynomials, and higher order degenerate Frobenius-Euler polynomials.

Quantitative estimation of chemical microheterogeneity through the determination of fuzzy entropy.

Chemical micro-heterogeneity is an attribute of all living systems and most of the soft and crystalline materials. Its characterization requires a plethora of techniques. This work proposes a strategy for quantifying the degree of chemical micro-heterogeneity. First of all, our approach needs the collection of time-evolving signals that can be fitted through poly-exponential functions. The best fit is determined through the Maximum Entropy Method. The pre-exponential terms of the poly-exponential fitting function are used to estimate Fuzzy Entropy. Related to the possibility of implementing Fuzzy sets through the micro-heterogeneity of chemical systems. Fuzzy Entropy becomes a quantitative estimation of the Fuzzy Information that can be processed through micro-heterogeneous chemical systems. We conclude that our definition of Fuzzy Entropy can be extended to other kinds of data, such as morphological and structural distributions, spectroscopic bands and chromatographic peaks. The chemical implementation of Fuzzy sets and Fuzzy logic will promote the development of Chemical Artificial Intelligence.

Degenerate Poly‐Lah‐Bell Polynomials and Numbers

Many mathematicians studied “poly” as a generalization of the well‐known special polynomials such as Bernoulli polynomials, Euler polynomials, Cauchy polynomials, and Genocchi polynomials. In this paper, we define the degenerate poly‐Lah‐Bell polynomials arising from the degenerate polyexponential functions which are reduced to degenerate Lah‐Bell polynomials when k = 1. In particular, we call these polynomials the “poly‐Lah‐Bell polynomials” when λ⟶0. We give their explicit expression, Dobinski‐like formulas, and recurrence relation. In addition, we obtain various algebraic identities including Lah numbers, the degenerate Stirling numbers of the first and second kind, the degenerate poly‐Bell polynomials, the degenerate poly‐Bernoulli numbers, and the degenerate poly‐Genocchi numbers.

Type 2 degenerate modified poly-Bernoulli polynomials arising from the degenerate poly-exponential functions

<abstract><p>We present a new type of degenerate poly-Bernoulli polynomials and numbers by modifying the polyexponential function in terms of the degenerate exponential functions and degenerate logarithm functions. Also, we introduce a new variation of the degenerate unipoly-Bernoulli polynomials by the similar modification. Based on these polynomials, we investigate some properties, new identities, and their relations to the known special functions and numbers such as the degenerate type 2-Bernoulli polynomials, the type 2 degenerate Euler polynomials, the degenerate Bernoulli polynomials and numbers, the degenerate Stirling numbers of the first kind, and $ \lambda $-falling factorial sequence. In addition, we compute some of the proposed polynomials and present their zeros and behaviors for different variables in specific cases.</p></abstract>

Poly-falling factorial sequences and poly-rising factorial sequences

Abstract In this paper, we introduce generalizations of rising factorials and falling factorials, respectively, and study their relations with the well-known Stirling numbers, Lah numbers, and so on. The first stage is to define poly-falling factorial sequences in terms of the polyexponential functions, reducing them to falling factorials if k = 1 k=1 , necessitating a demonstration of the relations: between poly-falling factorial sequences and the Stirling numbers of the first and second kind, respectively; between poly-falling factorial sequences and the poly-Bell polynomials; between poly-falling factorial sequences and the poly-Bernoulli numbers; between poly-falling factorial sequences and poly-Genocchi numbers; and recurrence formula of these sequences. The later part of the paper deals with poly-rising factorial sequences in terms of the polyexponential functions, reducing them to rising factorial if k = 1 k=1 . We study some relations: between poly-falling factorial sequences and poly-rising factorial sequences; between poly-rising factorial sequences and the Stirling numbers of the first kind and the power of x x ; and between poly-rising factorial sequences and Lah numbers and the poly-falling factorial sequences. We also derive recurrence formula of these sequences and reciprocal formula of the poly-falling factorial sequences.

Construction of Type 2 Poly-Changhee Polynomials and Its Applications

In this paper, we introduce type 2 poly-Changhee polynomials by using the polyexponential function. We derive some explicit expressions and identities for these polynomials, and we also prove some relationships between poly-Changhee polynomials and Stirling numbers of the first and second kind. Also, we introduce the unipoly-Changhee polynomials by employing unipoly function and give multifarious properties. Furthermore, we provide a correlation between the unipoly-Changhee polynomials and the classical Changhee polynomials.

Poly-central factorial sequences and poly-central-Bell polynomials

In this paper, we introduce poly-central factorial sequences and poly-central Bell polynomials arising from the polyexponential functions, reducing them to central factorials and central Bell polynomials of the second kind respectively when k = 1. We also show some relations: between poly-central factorial sequences and power of x; between poly-central Bell polynomials and power of x; between poly-central Bell polynomials and the poly-Bell polynomials; between poly-central Bell polynomials and higher order type 2 Bernoulli polynomials of second kind; recurrence formula of poly-central Bell polynomials.

Some Identities of the Degenerate Poly-Frobenius-Genocchi Polynomials of Complex Variables

The main of this paper is to define and investigate a new class of the degenerate poly-Frobenius-Genocchi polynomials with the help of the polyexponential functions. In this paper, we define the degenerate poly-Frobenius-Genocchi polynomials of complex variables arising from the modified polyexponential functions, and establish some explicit expressions for these polynomials. Meanwhile, some interesting connections between these polynomials and some other special polynomials are also showed.

Analytical properties of type 2 degenerate poly-Bernoulli polynomials associated with their applications

Recently, Kim et al. (Adv. Differ. Equ. 2020:168, 2020) considered the poly-Bernoulli numbers and polynomials resulting from the moderated version of degenerate polyexponential functions. In this paper, we investigate the degenerate type 2 poly-Bernoulli numbers and polynomials which are derived from the moderated version of degenerate polyexponential functions. Our degenerate type 2 degenerate poly-Bernoulli numbers and polynomials are different from those of Kim et al. (Adv. Differ. Equ. 2020:168, 2020) and Kim and Kim (Russ. J. Math. Phys. 26(1):40–49, 2019). Utilizing the properties of moderated degenerate poly-exponential function, we explore some properties of our type 2 degenerate poly-Bernoulli numbers and polynomials. From our investigation, we derive some explicit expressions for type 2 degenerate poly-Bernoulli numbers and polynomials. In addition, we also scrutinize type 2 degenerate unipoly-Bernoulli polynomials related to an arithmetic function and investigate some identities for those polynomials. In particular, we consider certain new explicit expressions and relations of type 2 degenerate unipoly-Bernoulli polynomials and numbers related to special numbers and polynomials. Further, some related beautiful zeros and graphical representations are displayed with the help of Mathematica.

On the type 2 poly-Bernoulli polynomials associated with umbral calculus

Abstract Type 2 poly-Bernoulli polynomials were introduced recently with the help of modified polyexponential functions. In this paper, we investigate several properties and identities associated with those polynomials arising from umbral calculus techniques. In particular, we express the type 2 poly-Bernoulli polynomials in terms of several special polynomials, like higher-order Cauchy polynomials, higher-order Euler polynomials, and higher-order Frobenius-Euler polynomials.

A note on polyexponential and unipoly Bernoulli polynomials of the second kind

Abstract In this paper, the authors study the poly-Bernoulli numbers of the second kind, which are defined by using polyexponential functions introduced by Kims. Also by using unipoly function, we study the unipoly Bernoulli numbers of the second kind, which are attached to an arithmetic function. We derive their explicit expressions and some identities involving poly-Bernoulli numbers of the second kind and unipoly Bernoulli numbers of the second kind.

Degenerate poly-Bell polynomials and numbers

Numerous mathematicians have studied ‘poly’ as one of the generalizations to special polynomials, such as Bernoulli, Euler, Cauchy, and Genocchi polynomials. In relation to this, in this paper, we introduce the degenerate poly-Bell polynomials emanating from the degenerate polyexponential functions which are called the poly-Bell polynomials when lambda rightarrow 0. Specifically, we demonstrate that they are reduced to the degenerate Bell polynomials if k = 1. We also provide explicit representations and combinatorial identities for these polynomials, including Dobinski-like formulas, recurrence relationships, etc.

Some identities and reciprocity relationsof unipoly-Dedekind type DC sums

Dedekind type DC sums and their generalizations are defined in terms of Euler functions and their generalization. Recently, Ma et al. (Adv. Differ. Equ. 2021:30 2021) introduced the poly-Dedekind type DC sums by replacing the Euler function appearing in Dedekind sums, and they were shown to satisfy a reciprocity relation. In this paper, we consider two kinds of new generalizations of the poly-Dedekind type DC sums. One is a unipoly-Dedekind type DC sum associated with the type 2 unipoly-Euler functions expressed in the type 2 unipoly-Euler polynomials using the modified polyexponential function, and we study some identities and the reciprocity relation for these unipoly-Dedekind type DC sums. The other is a unipoly-Dedekind sums type DC associated with the poly-Euler functions expressed in the unipoly-Euler polynomials using the polylogarithm function, and we derive some identities and the reciprocity relation for those unipoly-Dedekind type DC sums.

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

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.

Two-Variable Type 2 Poly-Fubini Polynomials

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.

HIGHER ORDER APOSTOL-TYPE POLY-GENOCCHI POLYNOMIALS WITH PARAMETERS a, b AND c

In this paper, a new form of poly-Genocchi polynomials is defined by means of poly-logarithm, namely, the Apostol-type poly-Genocchi polynomials of higher order with parameters a, b and c. Several properties of these polynomials are established including some recurrence relations and explicit formulas, which express these higher order Apostol-type poly-Genocchi polynomials in terms of Stirling numbers of the second kind, Apostol-type Bernoulli and Frobenius polynomials of higher order. Moreover, certain differential identity is obtained that leads this new form of poly-Genocchi polynomials to be classified as Appell polynomials and, consequently, draw more properties using some theorems on Appell polynomials. Furthermore, a symmetrized generalization of this new form of poly-Genocchi polynomials is introduced that possesses a double generating function. Finally, the type 2 Apostol-poly-Genocchi polynomials with parameters a, b and c are defined using the concept of polyexponential function and several identities are derived, two of which show the connections of these polynomials with Stirling numbers of the first kind and the type 2 Apostol-type poly-Bernoulli polynomials.

A new family of degenerate poly-Bernoulli polynomials of the second kind with its certain related properties

<abstract><p>The main object of this article is to present type 2 degenerate poly-Bernoulli polynomials of the second kind and numbers by arising from modified degenerate polyexponential function and investigate some properties of them. Thereafter, we treat the type 2 degenerate unipoly-Bernoulli polynomials of the second kind via modified degenerate polyexponential function and derive several properties of these polynomials. Furthermore, some new identities and explicit expressions for degenerate unipoly polynomials related to special numbers and polynomials are obtained. In addition, certain related beautiful zeros and graphical representations are displayed with the help of <italic>Mathematica</italic>.</p></abstract>

A Note on the Poly-Bernoulli Polynomials of the Second Kind

In this paper, we define the poly-Bernoulli polynomials of the second kind by using the polyexponential function and find some interesting identities of those polynomials. In addition, we define unipoly-Bernoulli polynomials of the second kind and study some properties of those polynomials.

Note on the Type 2 Degenerate Multi-Poly-Euler Polynomials

Kim and Kim (Russ. J. Math. Phys. 26, 2019, 40-49) introduced polyexponential function as an inverse to the polylogarithm function and by this, constructed a new type poly-Bernoulli polynomials. Recently, by using the polyexponential function, a number of generalizations of some polynomials and numbers have been presented and investigated. Motivated by these researches, in this paper, multi-poly-Euler polynomials are considered utilizing the degenerate multiple polyexponential functions and then, their properties and relations are investigated and studied. That the type 2 degenerate multi-poly-Euler polynomials equal a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the first kind is proved. Moreover, an addition formula and a derivative formula are derived. Furthermore, in a special case, a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers is shown.

Degenerate polyexponential-Genocchi numbers and polynomials

Recently, Kim et al. in [T. Kim, D. S. Kim, H. Y. Kim, L.-C. Jang, Informatica, \(\bf 3\) (2020), 8 pages] studied the degenerate poly-Bernoulli numbers and polynomials which are defined by using the polylogarithm function. In this paper, we study the degenerate polyexponential-Genocchi polynomials and numbers arising from polyexponential function and derive their explicit expressions and some identity involving them. In the final section, we introduce degenerate unipoly-Genocchi polynomials attached to an arithmetic function, by using polylogarithm function and investigate some identities for those polynomials.