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Special Polynomials
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Special Polynomials

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (30 March 2020) | Viewed by 47855

Special Issue Editor

Prof. Dr. Taekyun Kim

E-Mail Website1 Website2
Guest Editor
Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea
Interests: q-Calculus; p-adic Analysis; Analytic Number Theory; Complex Analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Special numbers and polynomials play an extremely important role in the development of several branches of mathematics, physics and engineering. The problems arising in the mathematical physics and engineering fields are mathematically framed in terms of differential equations. Most of them can only be solved using families of special functions which provide new means of mathematical analysis. Such families can be described in various ways, for instance, by orthogonality conditions, as solutions to differential equations, by generating functions, by recurrence relations, by operational formulas, and by bosonic and fermionic p-adic integrals. In last few years, attention has been centered on obtaining new representations of families of special functions and polynomials.

Each paper that will be published in this Special Issue aims to enrich the understanding of current research problems, theories and applications of some special functions and polynomials.

Prof. Dr. Taekyun Kim
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • the modern umbral calculus (binomial, Appell, and Sheffer polynomial sequences)
  • number theory and special functions
  • applications of special functions to statistics, physical sciences and engineering
  • applications of bosonic and fermionic p-adic integrals to special functions and polynomials

Published Papers (22 papers)

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Research

16 pages, 245 KiB  
Article
A Parametric Kind of Fubini Polynomials of a Complex Variable
by Sunil Kumar Sharma, Waseem A. Khan and Cheon Seoung Ryoo
Mathematics 2020, 8(4), 643; https://doi.org/10.3390/math8040643 - 22 Apr 2020
Cited by 5 | Viewed by 1740
Abstract
In this paper, we propose a parametric kind of Fubini polynomials by defining the two specific generating functions. We also investigate some analytical properties (for example, summation formulae, differential formulae and relationships with other well-known polynomials and numbers) for our introduced polynomials in [...] Read more.
In this paper, we propose a parametric kind of Fubini polynomials by defining the two specific generating functions. We also investigate some analytical properties (for example, summation formulae, differential formulae and relationships with other well-known polynomials and numbers) for our introduced polynomials in a systematic way. Furthermore, we consider some relationships for parametric kind of Fubini polynomials associated with Bernoulli, Euler, and Genocchi polynomials and Stirling numbers of the second kind. Full article
(This article belongs to the Special Issue Special Polynomials)
12 pages, 240 KiB  
Article
Generalized Tepper’s Identity and Its Application
by Dmitry Kruchinin, Vladimir Kruchinin and Yilmaz Simsek
Mathematics 2020, 8(2), 243; https://doi.org/10.3390/math8020243 - 14 Feb 2020
Cited by 4 | Viewed by 1964
Abstract
The aim of this paper is to study the Tepper identity, which is very important in number theory and combinatorial analysis. Using generating functions and compositions of generating functions, we derive many identities and relations associated with the Bernoulli numbers and polynomials, the [...] Read more.
The aim of this paper is to study the Tepper identity, which is very important in number theory and combinatorial analysis. Using generating functions and compositions of generating functions, we derive many identities and relations associated with the Bernoulli numbers and polynomials, the Euler numbers and polynomials, and the Stirling numbers. Moreover, we give applications related to the Tepper identity and these numbers and polynomials. Full article
(This article belongs to the Special Issue Special Polynomials)
17 pages, 278 KiB  
Article
Studies in Sums of Finite Products of the Second, Third, and Fourth Kind Chebyshev Polynomials
by Taekyun Kim, Dae San Kim, Hyunseok Lee and Jongkyum Kwon
Mathematics 2020, 8(2), 210; https://doi.org/10.3390/math8020210 - 07 Feb 2020
Cited by 3 | Viewed by 1548
Abstract
In this paper, we consider three sums of finite products of Chebyshev polynomials of two different kinds, namely sums of finite products of the second and third kind Chebyshev polynomials, those of the second and fourth kind Chebyshev polynomials, and those of the [...] Read more.
In this paper, we consider three sums of finite products of Chebyshev polynomials of two different kinds, namely sums of finite products of the second and third kind Chebyshev polynomials, those of the second and fourth kind Chebyshev polynomials, and those of the third and fourth kind Chebyshev polynomials. As a generalization of the classical linearization problem, we represent each of such sums of finite products as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials. These are done by explicit computations and the coefficients involve terminating hypergeometric functions 2 F 1 , 1 F 1 , 2 F 2 , and 4 F 3 . Full article
(This article belongs to the Special Issue Special Polynomials)
13 pages, 251 KiB  
Article
Some Identities Involving Certain Hardy Sums and General Kloosterman Sums
by Huifang Zhang and Tianping Zhang
Mathematics 2020, 8(1), 95; https://doi.org/10.3390/math8010095 - 07 Jan 2020
Cited by 2 | Viewed by 1329
Abstract
Using the properties of Gauss sums, the orthogonality relation of character sum and the mean value of Dirichlet L-function, we obtain some exact computational formulas for the hybrid mean value involving general Kloosterman sums [...] Read more.
Using the properties of Gauss sums, the orthogonality relation of character sum and the mean value of Dirichlet L-function, we obtain some exact computational formulas for the hybrid mean value involving general Kloosterman sums K ( r , l , λ ; p ) and certain Hardy sums S 1 ( h , q ) ∑ m = 1 p − 1 ∑ s = 1 p − 1 K ( m , n , λ ; p ) K ( s , t , λ ; p ) S 1 ( 2 m s ¯ , p ) , ∑ m = 1 p − 1 ∑ s = 1 p − 1 | K ( m , n , λ ; p ) | 2 | K ( s , t , λ ; p ) | 2 S 1 ( 2 m s ¯ , p ) . Our results not only cover the previous results, but also contain something quite new. Actually the previous authors just consider the case of the principal character λ modulo p, while we consider all the cases. Full article
(This article belongs to the Special Issue Special Polynomials)
8 pages, 234 KiB  
Article
Some Identities of Degenerate Bell Polynomials
by Taekyun Kim, Dae San Kim, Han Young Kim and Jongkyum Kwon
Mathematics 2020, 8(1), 40; https://doi.org/10.3390/math8010040 - 01 Jan 2020
Cited by 28 | Viewed by 2137
Abstract
The new type degenerate of Bell polynomials and numbers were recently introduced, which are a degenerate version of Bell polynomials and numbers and are different from the previously introduced partially degenerate Bell polynomials and numbers. Several expressions and identities on those polynomials and [...] Read more.
The new type degenerate of Bell polynomials and numbers were recently introduced, which are a degenerate version of Bell polynomials and numbers and are different from the previously introduced partially degenerate Bell polynomials and numbers. Several expressions and identities on those polynomials and numbers were obtained. In this paper, as a further investigation of the new type degenerate Bell polynomials, we derive several identities involving those degenerate Bell polynomials, Stirling numbers of the second kind and Carlitz’s degenerate Bernoulli or degenerate Euler polynomials. In addition, we obtain an identity connecting the degenerate Bell polynomials, Cauchy polynomials, Bernoulli numbers, Stirling numbers of the second kind and degenerate Stirling numbers of the second kind. Full article
(This article belongs to the Special Issue Special Polynomials)
12 pages, 231 KiB  
Article
A Note on Some Identities of New Type Degenerate Bell Polynomials
by Taekyun Kim, Dae San Kim, Hyunseok Lee and Jongkyum Kwon
Mathematics 2019, 7(11), 1086; https://doi.org/10.3390/math7111086 - 11 Nov 2019
Cited by 11 | Viewed by 1754
Abstract
Recently, the partially degenerate Bell polynomials and numbers, which are a degenerate version of Bell polynomials and numbers, were introduced. In this paper, we consider the new type degenerate Bell polynomials and numbers, and obtain several expressions and identities on those polynomials and [...] Read more.
Recently, the partially degenerate Bell polynomials and numbers, which are a degenerate version of Bell polynomials and numbers, were introduced. In this paper, we consider the new type degenerate Bell polynomials and numbers, and obtain several expressions and identities on those polynomials and numbers. In more detail, we obtain an expression involving the Stirling numbers of the second kind and the generalized falling factorial sequences, Dobinski type formulas, an expression connected with the Stirling numbers of the first and second kinds, and an expression involving the Stirling polynomials of the second kind. Full article
(This article belongs to the Special Issue Special Polynomials)
17 pages, 292 KiB  
Article
A Class of Sheffer Sequences of Some Complex Polynomials and Their Degenerate Types
by Dojin Kim
Mathematics 2019, 7(11), 1064; https://doi.org/10.3390/math7111064 - 06 Nov 2019
Cited by 3 | Viewed by 1757
Abstract
We study some properties of Sheffer sequences for some special polynomials with complex Changhee and Daehee polynomials introducing their complex versions of the polynomials and splitting them into real and imaginary parts using trigonometric polynomial sequences. Moreover, considering their degenerate types of Sheffer [...] Read more.
We study some properties of Sheffer sequences for some special polynomials with complex Changhee and Daehee polynomials introducing their complex versions of the polynomials and splitting them into real and imaginary parts using trigonometric polynomial sequences. Moreover, considering their degenerate types of Sheffer sequences based on umbral composition, we present some useful expressions, properties, and examples about complex versions of the degenerate polynomials. Full article
(This article belongs to the Special Issue Special Polynomials)
9 pages, 228 KiB  
Article
On the High-Power Mean of the Generalized Gauss Sums and Kloosterman Sums
by Xinyu Liu and Wenpeng Zhang
Mathematics 2019, 7(10), 907; https://doi.org/10.3390/math7100907 - 27 Sep 2019
Cited by 16 | Viewed by 1737
Abstract
The main aim of this paper is to use the properties of the trigonometric sums and character sums, and the number of the solutions of several symmetry congruence equations to research the computational problem of a certain sixth power mean of the generalized [...] Read more.
The main aim of this paper is to use the properties of the trigonometric sums and character sums, and the number of the solutions of several symmetry congruence equations to research the computational problem of a certain sixth power mean of the generalized Gauss sums and generalized Kloosterman sums, and to give two exact computational formulae for them. Full article
(This article belongs to the Special Issue Special Polynomials)
12 pages, 272 KiB  
Article
A Note on Type 2 w-Daehee Polynomials
by Minyoung Ma and Dongkyu Lim
Mathematics 2019, 7(8), 697; https://doi.org/10.3390/math7080697 - 02 Aug 2019
Viewed by 1881
Abstract
In the paper, by virtue of the p-adic invariant integral on Z p , the authors consider a type 2 w-Daehee polynomials and present some properties and identities of these polynomials related with well-known special polynomials. In addition, we present some [...] Read more.
In the paper, by virtue of the p-adic invariant integral on Z p , the authors consider a type 2 w-Daehee polynomials and present some properties and identities of these polynomials related with well-known special polynomials. In addition, we present some symmetric identities involving the higher order type 2 w-Daehee polynomials. These identities extend and generalize some known results. Full article
(This article belongs to the Special Issue Special Polynomials)
11 pages, 264 KiB  
Article
A Note on Type 2 Degenerate q-Euler Polynomials
by Taekyun Kim, Dae San Kim, Han Young Kim and Sung-Soo Pyo
Mathematics 2019, 7(8), 681; https://doi.org/10.3390/math7080681 - 30 Jul 2019
Cited by 1 | Viewed by 1850
Abstract
Recently, type 2 degenerate Euler polynomials and type 2 q-Euler polynomials were studied, respectively, as degenerate versions of the type 2 Euler polynomials as well as a q-analog of the type 2 Euler polynomials. In this paper, we consider the type [...] Read more.
Recently, type 2 degenerate Euler polynomials and type 2 q-Euler polynomials were studied, respectively, as degenerate versions of the type 2 Euler polynomials as well as a q-analog of the type 2 Euler polynomials. In this paper, we consider the type 2 degenerate q-Euler polynomials, which are derived from the fermionic p-adic q-integrals on Z p , and investigate some properties and identities related to these polynomials and numbers. In detail, we give for these polynomials several expressions, generating function, relations with type 2 q-Euler polynomials and the expression corresponding to the representation of alternating integer power sums in terms of Euler polynomials. One novelty about this paper is that the type 2 degenerate q-Euler polynomials arise naturally by means of the fermionic p-adic q-integrals so that it is possible to easily find some identities of symmetry for those polynomials and numbers, as were done previously. Full article
(This article belongs to the Special Issue Special Polynomials)
12 pages, 234 KiB  
Article
The General Model for Least Convex Disparity RIM Quantifier Problems
by Dug Hun Hong
Mathematics 2019, 7(7), 576; https://doi.org/10.3390/math7070576 - 28 Jun 2019
Viewed by 1445
Abstract
Hong (Mathematics 2019, 7, 326) recently introduced the general least squares deviation (LSD) model for ordered weighted averaging (OWA) operator weights. In this paper, we propose the corresponding generalized least square disparity model for regular increasing monotone (RIM) quantifier determination under a given [...] Read more.
Hong (Mathematics 2019, 7, 326) recently introduced the general least squares deviation (LSD) model for ordered weighted averaging (OWA) operator weights. In this paper, we propose the corresponding generalized least square disparity model for regular increasing monotone (RIM) quantifier determination under a given orness level. We prove this problem mathematically. Using this result, we provide the full solution of the least square disparity RIM quantifier model as an illustrative example. Full article
(This article belongs to the Special Issue Special Polynomials)
20 pages, 333 KiB  
Article
The General Least Square Deviation OWA Operator Problem
by Dug Hun Hong and Sangheon Han
Mathematics 2019, 7(4), 326; https://doi.org/10.3390/math7040326 - 03 Apr 2019
Cited by 5 | Viewed by 1990
Abstract
A crucial issue in applying the ordered weighted averaging (OWA) operator for decision making is the determination of the associated weights. This paper proposes a general least convex deviation model for OWA operators which attempts to obtain the desired OWA weight vector under [...] Read more.
A crucial issue in applying the ordered weighted averaging (OWA) operator for decision making is the determination of the associated weights. This paper proposes a general least convex deviation model for OWA operators which attempts to obtain the desired OWA weight vector under a given orness level to minimize the least convex deviation after monotone convex function transformation of absolute deviation. The model includes the least square deviation (LSD) OWA operators model suggested by Wang, Luo and Liu in Computers & Industrial Engineering, 2007, as a special class. We completely prove this constrained optimization problem analytically. Using this result, we also give solution of LSD model suggested by Wang, Luo and Liu as a function of n and α completely. We reconsider two numerical examples that Wang, Luo and Liu, 2007 and Sang and Liu, Fuzzy Sets and Systems, 2014, showed and consider another different type of the model to illustrate our results. Full article
(This article belongs to the Special Issue Special Polynomials)
16 pages, 771 KiB  
Article
Representing by Orthogonal Polynomials for Sums of Finite Products of Fubini Polynomials
by Dae San Kim, Dmitry V. Dolgy, Dojin Kim and Taekyun Kim
Mathematics 2019, 7(4), 319; https://doi.org/10.3390/math7040319 - 29 Mar 2019
Cited by 4 | Viewed by 2395
Abstract
In the classical connection problem, it is dealt with determining the coefficients in the expansion of the product of two polynomials with regard to any given sequence of polynomials. As a generalization of this problem, we will consider sums of finite products of [...] Read more.
In the classical connection problem, it is dealt with determining the coefficients in the expansion of the product of two polynomials with regard to any given sequence of polynomials. As a generalization of this problem, we will consider sums of finite products of Fubini polynomials and represent these in terms of orthogonal polynomials. Here, the involved orthogonal polynomials are Chebyshev polynomials of the first, second, third and fourth kinds, and Hermite, extended Laguerre, Legendre, Gegenbauer, and Jabcobi polynomials. These representations are obtained by explicit computations. Full article
(This article belongs to the Special Issue Special Polynomials)
12 pages, 297 KiB  
Article
On the (p, q)–Chebyshev Polynomials and Related Polynomials
by Can Kızılateş, Naim Tuğlu and Bayram Çekim
Mathematics 2019, 7(2), 136; https://doi.org/10.3390/math7020136 - 01 Feb 2019
Cited by 8 | Viewed by 2500
Abstract
In this paper, we introduce ( p , q ) –Chebyshev polynomials of the first and second kind that reduces the ( p , q ) –Fibonacci and the ( p , q ) –Lucas polynomials. These polynomials have explicit forms and generating [...] Read more.
In this paper, we introduce ( p , q ) –Chebyshev polynomials of the first and second kind that reduces the ( p , q ) –Fibonacci and the ( p , q ) –Lucas polynomials. These polynomials have explicit forms and generating functions are given. Then, derivative properties between these first and second kind polynomials, determinant representations, multilateral and multilinear generating functions are derived. Full article
(This article belongs to the Special Issue Special Polynomials)
14 pages, 324 KiB  
Article
On Gould–Hopper-Based Fully Degenerate Poly-Bernoulli Polynomials with a q-Parameter
by Ugur Duran and Patrick Njionou Sadjang
Mathematics 2019, 7(2), 121; https://doi.org/10.3390/math7020121 - 23 Jan 2019
Cited by 7 | Viewed by 2045
Abstract
We firstly consider the fully degenerate Gould–Hopper polynomials with a q parameter and investigate some of their properties including difference rule, inversion formula and addition formula. We then introduce the Gould–Hopper-based fully degenerate poly-Bernoulli polynomials with a q parameter and provide some of [...] Read more.
We firstly consider the fully degenerate Gould–Hopper polynomials with a q parameter and investigate some of their properties including difference rule, inversion formula and addition formula. We then introduce the Gould–Hopper-based fully degenerate poly-Bernoulli polynomials with a q parameter and provide some of their diverse basic identities and properties including not only addition property, but also difference rule properties. By the same way of mentioned polynomials, we define the Gould–Hopper-based fully degenerate ( α , q ) -Stirling polynomials of the second kind, and then give many relations. Moreover, we derive multifarious correlations and identities for foregoing polynomials and numbers, including recurrence relations and implicit summation formulas. Full article
(This article belongs to the Special Issue Special Polynomials)
9 pages, 220 KiB  
Article
Change of Basis Transformation from the Bernstein Polynomials to the Chebyshev Polynomials of the Fourth Kind
by Abedallah Rababah and Esraa Hijazi
Mathematics 2019, 7(2), 120; https://doi.org/10.3390/math7020120 - 23 Jan 2019
Viewed by 2084
Abstract
In this paper, the change of bases transformations between the Bernstein polynomial basis and the Chebyshev polynomial basis of the fourth kind are studied and the matrices of transformation among these bases are constructed. Some examples are given. Full article
(This article belongs to the Special Issue Special Polynomials)
11 pages, 254 KiB  
Article
Some Identities for the Two Variable Fubini Polynomials
by Dan-Dan Su and Yuan He
Mathematics 2019, 7(2), 115; https://doi.org/10.3390/math7020115 - 22 Jan 2019
Cited by 9 | Viewed by 2017
Abstract
In this paper, we perform a further investigation for the Fubini polynomials. By making use of the generating function methods and Padé approximation techniques, we establish some new identities for the two variable Fubini polynomials. Some special cases as well as immediate consequences [...] Read more.
In this paper, we perform a further investigation for the Fubini polynomials. By making use of the generating function methods and Padé approximation techniques, we establish some new identities for the two variable Fubini polynomials. Some special cases as well as immediate consequences of the main results presented here are also considered. Full article
(This article belongs to the Special Issue Special Polynomials)
7 pages, 242 KiB  
Article
Some Types of Identities Involving the Legendre Polynomials
by Shimeng Shen and Li Chen
Mathematics 2019, 7(2), 114; https://doi.org/10.3390/math7020114 - 22 Jan 2019
Cited by 7 | Viewed by 3152
Abstract
In this paper, a new non-linear recursive sequence is firstly introduced. Then, using this sequence, a computational problem involving the convolution of the Legendre polynomial is studied using the basic and combinatorial methods. Finally, we give an interesting identity. Full article
(This article belongs to the Special Issue Special Polynomials)
15 pages, 265 KiB  
Article
Representing Sums of Finite Products of Chebyshev Polynomials of the First Kind and Lucas Polynomials by Chebyshev Polynomials
by Taekyun Kim, Dae San Kim, Dmitry V. Dolgy and Jongkyum Kwon
Mathematics 2019, 7(1), 26; https://doi.org/10.3390/math7010026 - 27 Dec 2018
Cited by 15 | Viewed by 2801
Abstract
In this paper, we study sums of finite products of Chebyshev polynomials of the first kind and Lucas polynomials and represent each of them in terms of Chebyshev polynomials of all kinds. Here, the coefficients involve terminating hypergeometric functions 2 F 1 and [...] Read more.
In this paper, we study sums of finite products of Chebyshev polynomials of the first kind and Lucas polynomials and represent each of them in terms of Chebyshev polynomials of all kinds. Here, the coefficients involve terminating hypergeometric functions 2 F 1 and these representations are obtained by explicit computations. Full article
(This article belongs to the Special Issue Special Polynomials)
8 pages, 223 KiB  
Article
Some Identities Involving Fibonacci Polynomials and Fibonacci Numbers
by Yuankui Ma and Wenpeng Zhang
Mathematics 2018, 6(12), 334; https://doi.org/10.3390/math6120334 - 18 Dec 2018
Cited by 17 | Viewed by 4134
Abstract
The aim of this paper is to research the structural properties of the Fibonacci polynomials and Fibonacci numbers and obtain some identities. To achieve this purpose, we first introduce a new second-order nonlinear recursive sequence. Then, we obtain our main results by using [...] Read more.
The aim of this paper is to research the structural properties of the Fibonacci polynomials and Fibonacci numbers and obtain some identities. To achieve this purpose, we first introduce a new second-order nonlinear recursive sequence. Then, we obtain our main results by using this new sequence, the properties of the power series, and the combinatorial methods. Full article
(This article belongs to the Special Issue Special Polynomials)
6 pages, 209 KiB  
Article
A Note on the Classical Gauss Sums
by Tingting Wang and Guohui Chen
Mathematics 2018, 6(12), 313; https://doi.org/10.3390/math6120313 - 08 Dec 2018
Cited by 5 | Viewed by 2337
Abstract
The main purpose of this paper is to study the computational problem of one kind rational polynomials of the classical Gauss sums, and using the purely algebraic methods and the properties of the character sums mod p (a prime with [...] Read more.
The main purpose of this paper is to study the computational problem of one kind rational polynomials of the classical Gauss sums, and using the purely algebraic methods and the properties of the character sums mod p (a prime with p ≡ 1 mod 12 ) to give an exact evaluation formula for it. Full article
(This article belongs to the Special Issue Special Polynomials)
6 pages, 200 KiB  
Article
Some Identities Involving the Fubini Polynomials and Euler Polynomials
by Guohui Chen and Li Chen
Mathematics 2018, 6(12), 300; https://doi.org/10.3390/math6120300 - 04 Dec 2018
Cited by 7 | Viewed by 2349
Abstract
In this paper, we first introduce a new second-order non-linear recursive polynomials U h , i ( x ) , and then use these recursive polynomials, the properties of the power series and the combinatorial methods to prove some identities involving the Fubini [...] Read more.
In this paper, we first introduce a new second-order non-linear recursive polynomials U h , i ( x ) , and then use these recursive polynomials, the properties of the power series and the combinatorial methods to prove some identities involving the Fubini polynomials, Euler polynomials and Euler numbers. Full article
(This article belongs to the Special Issue Special Polynomials)
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