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14 pages, 241 KB

Open AccessArticle

A Note on Higher Order Degenerate Changhee–Genocchi Numbers and Polynomials of the Second Kind

by Liwei Liu and Wuyungaowa

Symmetry 2023, 15(1), 56; https://doi.org/10.3390/sym15010056 - 26 Dec 2022

Viewed by 1231

In this paper, we consider the higher order degenerate Changhee–Genocchi polynomials of the second kind by using generating functions and the Riordan matrix methods. At the same time, we give some properties of the higher order degenerate Changhee–Genocchi polynomials of the second kind. [...] Read more.

In this paper, we consider the higher order degenerate Changhee–Genocchi polynomials of the second kind by using generating functions and the Riordan matrix methods. At the same time, we give some properties of the higher order degenerate Changhee–Genocchi polynomials of the second kind. In addition, we establish some new equalities involving the higher order degenerate Changhee–Genocchi polynomials of the second kind, the generalized Bell polynomials, higher order Changhee polynomials, the higher order degenerate Daehee polynomials of the second kind, Lah numbers and Stirling numbers, etc. Full article

14 pages, 282 KB

Open AccessArticle

A Note on Degenerate Catalan-Daehee Numbers and Polynomials

by Waseem Ahmad Khan, Maryam Salem Alatawi and Ugur Duran

Symmetry 2022, 14(10), 2169; https://doi.org/10.3390/sym14102169 - 16 Oct 2022

Cited by 4 | Viewed by 1545

Abstract

In this paper, we consider the degenerate forms of the Catalan–Daehee polynomials and numbers by the Volkenborn integrals and obtain diverse explicit expressions and formulas. Moreover, we show the expressions of the degenerate Catalan–Daehee numbers in terms of

λ

-Daehee numbers, Stirling numbers [...] Read more.

In this paper, we consider the degenerate forms of the Catalan–Daehee polynomials and numbers by the Volkenborn integrals and obtain diverse explicit expressions and formulas. Moreover, we show the expressions of the degenerate Catalan–Daehee numbers in terms of

λ

-Daehee numbers, Stirling numbers of the first kind and Bernoulli polynomials, and we also obtain a relation covering the Bernoulli numbers, the degenerate Catalan–Daehee numbers and Stirling numbers of the second kind. In addition, we prove an implicit summation formula and a symmetric identity, and we derive an explicit expression for the degenerate Catalan–Daehee polynomials including the Stirling numbers of the first kind and Bernoulli polynomials. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

13 pages, 273 KB

Open AccessArticle

Fully Degenerating of Daehee Numbers and Polynomials

by Sahar Albosaily, Waseem Ahmad Khan, Serkan Araci and Azhar Iqbal

Mathematics 2022, 10(14), 2528; https://doi.org/10.3390/math10142528 - 20 Jul 2022

Cited by 2 | Viewed by 1735

Abstract

In this paper, we consider fully degenerate Daehee numbers and polynomials by using degenerate logarithm function. We investigate some properties of these numbers and polynomials. We also introduce higher-order multiple fully degenerate Daehee polynomials and numbers which can be represented in terms of [...] Read more.

In this paper, we consider fully degenerate Daehee numbers and polynomials by using degenerate logarithm function. We investigate some properties of these numbers and polynomials. We also introduce higher-order multiple fully degenerate Daehee polynomials and numbers which can be represented in terms of Riemann integrals on the interval

[0, 1]

. Finally, we derive their summation formulae. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

15 pages, 342 KB

Open AccessArticle

Some Generalized Properties of Poly-Daehee Numbers and Polynomials Based on Apostol–Genocchi Polynomials

by Talha Usman, Nabiullah Khan, Mohd Aman, Shrideh Al-Omari, Kamsing Nonlaopon and Junesang Choi

Mathematics 2022, 10(14), 2502; https://doi.org/10.3390/math10142502 - 18 Jul 2022

Cited by 4 | Viewed by 1953

Abstract

Numerous polynomial variations and their extensions have been explored extensively and found applications in a variety of research fields. The purpose of this research is to establish a unified class of Apostol–Genocchi polynomials based on poly-Daehee polynomials and to explore some of their [...] Read more.

Numerous polynomial variations and their extensions have been explored extensively and found applications in a variety of research fields. The purpose of this research is to establish a unified class of Apostol–Genocchi polynomials based on poly-Daehee polynomials and to explore some of their features and identities. We investigate these polynomials via generating functions and deduce various identities, summation formulae, differential and integral formulas, implicit summation formulae, and several characterized generating functions for new numbers and polynomials. Finally, by using an operational version of Apostol–Genocchi polynomials, we derive some results in terms of new special polynomials. Due to the generic nature of the findings described here, they are used to reduce and generate certain known or novel formulae and identities for relatively simple polynomials and numbers. Full article

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

11 pages, 264 KB

Open AccessArticle

Some Explicit Expressions for Twisted Catalan-Daehee Numbers

by Dongkyu Lim

Symmetry 2022, 14(2), 189; https://doi.org/10.3390/sym14020189 - 19 Jan 2022

Cited by 3 | Viewed by 1591

Abstract

In this paper, the author considers the twisted Catalan numbers and the twisted Catalan-Daehee numbers, which are arisen from p-adic fermionic integrals and p-adic invariant integrals, respectively. We give some explicit identities and properties for those twisted numbers and polynomials by [...] Read more.

In this paper, the author considers the twisted Catalan numbers and the twisted Catalan-Daehee numbers, which are arisen from p-adic fermionic integrals and p-adic invariant integrals, respectively. We give some explicit identities and properties for those twisted numbers and polynomials by using p-adic integrals or generating functions. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Their Applications)

7 pages, 248 KB

Open AccessArticle

Some Identities on the Twisted q-Analogues of Catalan-Daehee Numbers and Polynomials

by Dongkyu Lim

Axioms 2022, 11(1), 9; https://doi.org/10.3390/axioms11010009 - 23 Dec 2021

Viewed by 2483

Abstract

In this paper, the author considers twisted q-analogues of Catalan-Daehee numbers and polynomials by using p-adic q-integral on

Z_{p}

. We derive some explicit identities for those twisted numbers and polynomials related to various special numbers and polynomials. Full article

(This article belongs to the Special Issue p-adic Analysis and q-Calculus with Their Applications)

13 pages, 311 KB

Open AccessArticle

New Families of Special Polynomial Identities Based upon Combinatorial Sums Related to p-Adic Integrals

by Yilmaz Simsek

Symmetry 2021, 13(8), 1484; https://doi.org/10.3390/sym13081484 - 13 Aug 2021

Cited by 1 | Viewed by 1834

Abstract

The aim of this paper is to study and investigate generating-type functions, which have been recently constructed by the author, with the aid of the Euler’s identity, combinatorial sums, and p-adic integrals. Using these generating functions with their functional equation, we derive [...] Read more.

The aim of this paper is to study and investigate generating-type functions, which have been recently constructed by the author, with the aid of the Euler’s identity, combinatorial sums, and p-adic integrals. Using these generating functions with their functional equation, we derive various interesting combinatorial sums and identities including new families of numbers and polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, the Changhee numbers, and other numbers and polynomials. Moreover, we present some revealing remarks and comments on the results of this paper. Full article

(This article belongs to the Special Issue The 33rd International Conference of The Jangjeon Mathematical Society (ICJMS2020) will be held at Hasanuddin University, Makassar, Indonesia)

13 pages, 282 KB

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 18 | Viewed by 2620

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)

16 pages, 319 KB

Open AccessArticle

Some Identities of the Degenerate Higher Order Derangement Polynomials and Numbers

by Hye Kyung Kim

Symmetry 2021, 13(2), 176; https://doi.org/10.3390/sym13020176 - 22 Jan 2021

Cited by 2 | Viewed by 1694

Abstract

Recently, Kim-Kim (J. Math. Anal. Appl. (2021), Vol. 493(1), 124521) introduced the

λ

-Sheffer sequence and the degenerate Sheffer sequence. In addition, Kim et al. (arXiv:2011.08535v1 17 November 2020) studied the degenerate derangement polynomials and numbers, and investigated some properties of those polynomials [...] Read more.

Recently, Kim-Kim (J. Math. Anal. Appl. (2021), Vol. 493(1), 124521) introduced the

λ

-Sheffer sequence and the degenerate Sheffer sequence. In addition, Kim et al. (arXiv:2011.08535v1 17 November 2020) studied the degenerate derangement polynomials and numbers, and investigated some properties of those polynomials without using degenerate umbral calculus. In this paper, the y the degenerate derangement polynomials of order s (

s \in N

) and give a combinatorial meaning about higher order derangement numbers. In addition, the author gives some interesting identities related to the degenerate derangement polynomials of order s and special polynomials and numbers by using degenerate Sheffer sequences, and at the same time derive the inversion formulas of these identities. Full article

(This article belongs to the Special Issue The 33rd International Conference of The Jangjeon Mathematical Society (ICJMS2020) will be held at Hasanuddin University, Makassar, Indonesia)

13 pages, 306 KB

Open AccessArticle

Some New Families of Special Polynomials and Numbers Associated with Finite Operators

by Yilmaz Simsek

Symmetry 2020, 12(2), 237; https://doi.org/10.3390/sym12020237 - 4 Feb 2020

Cited by 12 | Viewed by 2383

Abstract

The aim of this study was to define a new operator. This operator unify and modify many known operators, some of which were introduced by the author. Many properties of this operator are given. Using this operator, two new classes of special polynomials [...] Read more.

The aim of this study was to define a new operator. This operator unify and modify many known operators, some of which were introduced by the author. Many properties of this operator are given. Using this operator, two new classes of special polynomials and numbers are defined. Many identities and relationships are derived, including these new numbers and polynomials, combinatorial sums, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, and the Changhee numbers. By applying the derivative operator to these new polynomials, derivative formulas are found. Integral representations, including the Volkenborn integral, the fermionic p-adic integral, and the Riemann integral, are given for these new polynomials. Full article

(This article belongs to the Special Issue The 32th Congress of The Jangjeon Mathematical Society (ICJMS2019) will be Held at Far Eastern Federal Universit, Vladivostok Russia)

16 pages, 272 KB

Open AccessArticle

Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function

by Irem Kucukoglu, Burcin Simsek and Yilmaz Simsek

Axioms 2019, 8(4), 112; https://doi.org/10.3390/axioms8040112 - 11 Oct 2019

Cited by 24 | Viewed by 3982

Abstract

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new [...] Read more.

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

13 pages, 751 KB

Open AccessArticle

On Generating Functions for Boole Type Polynomials and Numbers of Higher Order and Their Applications

by Yilmaz Simsek and Ji Suk So

Symmetry 2019, 11(3), 352; https://doi.org/10.3390/sym11030352 - 8 Mar 2019

Cited by 14 | Viewed by 2343

Abstract

The purpose of this manuscript is to study and investigate generating functions for Boole type polynomials and numbers of higher order. With the help of these generating functions, many properties of Boole type polynomials and numbers are presented. By applications of partial derivative [...] Read more.

The purpose of this manuscript is to study and investigate generating functions for Boole type polynomials and numbers of higher order. With the help of these generating functions, many properties of Boole type polynomials and numbers are presented. By applications of partial derivative and functional equations for these functions, derivative formulas, recurrence relations and finite combinatorial sums involving the Apostol-Euler polynomials, the Stirling numbers and the Daehee numbers are given. Full article

10 pages, 236 KB

Open AccessArticle

Degenerate Daehee Numbers of the Third Kind

by Sung-Soo Pyo, Taekyun Kim and Seog-Hoon Rim

Mathematics 2018, 6(11), 239; https://doi.org/10.3390/math6110239 - 6 Nov 2018

Cited by 6 | Viewed by 2623

Abstract

In this paper, we define new Daehee numbers, the degenerate Daehee numbers of the third kind, using the degenerate log function as generating function. We obtain some identities for the degenerate Daehee numbers of the third kind associated with the Daehee, degenerate Daehee, [...] Read more.

In this paper, we define new Daehee numbers, the degenerate Daehee numbers of the third kind, using the degenerate log function as generating function. We obtain some identities for the degenerate Daehee numbers of the third kind associated with the Daehee, degenerate Daehee, and degenerate Daehee numbers of the second kind. In addition, we derive a differential equation associated with the degenerate log function. We deduce some identities from the differential equation. Full article

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