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20 pages, 338 KB

Open AccessArticle

Probabilistic Multiple-Integral Evaluation of Odd Dirichlet Beta and Even Zeta Functions and Proof of Digamma-Trigamma Reflections

by Antonio E. Bargellini, Daniele Ritelli and Giulia Spaletta

Foundations 2025, 5(3), 27; https://doi.org/10.3390/foundations5030027 - 11 Aug 2025

Viewed by 340

Abstract

The aim of this work was to construct explicit expressions for the summation of Dirichlet Beta functions with odd arguments and Zeta functions with even arguments. In the established literature, this is typically done using Fourier series expansions or Bernoulli numbers and polynomials. [...] Read more.

The aim of this work was to construct explicit expressions for the summation of Dirichlet Beta functions with odd arguments and Zeta functions with even arguments. In the established literature, this is typically done using Fourier series expansions or Bernoulli numbers and polynomials. Here, instead, we achieve our goal by employing tools from probability: specifically, we introduce a generalisation of a technique based on multiple integrals and the algebra of random variables. This also allows us to increase the number of nested integrals and Cauchy random variables involved. Another key contribution is that, by generalising the exponent of Cauchy random variables, we obtain an original proof of the reflection formulae for the Digamma and Trigamma functions. These probabilistic proofs crucially utilise the Mellin transform to compute the integrals needed to determine probability density functions. It is noteworthy that, while understanding the presented topic requires knowledge of the rules for calculating multiple integrals (Fubini’s Theorem) and the algebra of continuous random variables, these are concepts commonly acquired by second-year university students in STEM disciplines. Our study thus offers new perspectives on how the mathematical functions considered relate and shows the significant role of probabilistic methods in promoting comprehension of this research area, in a way accessible to a broad and non-specialist audience. Full article

(This article belongs to the Collection Editorial Board Members’ Collection Series: Feature Papers in Mathematical Sciences)

21 pages, 413 KB

Open AccessArticle

Construction of a Hybrid Class of Special Polynomials: Fubini–Bell-Based Appell Polynomials and Their Properties

by Yasir A. Madani, Abdulghani Muhyi, Khaled Aldwoah, Amel Touati, Khidir Shaib Mohamed and Ria H. Egami

Mathematics 2025, 13(6), 1009; https://doi.org/10.3390/math13061009 - 20 Mar 2025

Viewed by 473

Abstract

This paper aims to establish a new hybrid class of special polynomials, namely, the Fubini–Bell-based Appell polynomials. The monomiality principle is used to derive the generating function for these polynomials. Several related identities and properties, including symmetry identities, are explored. The determinant representation [...] Read more.

This paper aims to establish a new hybrid class of special polynomials, namely, the Fubini–Bell-based Appell polynomials. The monomiality principle is used to derive the generating function for these polynomials. Several related identities and properties, including symmetry identities, are explored. The determinant representation of the Fubini–Bell-based Appell polynomials is also established. Furthermore, some special members of the Fubini–Bell-based Appell family—such as the Fubini–Bell-based Bernoulli polynomials and the Fubini–Bell-based Euler polynomials—are derived, with analogous results presented for each. Finally, computational results and graphical representations of the zero distributions of these members are investigated. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications, 2nd Edition)

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12 pages, 275 KB

Open AccessArticle

Symmetric Identities Involving the Extended Degenerate Central Fubini Polynomials Arising from the Fermionic p-Adic Integral on ℤ_p

by Maryam Salem Alatawi, Waseem Ahmad Khan and Ugur Duran

Axioms 2024, 13(7), 421; https://doi.org/10.3390/axioms13070421 - 22 Jun 2024

Cited by 1 | Viewed by 938

Abstract

Since the constructions of p-adic q-integrals, these integrals as well as particular cases have been used not only as integral representations of many special functions, polynomials, and numbers, but they also allow for deep examinations of many families of special numbers [...] Read more.

Since the constructions of p-adic q-integrals, these integrals as well as particular cases have been used not only as integral representations of many special functions, polynomials, and numbers, but they also allow for deep examinations of many families of special numbers and polynomials, such as central Fubini, Bernoulli, central Bell, and Changhee numbers and polynomials. One of the key applications of these integrals is for obtaining the symmetric identities of certain special polynomials. In this study, we focus on a novel generalization of degenerate central Fubini polynomials. First, we introduce two variable degenerate w-torsion central Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. Through this representation, we investigate several symmetric identities for these polynomials using special p-adic integral techniques. Also, using series manipulation methods, we obtain an identity of symmetry for the two variable degenerate w-torsion central Fubini polynomials. Finally, we provide a representation of the degenerate differential operator on the two variable degenerate w-torsion central Fubini polynomials related to the degenerate central factorial polynomials of the second kind. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications, 2nd Edition)

11 pages, 264 KB

Open AccessArticle

Several Symmetric Identities of the Generalized Degenerate Fubini Polynomials by the Fermionic p-Adic Integral on

Z_{p}

by Maryam Salem Alatawi, Waseem Ahmad Khan and Ugur Duran

Symmetry 2024, 16(6), 686; https://doi.org/10.3390/sym16060686 - 3 Jun 2024

Cited by 1 | Viewed by 715

Abstract

After constructions of p-adic q-integrals, in recent years, these integrals with some of their special cases have not only been utilized as integral representations of many special numbers, polynomials, and functions but have also given the chance for deep analysis of [...] Read more.

After constructions of p-adic q-integrals, in recent years, these integrals with some of their special cases have not only been utilized as integral representations of many special numbers, polynomials, and functions but have also given the chance for deep analysis of many families of special polynomials and numbers, such as Bernoulli, Fubini, Bell, and Changhee polynomials and numbers. One of the main applications of these integrals is to obtain symmetric identities for the special polynomials. In this study, we focus on a novel extension of the degenerate Fubini polynomials and on obtaining some symmetric identities for them. First, we introduce the two-variable degenerate w-torsion Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. By this representation, we derive some new symmetric identities for these polynomials, using some special p-adic integral techniques. Lastly, by using some series manipulation techniques, we obtain more identities of symmetry for the two variable degenerate w-torsion Fubini polynomials. Full article

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

18 pages, 390 KB

Open AccessArticle

On Generalized Class of Bell Polynomials Associated with Geometric Applications

by Rashad A. Al-Jawfi, Abdulghani Muhyi and Wadia Faid Hassan Al-shameri

Axioms 2024, 13(2), 73; https://doi.org/10.3390/axioms13020073 - 23 Jan 2024

Cited by 5 | Viewed by 1540

Abstract

In this paper, we introduce a new class of special polynomials called the generalized Bell polynomials, constructed by combining two-variable general polynomials with two-variable Bell polynomials. The concept of the monomiality principle was employed to establish the generating function and obtain various results [...] Read more.

In this paper, we introduce a new class of special polynomials called the generalized Bell polynomials, constructed by combining two-variable general polynomials with two-variable Bell polynomials. The concept of the monomiality principle was employed to establish the generating function and obtain various results for these polynomials. We explore certain related identities, properties, as well as differential and integral formulas. Further, specific members within the generalized Bell family—such as the Gould-Hopper-Bell polynomials, Laguerre-Bell polynomials, truncated-exponential-Bell polynomials, Hermite-Appell-Bell polynomials, and Fubini-Bell polynomials—were examined, unveiling analogous outcomes for each. Finally, Mathematica was utilized to investigate the zero distributions of the Gould-Hopper-Bell polynomials. Full article

(This article belongs to the Section Mathematical Analysis)

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18 pages, 1242 KB

Open AccessArticle

Novel Properties of q-Sine-Based and q-Cosine-Based q-Fubini Polynomials

by Waseem Ahmad Khan, Maryam Salem Alatawi, Cheon Seoung Ryoo and Ugur Duran

Symmetry 2023, 15(2), 356; https://doi.org/10.3390/sym15020356 - 28 Jan 2023

Cited by 8 | Viewed by 1398

Abstract

The main purpose of this paper is to consider q-sine-based and q-cosine-Based q-Fubini polynomials and is to investigate diverse properties of these polynomials. Furthermore, multifarious correlations including q-analogues of the Genocchi, Euler and Bernoulli polynomials, and the q-Stirling [...] Read more.

The main purpose of this paper is to consider q-sine-based and q-cosine-Based q-Fubini polynomials and is to investigate diverse properties of these polynomials. Furthermore, multifarious correlations including q-analogues of the Genocchi, Euler and Bernoulli polynomials, and the q-Stirling numbers of the second kind are derived. Moreover, some approximate zeros of the q-sinebased and q-cosine-Based q-Fubini polynomials in a complex plane are examined, and lastly, these zeros are shown using figures. Full article

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

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10 pages, 276 KB

Open AccessArticle

Degenerate Fubini-Type Polynomials and Numbers, Degenerate Apostol–Bernoulli Polynomials and Numbers, and Degenerate Apostol–Euler Polynomials and Numbers

by Siqintuya Jin, Muhammet Cihat Dağli and Feng Qi

Axioms 2022, 11(9), 477; https://doi.org/10.3390/axioms11090477 - 17 Sep 2022

Cited by 4 | Viewed by 2654

Abstract

In this paper, by introducing degenerate Fubini-type polynomials, with the help of the Faà di Bruno formula and some properties of partial Bell polynomials, the authors provide several new explicit formulas and recurrence relations for Fubini-type polynomials and numbers, associate the newly defined [...] Read more.

In this paper, by introducing degenerate Fubini-type polynomials, with the help of the Faà di Bruno formula and some properties of partial Bell polynomials, the authors provide several new explicit formulas and recurrence relations for Fubini-type polynomials and numbers, associate the newly defined degenerate Fubini-type polynomials with degenerate Apostol–Bernoulli polynomials and degenerate Apostol–Euler polynomials of order

α

. These results enable one to present additional relations for some degenerate special polynomials and numbers. Full article

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)

12 pages, 290 KB

Open AccessArticle

On (p, q)-Sine and (p, q)-Cosine Fubini Polynomials

by Waseem Ahmad Khan, Ghulam Muhiuddin, Ugur Duran and Deena Al-Kadi

Symmetry 2022, 14(3), 527; https://doi.org/10.3390/sym14030527 - 4 Mar 2022

Cited by 6 | Viewed by 2213

Abstract

In recent years,

(p, q)

-special polynomials, such as

(p, q)

-Euler,

(p, q)

-Genocchi,

(p, q)

-Bernoulli, and

(p, q)

-Frobenius-Euler, have been studied and investigated by many mathematicians, as well physicists. It is important [...] Read more.

In recent years,

(p, q)

-special polynomials, such as

(p, q)

-Euler,

(p, q)

-Genocchi,

(p, q)

-Bernoulli, and

(p, q)

-Frobenius-Euler, have been studied and investigated by many mathematicians, as well physicists. It is important that any polynomial have explicit formulas, symmetric identities, summation formulas, and relations with other polynomials. In this work, the

(p, q)

-sine and

(p, q)

-cosine Fubini polynomials are introduced and multifarious abovementioned properties for these polynomials are derived by utilizing some series manipulation methods.

(p, q)

-derivative operator rules and

(p, q)

-integral representations for the

(p, q)

-sine and

(p, q)

-cosine Fubini polynomials are also given. Moreover, several correlations related to both the

(p, q)

-Bernoulli, Euler, and Genocchi polynomials and the

(p, q)

-Stirling numbers of the second kind are developed. Full article

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

14 pages, 296 KB

Open AccessArticle

Degenerate Derangement Polynomials and Numbers

by Minyoung Ma and Dongkyu Lim

Fractal Fract. 2021, 5(3), 59; https://doi.org/10.3390/fractalfract5030059 - 22 Jun 2021

Cited by 3 | Viewed by 2387

Abstract

In this paper, we consider a new type of degenerate derangement polynomial and number, which shall be called the degenerate derangement polynomials and numbers of the second kind. These concepts are motivated by Kim et al.’s work on degenerate derangement polynomials and numbers. [...] Read more.

In this paper, we consider a new type of degenerate derangement polynomial and number, which shall be called the degenerate derangement polynomials and numbers of the second kind. These concepts are motivated by Kim et al.’s work on degenerate derangement polynomials and numbers. We investigate some properties of these new degenerate derangement polynomials and numbers and explore their connections with the degenerate gamma distributions for the case

λ \in (- 1, 0)

. In more detail, we derive their explicit expressions, recurrence relations, and some identities involving our degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials, and the degenerate Stirling numbers of the first and the second kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions. Moreover, we compare the degenerate derangement polynomials and numbers of the second kind to those of Kim et al. Full article

(This article belongs to the Special Issue Advanced Trends of Special Functions and Analysis of PDEs)

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 2578

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, 245 KB

Open AccessArticle

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 2424

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)

13 pages, 236 KB

Open AccessArticle

A Parametric Kind of the Degenerate Fubini Numbers and Polynomials

by Sunil Kumar Sharma, Waseem A. Khan and Cheon Seoung Ryoo

Mathematics 2020, 8(3), 405; https://doi.org/10.3390/math8030405 - 12 Mar 2020

Cited by 16 | Viewed by 2470

Abstract

In this article, we introduce the parametric kinds of degenerate type Fubini polynomials and numbers. We derive recurrence relations, identities and summation formulas of these polynomials with the aid of generating functions and trigonometric functions. Further, we show that the parametric kind of [...] Read more.

In this article, we introduce the parametric kinds of degenerate type Fubini polynomials and numbers. We derive recurrence relations, identities and summation formulas of these polynomials with the aid of generating functions and trigonometric functions. Further, we show that the parametric kind of the degenerate type Fubini polynomials are represented in terms of the Stirling numbers. Full article

(This article belongs to the Special Issue Polynomials: Theory and Applications)

43 pages, 1041 KB

Open AccessArticle

On Degenerate Truncated Special Polynomials

by Ugur Duran and Mehmet Acikgoz

Mathematics 2020, 8(1), 144; https://doi.org/10.3390/math8010144 - 20 Jan 2020

Cited by 14 | Viewed by 4067

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Special Functions and Applications)

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19 pages, 282 KB

Open AccessArticle

Two Parametric Kinds of Eulerian-Type Polynomials Associated with Euler’s Formula

by Neslihan Kilar and Yilmaz Simsek

Symmetry 2019, 11(9), 1097; https://doi.org/10.3390/sym11091097 - 2 Sep 2019

Cited by 16 | Viewed by 2667

Abstract

The purpose of this article is to construct generating functions for new families of special polynomials including two parametric kinds of Eulerian-type polynomials. Some fundamental properties of these functions are given. By using these generating functions and the Euler’s formula, some identities and [...] Read more.

The purpose of this article is to construct generating functions for new families of special polynomials including two parametric kinds of Eulerian-type polynomials. Some fundamental properties of these functions are given. By using these generating functions and the Euler’s formula, some identities and relations among trigonometric functions, two parametric kinds of Eulerian-type polynomials, Apostol-type polynomials, the Stirling numbers and Fubini-type polynomials are presented. Computational formulae for these polynomials are obtained. Applying a partial derivative operator to these generating functions, some derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers are also obtained. In addition, some remarks and observations on these polynomials are given. 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)

15 pages, 259 KB

Open AccessArticle

Truncated Fubini Polynomials

by Ugur Duran and Mehmet Acikgoz

Mathematics 2019, 7(5), 431; https://doi.org/10.3390/math7050431 - 15 May 2019

Cited by 13 | Viewed by 3028

Abstract

In this paper, we introduce the two-variable truncated Fubini polynomials and numbers and then investigate many relations and formulas for these polynomials and numbers, including summation formulas, recurrence relations, and the derivative property. We also give some formulas related to the truncated Stirling [...] Read more.

In this paper, we introduce the two-variable truncated Fubini polynomials and numbers and then investigate many relations and formulas for these polynomials and numbers, including summation formulas, recurrence relations, and the derivative property. We also give some formulas related to the truncated Stirling numbers of the second kind and Apostol-type Stirling numbers of the second kind. Moreover, we derive multifarious correlations associated with the truncated Euler polynomials and truncated Bernoulli polynomials. Full article

(This article belongs to the Special Issue Polynomials: Theory and Applications)

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