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27 pages, 341 KiB

Open AccessFeature PaperArticle

Symbolic Methods Applied to a Class of Identities Involving Appell Polynomials and Stirling Numbers

by Tian-Xiao He and Emanuele Munarini

Mathematics 2025, 13(11), 1732; https://doi.org/10.3390/math13111732 - 24 May 2025

Viewed by 321

In this paper, we present two symbolic methods, in particular, the method starting from the source identity, umbra identity, for constructing identities of s-Appell polynomials related to Stirling numbers and binomial coefficients. We discuss some properties of s-Appell polynomial sequences related [...] Read more.

In this paper, we present two symbolic methods, in particular, the method starting from the source identity, umbra identity, for constructing identities of s-Appell polynomials related to Stirling numbers and binomial coefficients. We discuss some properties of s-Appell polynomial sequences related to Riordan arrays, Sheffer matrices, and their q analogs. Full article

(This article belongs to the Special Issue Combinatorics, Riordan Matrices and Umbral Calculus—in Memory of Prof. Emanuele Munarini)

11 pages, 285 KiB

Open AccessArticle

Some Identities with Multi-Generalized q-Hyperharmonic Numbers of Order r

by Zhihua Chen, Neşe Ömür, Sibel Koparal and Waseem Ahmad Khan

Symmetry 2023, 15(4), 917; https://doi.org/10.3390/sym15040917 - 14 Apr 2023

Cited by 5 | Viewed by 1366

Abstract

The main purpose of this paper is to define multiple alternative q-harmonic numbers,

H_{n} (k; q)

and multi-generalized q-hyperharmonic numbers of order r,

H_{n}^{r} (k; q)

by using q-multiple zeta star values (q [...] Read more.

The main purpose of this paper is to define multiple alternative q-harmonic numbers,

H_{n} (k; q)

and multi-generalized q-hyperharmonic numbers of order r,

H_{n}^{r} (k; q)

by using q-multiple zeta star values (q-MZSVs). We obtain some finite sum identities and give some applications of them for certain combinations of q-multiple polylogarithms

L i_{q; k_{1}, k_{2}, \dots, k_{d}} (t_{1}, t_{2}, \dots, t_{d})

with the help of generating functions. Additionally, one of the applications is the sum involving q-Stirling numbers and q-Bernoulli numbers. Full article

(This article belongs to the Special Issue Special Functions, Integral Transforms and Polynomial Sequences in Real World with Symmetry)

18 pages, 1242 KiB

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 1372

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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18 pages, 1286 KiB

Open AccessArticle

Explicit Properties of q-Cosine and q-Sine Array-Type Polynomials Containing Symmetric Structures

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

Symmetry 2022, 14(8), 1675; https://doi.org/10.3390/sym14081675 - 12 Aug 2022

Cited by 7 | Viewed by 1508

Abstract

The main aim of this study is to define parametric kinds of

λ

-Array-type polynomials by using q-trigonometric polynomials and to investigate some of their analytical properties and applications. For this purpose, many formulas and relations for these polynomials, including some implicit [...] Read more.

The main aim of this study is to define parametric kinds of

λ

-Array-type polynomials by using q-trigonometric polynomials and to investigate some of their analytical properties and applications. For this purpose, many formulas and relations for these polynomials, including some implicit summation formulas, differentiation rules, and relations with the earlier polynomials by utilizing some series manipulation method are derived. Additionally, as an application, the zero values of q-Array-type polynomials are presented by the tables and multifarious graphical representations for these zero values are drawn. Full article

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

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17 pages, 319 KiB

Open AccessFeature PaperArticle

Multiple q-Integral and Records from Geometrically Distributed Sequences

by Fernando López-Blázquez and Begoña Salamanca-Miño

Mathematics 2022, 10(13), 2313; https://doi.org/10.3390/math10132313 - 1 Jul 2022

Cited by 1 | Viewed by 1316

Abstract

We study the distribution theory of some statistics related to records from a sequence of geometrically distributed random variables. The main novelty in this study is the introduction of q-calculus techniques. We obtain representations for the probability functions of some variables of [...] Read more.

We study the distribution theory of some statistics related to records from a sequence of geometrically distributed random variables. The main novelty in this study is the introduction of q-calculus techniques. We obtain representations for the probability functions of some variables of interest, such as, for instance, record times, number of records, inter-record times, record indicators, etc., in terms of multiple q-integrals. It is remarkable to note that the expressions thus obtained are q-analogues of the corresponding well-known results in the absolutely continuous case. We also explore a duality between the results in the case of weak and ordinary records. Full article

(This article belongs to the Section D1: Probability and Statistics)

10 pages, 256 KiB

Open AccessArticle

A Note on q-analogue of Degenerate Catalan Numbers Associated with p-adic Integral on

Z_{p}

by Waseem A. Khan

Symmetry 2022, 14(6), 1119; https://doi.org/10.3390/sym14061119 - 29 May 2022

Cited by 18 | Viewed by 1616

Abstract

In this paper, we introduce q-analogues of degenerate Catalan numbers and polynomials with the help of a fermionic p-adic q-integrals on

Z_{p}

and establish some new connections with the degenerate Stirling numbers of the first and second kinds. Furthermore, [...] Read more.

In this paper, we introduce q-analogues of degenerate Catalan numbers and polynomials with the help of a fermionic p-adic q-integrals on

Z_{p}

and establish some new connections with the degenerate Stirling numbers of the first and second kinds. Furthermore, we also find a few new identities and results of this type of polynomials and numbers. Full article

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

12 pages, 290 KiB

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 2189

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)

18 pages, 897 KiB

Open AccessArticle

Certain Identities Associated with (p,q)-Binomial Coefficients and (p,q)-Stirling Polynomials of the Second Kind

by Talha Usman, Mohd Saif and Junesang Choi

Symmetry 2020, 12(9), 1436; https://doi.org/10.3390/sym12091436 - 31 Aug 2020

Cited by 17 | Viewed by 3265

Abstract

The q-Stirling numbers (polynomials) of the second kind have been investigated and applied in a variety of research subjects including, even, the q-analogue of Bernstein polynomials. The

(p, q)

-Stirling numbers (polynomials) of the second kind have been [...] Read more.

The q-Stirling numbers (polynomials) of the second kind have been investigated and applied in a variety of research subjects including, even, the q-analogue of Bernstein polynomials. The

(p, q)

-Stirling numbers (polynomials) of the second kind have been studied, particularly, in relation to combinatorics. In this paper, we aim to introduce new

(p, q)

-Stirling polynomials of the second kind which are shown to be fit for the

(p, q)

-analogue of Bernstein polynomials. We also present some interesting identities involving the

(p, q)

-binomial coefficients. We further discuss certain vanishing identities associated with the q-and

(p, q)

-Stirling polynomials of the second kind. Full article

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

14 pages, 324 KiB

Open AccessArticle

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 9 | Viewed by 2540

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)

28 pages, 4861 KiB

Open AccessArticle

Definition and Counting of Configurational Microstates in Steady-State Two-Phase Flows in Pore Networks

by Marios S. Valavanides and Tryfon Daras

Entropy 2016, 18(2), 54; https://doi.org/10.3390/e18020054 - 6 Feb 2016

Cited by 6 | Viewed by 8358

Abstract

Steady-state two-phase flow in porous media is a process whereby a wetting phase displaces a non-wetting phase within a pore network. It is an off-equilibrium stationary process—in the sense that it is maintained in dynamic equilibrium at the expense of energy supplied to [...] Read more.

Steady-state two-phase flow in porous media is a process whereby a wetting phase displaces a non-wetting phase within a pore network. It is an off-equilibrium stationary process—in the sense that it is maintained in dynamic equilibrium at the expense of energy supplied to the system. The efficiency of the process depends on its spontaneity, measurable by the rate of global entropy production. The latter has been proposed to comprise two components: the rate of mechanical energy dissipation at constant temperature (a thermal entropy component, Q/T, in the continuum mechanics scale) and the configurational entropy (a Boltzmann–Gibbs entropy component, klnW), due to the existence of a canonical ensemble of flow configurations, physically admissible to the externally imposed macrostate conditions. Here, we propose an analytical model to account the number of microstates, lnW, in two-phase flows in pore networks. Combinatorial analysis is implemented to evaluate the number of identified microstates per physically admissible internal flow arrangement, compatible with the imposed steady-state flow conditions. Then, Stirling’s approximation is applied to downscale the large factorial numbers. Finally, the number of microstates is estimated by contriving an appropriate mixing scheme over the canonical ensemble of the physically admissible flow configurations. Indicative computations are furnished. Full article

(This article belongs to the Section Statistical Physics)

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22 pages, 259 KiB

Open AccessArticle

Convergence Aspects for Generalizations of q-Hypergeometric Functions

by Thomas Ernst

Axioms 2015, 4(2), 134-155; https://doi.org/10.3390/axioms4020134 - 8 Apr 2015

Cited by 1 | Viewed by 4323

Abstract

In an earlier paper, we found transformation and summation formulas for 43 q-hypergeometric functions of 2n variables. The aim of the present article is to find convergence regions and a few conjectures of convergence regions for these functions based on a vector [...] Read more.

In an earlier paper, we found transformation and summation formulas for 43 q-hypergeometric functions of 2n variables. The aim of the present article is to find convergence regions and a few conjectures of convergence regions for these functions based on a vector version of the Nova q-addition. These convergence regions are given in a purely formal way, extending the results of Karlsson (1976). The Γq-function and the q-binomial coefficients, which are used in the proofs, are adjusted accordingly. Furthermore, limits and special cases for the new functions, e.g., q-Lauricella functions and q-Horn functions, are pointed out. Full article

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2014)

13 pages, 253 KiB

Open AccessArticle

Continuous Stieltjes-Wigert Limiting Behaviour of a Family of Confluent q-Chu-Vandermonde Distributions

by Andreas Kyriakoussis and Malvina Vamvakari

Axioms 2014, 3(2), 140-152; https://doi.org/10.3390/axioms3020140 - 10 Apr 2014

Cited by 3 | Viewed by 4413

Abstract

From Kemp [1], we have a family of confluent q-Chu- Vandermonde distributions, consisted by three members I, II and III, interpreted as a family of q-steady-state distributions from Markov chains. In this article, we provide the moments of the distributions of [...] Read more.

From Kemp [1], we have a family of confluent q-Chu- Vandermonde distributions, consisted by three members I, II and III, interpreted as a family of q-steady-state distributions from Markov chains. In this article, we provide the moments of the distributions of this family and we establish a continuous limiting behavior for the members I and II, in the sense of pointwise convergence, by applying a q-analogue of the usual Stirling asymptotic formula for the factorial number of order n. Specifically, we initially give the q-factorial moments and the usual moments for the family of confluent q-Chu- Vandermonde distributions and then we designate as a main theorem the conditions under which the confluent q-Chu-Vandermonde distributions I and II converge to a continuous Stieltjes-Wigert distribution. For the member III we give a continuous analogue. Moreover, as applications of this study we present a modified q-Bessel distribution, a generalized q-negative Binomial distribution and a generalized over/underdispersed (O/U) distribution. Note that in this article we prove the convergence of a family of discrete distributions to a continuous distribution which is not of a Gaussian type. Full article

10 pages, 164 KiB

Open AccessArticle

Generalized q-Stirling Numbers and Their Interpolation Functions

by Hacer Ozden, Ismail Naci Cangul and Yilmaz Simsek

Axioms 2013, 2(1), 10-19; https://doi.org/10.3390/axioms2010010 - 8 Feb 2013

Cited by 4 | Viewed by 5563

Abstract

In this paper, we define the generating functions for the generalized q-Stirling numbers of the second kind. By applying Mellin transform to these functions, we construct interpolation functions of these numbers at negative integers. We also derive some identities and relations related to [...] Read more.

In this paper, we define the generating functions for the generalized q-Stirling numbers of the second kind. By applying Mellin transform to these functions, we construct interpolation functions of these numbers at negative integers. We also derive some identities and relations related to q-Bernoulli numbers and polynomials and q-Stirling numbers of the second kind. Full article

(This article belongs to the Special Issue q-Series and Related Topics in Special Functions and Analytic Number Theory)

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