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Axioms
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p-adic Analysis and q-Calculus with Their Applications
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p-adic Analysis and q-Calculus with Their Applications

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (30 November 2021) | Viewed by 24626

Special Issue Editors

Dr. Serkan Araci

E-Mail Website
Guest Editor
Department of Basic Sciences, Faculty of Engineering, Hasan Kalyoncu University, TR-27010 Gaziantep, Turkey
Interests: q-special functions and q-special polynomials; q-series; analytic number theory; umbral theory; p-adic q-analysis; fractional calculus and its applications
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Martin Bohner

E-Mail Website
Guest Editor
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA
Interests: Hamiltonian systems; Sturm–Liouville equations; boundary value problems; difference equations; variational analysis; control theory; optimization; dynamical systems; oscillation; fractional differentiation equations; positivity; matrix analysis; eigenvalue problems; computational mathematics; time scales
Special Issues, Collections and Topics in MDPI journals
Dr. Roberto B. Corcino

E-Mail Website
Guest Editor
1. Research Institute for Computational Mathematics and Physics (RICMP), Cebu Normal University, Cebu City, Philippines
2. Mathematics Department, Cebu Normal University, Cebu City, Philippines
Interests: enumerative and analytic combinatorics; q-series and q-polynomials; special functions
Dr. Sunil Dutt Purohit

E-Mail Website
Guest Editor
Department of HEAS (Mathematics), Rajasthan Technical University, Kota, India
Interests: special functions; fractional calculus; geometric function theory; mathematical physics

Special Issue Information

Dear Colleagues,

The idea of p-adic numbers traces back to Kurt Hensel (1861–1941). Motivated by this fruitful idea, many scientists have begun to study new scientific tools using good and useful properties of them. The subject of quantum calculus (or q-calculus) was launched in the 1920s. It leads to a new method for computations and classifications of q-special functions and q-special polynomials. However, it has only gained importance and considerable popularity during the last three decades. Especially in the last few decades, q-calculus has been developed into an interdisciplinary subject and served as a bridge between physics and mathematics.

p-adic analysis and q-calculus encompass several domains in mathematics and physics, including number theory, algebraic geometry, algebraic topology, mathematical analysis, mathematical physics, string theory, field theory, stochastic differential equations, quantum groups, and other parts of the natural sciences.

This Special Issue focuses on the applications of the p-adic analysis and q-calculus to various fields of number theory that deal mainly with mathematical analysis of functions of p-adic numbers in mathematics and the theory of p-adic strings and quantum mechanics, and the theory of complex disordered systems-spin glasses in physics.

Potential topics include but are not limited to the following:
  • q-umbral analysis
  • q-Sheffer polynomials
  • q-difference equations
  • q-calculus in operator theory
  • p-adic zeta functions
  • p-adic q-integrals with applications
  • p-adic mathematical physics

Dr. Serkan Araci
Prof. Dr. Martin Bohner
Dr. Roberto B. Corcino
Dr. Sunil Dutt Purohit
Guest Editors

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. Axioms is an international peer-reviewed open access monthly 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 2400 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.

Published Papers (10 papers)

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Research

14 pages, 324 KiB  
Article
New Estimation Method of an Error for J Iteration
by Aftab Hussain, Danish Ali and Nawab Hussain
Axioms 2022, 11(12), 677; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms11120677 - 28 Nov 2022
Viewed by 872
Abstract
The major aim of this article is to show how to estimate direct errors using the J iteration method. Direct error estimation of iteration processes is being investigated in different journals. We also illustrate that an error in the J iteration process can [...] Read more.
The major aim of this article is to show how to estimate direct errors using the J iteration method. Direct error estimation of iteration processes is being investigated in different journals. We also illustrate that an error in the J iteration process can be controlled. Furthermore, we express J iteration convergence by using distinct initial values. Full article
(This article belongs to the Special Issue p-adic Analysis and q-Calculus with Their Applications)
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7 pages, 248 KiB  
Article
Some Identities on the Twisted q-Analogues of Catalan-Daehee Numbers and Polynomials
by Dongkyu Lim
Axioms 2022, 11(1), 9; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms11010009 - 23 Dec 2021
Viewed by 1769
Abstract
In this paper, the author considers twisted q-analogues of Catalan-Daehee numbers and polynomials by using p-adic q-integral on Zp. 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)
17 pages, 320 KiB  
Article
Hankel Transform of the Type 2 (p,q)-Analogue of r-Dowling Numbers
by Roberto Corcino, Mary Ann Ritzell Vega and Amerah Dibagulun
Axioms 2021, 10(4), 343; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms10040343 - 16 Dec 2021
Viewed by 1844
Abstract
In this paper, type 2 (p,q)-analogues of the r-Whitney numbers of the second kind is defined and a combinatorial interpretation in the context of the A-tableaux is given. Moreover, some convolution-type identities, which are useful in [...] Read more.
In this paper, type 2 (p,q)-analogues of the r-Whitney numbers of the second kind is defined and a combinatorial interpretation in the context of the A-tableaux is given. Moreover, some convolution-type identities, which are useful in deriving the Hankel transform of the type 2 (p,q)-analogue of the r-Whitney numbers of the second kind are obtained. Finally, the Hankel transform of the type 2 (p,q)-analogue of the r-Dowling numbers are established. Full article
(This article belongs to the Special Issue p-adic Analysis and q-Calculus with Their Applications)
12 pages, 854 KiB  
Article
Generalized Quantum Integro-Differential Fractional Operator with Application of 2D-Shallow Water Equation in a Complex Domain
by Rabha W. Ibrahim and Dumitru Baleanu
Axioms 2021, 10(4), 342; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms10040342 - 12 Dec 2021
Cited by 1 | Viewed by 1902
Abstract
In this paper, we aim to generalize a fractional integro-differential operator in the open unit disk utilizing Jackson calculus (quantum calculus or q-calculus). Next, by consuming the generalized operator to define a formula of normalized analytic functions, we present a set of integral [...] Read more.
In this paper, we aim to generalize a fractional integro-differential operator in the open unit disk utilizing Jackson calculus (quantum calculus or q-calculus). Next, by consuming the generalized operator to define a formula of normalized analytic functions, we present a set of integral inequalities using the concepts of subordination and superordination. In addition, as an application, we determine the maximum and minimum solutions of the extended fractional 2D-shallow water equation in a complex domain. Full article
(This article belongs to the Special Issue p-adic Analysis and q-Calculus with Their Applications)
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Figure 1

19 pages, 300 KiB  
Article
On q-Horn Hypergeometric Functions H6 and H7
by Ayman Shehata
Axioms 2021, 10(4), 336; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms10040336 - 08 Dec 2021
Cited by 3 | Viewed by 2173
Abstract
This work aims to construct various properties for basic Horn functions H6 and H7 under conditions on the numerator and denominator parameters, such as several q-contiguous function relations, q-differential relations, and q-differential equations. Special cases of our main [...] Read more.
This work aims to construct various properties for basic Horn functions H6 and H7 under conditions on the numerator and denominator parameters, such as several q-contiguous function relations, q-differential relations, and q-differential equations. Special cases of our main results are also demonstrated. Full article
(This article belongs to the Special Issue p-adic Analysis and q-Calculus with Their Applications)
8 pages, 237 KiB  
Article
New Expressions for Sums of Products of the Catalan Numbers
by Conghui Xie and Yuan He
Axioms 2021, 10(4), 330; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms10040330 - 01 Dec 2021
Cited by 1 | Viewed by 1965
Abstract
In this paper, we perform a further investigation for the Catalan numbers. By making use of the method of derivatives and some properties of the Bell polynomials, we establish two new expressions for sums of products of arbitrary number of the Catalan numbers. [...] Read more.
In this paper, we perform a further investigation for the Catalan numbers. By making use of the method of derivatives and some properties of the Bell polynomials, we establish two new expressions for sums of products of arbitrary number of the Catalan numbers. The results presented here can be regarded as the development of some known formulas. Full article
(This article belongs to the Special Issue p-adic Analysis and q-Calculus with Their Applications)
6 pages, 235 KiB  
Article
Quadruple Integral Involving the Logarithm and Product of Bessel Functions Expressed in Terms of the Lerch Function
by Robert Reynolds and Allan Stauffer
Axioms 2021, 10(4), 324; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms10040324 - 30 Nov 2021
Cited by 3 | Viewed by 1969
Abstract
In this paper, we have derived and evaluated a quadruple integral whose kernel involves the logarithm and product of Bessel functions of the first kind. A new quadruple integral representation of Catalan’s G and Apéry’s ζ(3) constants are produced. Some [...] Read more.
In this paper, we have derived and evaluated a quadruple integral whose kernel involves the logarithm and product of Bessel functions of the first kind. A new quadruple integral representation of Catalan’s G and Apéry’s ζ(3) constants are produced. Some special cases of the result in terms of fundamental constants are evaluated. All the results in this work are new. Full article
(This article belongs to the Special Issue p-adic Analysis and q-Calculus with Their Applications)
13 pages, 318 KiB  
Article
q-Binomial Convolution and Transformations of q-Appell Polynomials
by Alaa Mohammed Obad, Asif Khan, Kottakkaran Sooppy Nisar and Ahmed Morsy
Axioms 2021, 10(2), 70; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms10020070 - 19 Apr 2021
Cited by 5 | Viewed by 1735
Abstract
In this paper, binomial convolution in the frame of quantum calculus is studied for the set Aq of q-Appell sequences. It has been shown that the set Aq of q-Appell sequences forms an Abelian group under the operation of [...] Read more.
In this paper, binomial convolution in the frame of quantum calculus is studied for the set Aq of q-Appell sequences. It has been shown that the set Aq of q-Appell sequences forms an Abelian group under the operation of binomial convolution. Several properties for this Abelian group structure Aq have been studied. A new definition of the q-Appell polynomials associated with a random variable is proposed. Scale transformation as well as transformation based on expectation with respect to a random variable is used to present the determinantal form of q-Appell sequences. Full article
(This article belongs to the Special Issue p-adic Analysis and q-Calculus with Their Applications)
9 pages, 265 KiB  
Article
Explicit, Determinantal, and Recurrent Formulas of Generalized Eulerian Polynomials
by Yan Wang, Muhammet Cihat Dağli, Xi-Min Liu and Feng Qi
Axioms 2021, 10(1), 37; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms10010037 - 18 Mar 2021
Cited by 11 | Viewed by 3255
Abstract
In the paper, by virtue of the Faà di Bruno formula, with the aid of some properties of the Bell polynomials of the second kind, and by means of a general formula for derivatives of the ratio between two differentiable functions, the authors [...] Read more.
In the paper, by virtue of the Faà di Bruno formula, with the aid of some properties of the Bell polynomials of the second kind, and by means of a general formula for derivatives of the ratio between two differentiable functions, the authors establish explicit, determinantal, and recurrent formulas for generalized Eulerian polynomials. Full article
(This article belongs to the Special Issue p-adic Analysis and q-Calculus with Their Applications)
23 pages, 340 KiB  
Article
Bell-Based Bernoulli Polynomials with Applications
by Ugur Duran, Serkan Araci and Mehmet Acikgoz
Axioms 2021, 10(1), 29; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms10010029 - 02 Mar 2021
Cited by 11 | Viewed by 2423
Abstract
In this paper, we consider Bell-based Stirling polynomials of the second kind and derive some useful relations and properties including some summation formulas related to the Bell polynomials and Stirling numbers of the second kind. Then, we introduce Bell-based Bernoulli polynomials of order [...] Read more.
In this paper, we consider Bell-based Stirling polynomials of the second kind and derive some useful relations and properties including some summation formulas related to the Bell polynomials and Stirling numbers of the second kind. Then, we introduce Bell-based Bernoulli polynomials of order α and investigate multifarious correlations and formulas including some summation formulas and derivative properties. Also, we acquire diverse implicit summation formulas and symmetric identities for Bell-based Bernoulli polynomials of order α. Moreover, we attain several interesting formulas of Bell-based Bernoulli polynomials of order α arising from umbral calculus. Full article
(This article belongs to the Special Issue p-adic Analysis and q-Calculus with Their Applications)
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