Axioms

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14 pages, 324 KiB

Open AccessArticle

New Estimation Method of an Error for J Iteration

by Aftab Hussain, Danish Ali and Nawab Hussain

Axioms 2022, 11(12), 677; https://doi.org/10.3390/axioms11120677 - 28 Nov 2022

Cited by 1 | Viewed by 1185

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

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 2154

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)

17 pages, 320 KiB

Open AccessArticle

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://doi.org/10.3390/axioms10040343 - 16 Dec 2021

Viewed by 2279

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

Open AccessArticle

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://doi.org/10.3390/axioms10040342 - 12 Dec 2021

Cited by 1 | Viewed by 2184

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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19 pages, 300 KiB

Open AccessArticle

On q-Horn Hypergeometric Functions H₆ and H₇

by Ayman Shehata

Axioms 2021, 10(4), 336; https://doi.org/10.3390/axioms10040336 - 8 Dec 2021

Cited by 3 | Viewed by 2536

Abstract

This work aims to construct various properties for basic Horn functions

H_{6}

and

H_{7}

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

H_{6}

and

H_{7}

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

Open AccessArticle

New Expressions for Sums of Products of the Catalan Numbers

by Conghui Xie and Yuan He

Axioms 2021, 10(4), 330; https://doi.org/10.3390/axioms10040330 - 1 Dec 2021

Cited by 1 | Viewed by 2372

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

Open AccessArticle

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://doi.org/10.3390/axioms10040324 - 30 Nov 2021

Cited by 3 | Viewed by 2274

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

Open AccessArticle

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://doi.org/10.3390/axioms10020070 - 19 Apr 2021

Cited by 7 | Viewed by 2160

Abstract

In this paper, binomial convolution in the frame of quantum calculus is studied for the set

A_{q}

of q-Appell sequences. It has been shown that the set

A_{q}

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

A_{q}

of q-Appell sequences. It has been shown that the set

A_{q}

of q-Appell sequences forms an Abelian group under the operation of binomial convolution. Several properties for this Abelian group structure

A_{q}

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

Open AccessArticle

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://doi.org/10.3390/axioms10010037 - 18 Mar 2021

Cited by 11 | Viewed by 4006

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

Open AccessFeature PaperArticle

Bell-Based Bernoulli Polynomials with Applications

by Ugur Duran, Serkan Araci and Mehmet Acikgoz

Axioms 2021, 10(1), 29; https://doi.org/10.3390/axioms10010029 - 2 Mar 2021

Cited by 12 | Viewed by 2932

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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