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16 pages, 417 KB

Open AccessArticle

Analysis of Finite Integrals with Incomplete Aleph Functions, Mittag-Leffler Generalizations, and the Error Function

by Dinesh Kumar, Frédéric Ayant, Meena Kumari Gurjar, Anil Kumar Vishnoi and Saroj Solanki

Fractal Fract. 2025, 9(11), 734; https://doi.org/10.3390/fractalfract9110734 - 13 Nov 2025

Cited by 1 | Viewed by 704

In this paper, we evaluate a general class of finite integrals involving the error function, generalized Mittag-Leffler functions, and incomplete Aleph functions. The main result provides a unified framework that extends several known formulas related to the incomplete Gamma, I-, and H [...] Read more.

In this paper, we evaluate a general class of finite integrals involving the error function, generalized Mittag-Leffler functions, and incomplete Aleph functions. The main result provides a unified framework that extends several known formulas related to the incomplete Gamma, I-, and H-functions. Under suitable conditions, these results reduce to many classical special cases. We discuss convergence conditions that justify the validity of the obtained formulas and include explicit corollaries that highlight connections with earlier results in the literature. To illustrate applicability, we present numerical examples and graphs, demonstrating the behavior of the error function integral and Mittag-Leffler functions for specific parameter values. These integrals arise naturally in fractional calculus, probability theory, viscoelasticity, and anomalous diffusion, underscoring the importance of the present work in both mathematical analysis and applications. Full article

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16 pages, 363 KB

Open AccessArticle

Certain q-Analogue of Fractional Integrals and Derivatives Involving Basic Analogue of the Several Variable Aleph-Function

by Dinesh Kumar, Frédéric Ayant, Norbert Südland and Junesang Choi

Axioms 2023, 12(1), 51; https://doi.org/10.3390/axioms12010051 - 3 Jan 2023

Cited by 7 | Viewed by 2461

Abstract

Using Mellin-Barnes contour integrals, we aim at suggesting a q-analogue (q-extension) of the several variable Aleph-function. Then we present Riemann Liouville fractional q-integral and q-differential formulae for the q-extended several variable Aleph-function. Using the q-analogue of [...] Read more.

Using Mellin-Barnes contour integrals, we aim at suggesting a q-analogue (q-extension) of the several variable Aleph-function. Then we present Riemann Liouville fractional q-integral and q-differential formulae for the q-extended several variable Aleph-function. Using the q-analogue of the Leibniz rule for the fractional q-derivative of a product of two basic functions, we also provide a formula for the q-extended several variable Aleph-function, which is expressed in terms of an infinite series of the q-extended several variable Aleph-function. Since the three main formulas presented in this article are so general, they can be reduced to yield a number of identities involving q-extended simpler special functions. In this connection, we choose only one main formula to offer some of its particular instances involving diverse q-extended special functions, for example, the q-extended I-function, the q-extended H-function, and the q-extended Meijer’s G-function. The results presented here are hoped and believed to find some applications, in particular, in quantum mechanics. Full article

(This article belongs to the Special Issue Advances in Quantum Theory and Quantum Computing)

24 pages, 1440 KB

Open AccessArticle

Sinc Numeric Methods for Fox-H, Aleph (ℵ), and Saxena-I Functions

by Gerd Baumann and Norbert Südland

Fractal Fract. 2022, 6(8), 449; https://doi.org/10.3390/fractalfract6080449 - 18 Aug 2022

Viewed by 2863

Abstract

The purpose of this study is to offer a systematic, unified approach to the Mellin-Barnes integrals and associated special functions as Fox H, Aleph ℵ, and Saxena I function, encompassing the fundamental features and important conclusions under natural minimal assumptions on [...] Read more.

The purpose of this study is to offer a systematic, unified approach to the Mellin-Barnes integrals and associated special functions as Fox H, Aleph ℵ, and Saxena I function, encompassing the fundamental features and important conclusions under natural minimal assumptions on the functions in question. The approach’s pillars are the concept of a Mellin-Barnes integral and the Mellin representation of the given function. A Sinc quadrature is used in conjunction with a Sinc approximation of the function to achieve the numerical approximation of the Mellin-Barnes integral. The method converges exponentially and can handle endpoint singularities. We give numerical representations of the Aleph ℵ and Saxena I functions for the first time. Full article

(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)

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11 pages, 317 KB

Open AccessArticle

The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals

by Mohsen Soltanifar

Mathematics 2022, 10(5), 706; https://doi.org/10.3390/math10050706 - 24 Feb 2022

Viewed by 3004

Abstract

In this paper, we present a second partial solution for the problem of cardinality calculation of the set of fractals for its subcategory of the random virtual ones. Consistent with the deterministic case, we show that for the given quantities of the Hausdorff [...] Read more.

In this paper, we present a second partial solution for the problem of cardinality calculation of the set of fractals for its subcategory of the random virtual ones. Consistent with the deterministic case, we show that for the given quantities of the Hausdorff dimension and the Lebesgue measure, there are aleph-two virtual random fractals with, almost surely, a Hausdorff dimension of a bivariate function of them and the expected Lebesgue measure equal to the latter one. The associated results for three other fractal dimensions are similar to the case given for the Hausdorff dimension. The problem remains unsolved in the case of non-Euclidean abstract fractal spaces. Full article

(This article belongs to the Special Issue Advances in Fractals)

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8 pages, 292 KB

Open AccessArticle

On Transformation Involving Basic Analogue to the Aleph-Function of Two Variables

by Dinesh Kumar, Dumitru Baleanu, Frédéric Ayant and Norbert Südland

Fractal Fract. 2022, 6(2), 71; https://doi.org/10.3390/fractalfract6020071 - 28 Jan 2022

Cited by 5 | Viewed by 3080

Abstract

In our work, we derived the fractional order q-integrals and q-derivatives concerning a basic analogue to the Aleph-function of two variables (AFTV). We discussed a related application and the q-extension of the corresponding Leibniz rule. Finally, we presented two corollaries [...] Read more.

In our work, we derived the fractional order q-integrals and q-derivatives concerning a basic analogue to the Aleph-function of two variables (AFTV). We discussed a related application and the q-extension of the corresponding Leibniz rule. Finally, we presented two corollaries concerning the basic analogue to the I-function of two variables and the basic analogue to the Aleph-function of one variable. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

11 pages, 284 KB

Open AccessArticle

Certain Finite Integrals Related to the Products of Special Functions

by Dinesh Kumar, Frédéric Ayant, Suphawat Asawasamrit and Jessada Tariboon

Symmetry 2021, 13(11), 2013; https://doi.org/10.3390/sym13112013 - 23 Oct 2021

Cited by 3 | Viewed by 2305

Abstract

The aim of this paper is to establish a theorem associated with the product of the Aleph-function, the multivariable Aleph-function, and the general class of polynomials. The results of this theorem are unified in nature and provide a very large number of analogous [...] Read more.

The aim of this paper is to establish a theorem associated with the product of the Aleph-function, the multivariable Aleph-function, and the general class of polynomials. The results of this theorem are unified in nature and provide a very large number of analogous results (new or known) involving simpler special functions and polynomials (of one or several variables) as special cases. The derived results lead to significant applications in physics and engineering sciences. Full article

(This article belongs to the Special Issue Theory and Applications of Special Functions in Mathematical Physics)

9 pages, 779 KB

Open AccessArticle

A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals

by Mohsen Soltanifar

Mathematics 2021, 9(13), 1546; https://doi.org/10.3390/math9131546 - 1 Jul 2021

Cited by 6 | Viewed by 5731

Abstract

How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual [...] Read more.

How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions. Full article

(This article belongs to the Special Issue Advances in Fractals)

17 pages, 355 KB

Open AccessArticle

On (p,q)-Sumudu and (p,q)-Laplace Transforms of the Basic Analogue of Aleph-Function

by Asifa Tassaddiq, Altaf Ahmad Bhat, D. K. Jain and Farhad Ali

Symmetry 2020, 12(3), 390; https://doi.org/10.3390/sym12030390 - 3 Mar 2020

Cited by 7 | Viewed by 3104

Abstract

In this paper, we introduce the definitions of Sumudu and Laplace transforms of first and second kind in quantum calculus by using functions of several variables. On account of the general nature of the

(p, q)

-analogue of Aleph-function, a [...] Read more.

In this paper, we introduce the definitions of Sumudu and Laplace transforms of first and second kind in quantum calculus by using functions of several variables. On account of the general nature of the

(p, q)

-analogue of Aleph-function, a large number of new and known results for these transforms were obtained. Also, we obtain some interesting relationships and identities for these transforms. We also derive some correlations among Aleph function and the above-mentioned integral transforms in quantum calculus. Full article

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