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

**Gerd Baumann 1,\* and Norbert Südland <sup>2</sup>**


**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 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.

**Keywords:** Mellin-Barnes integrals; Sinc methods; Sinc quadrature; Fox functions; Aleph functions; Saxena function; definite integrals; fractional calculus

### **1. Introduction**

In the past 40 years, the field of fractional calculus has undergone extraordinary development. The analytic and numeric approaches in fractional calculus created tremendous progress, especially the analytic side generated diverse directions which increased the improvement tremendously [1]. A large number of methods and approaches were developed, generating a consistent framework for analysis and symbolic computations [1,2]. However, the numeric developments are far behind the analytic achievements especially the numeric representation of special functions like Fox *H*, Aleph (ℵ), and Saxena *I* functions [3]. Such kinds of functions exist nowadays utilized in the analysis of fractional calculus. The special functions also found its way to applications in physics, engineering, and computer science [4]. It turned out during the years that linear transforms like Laplace-, Fourier-, and Mellin transforms play a vital role to generate special functions like Fox-H, ℵ, and Saxena's I function [1]. For the generation of such function, it is always essential to use the inverse of linear transforms which analytically exists but finally are difficult to compute numerically. The issue with unknown functions is that they could have previously unknown singularities. Naturally, this affects both the choice of the numerical method and the convergence at these singularities. The convergence of the employed numerical technique itself may also be a concern. This was the case if the approximation was calculated using an inverse Laplace transform, as mentioned in [5,6]. In a recent paper, we demonstrated by using Sinc methods that it becomes pretty efficient when Mittag-Leffler functions, a subset of Fox *H* functions, are the target in connection with an inverse Laplace transform [7]. Mittag-Leffler functions are frequently used in representing solutions of fractional differential or integral equations [8,9]. However, these functions are only a subset of the analytic functions needed to represent the large assortment of possible solutions to fractional equations.

Generalizations of Fox *H* functions are ℵ functions which also include the class of Saxena *I* functions [10]. These exceptional functions, which have been investigated analytically but are difficult to obtain numerically, are still a painstaking foundation for fractional calculus today. We aim to offer a numerical technique that solves most numerical difficulties like

**Citation:** Baumann, G.; Südland, N. Sinc Numeric Methods for Fox-H, Aleph (ℵ), and Saxena-I Functions. *Fractal Fract.* **2022**, *6*, 449. https://doi.org/10.3390/ fractalfract6080449

Academic Editors: Angelo B. Mingarelli, Leila Gholizadeh Zivlaei and Mohammad Dehghan

Received: 10 July 2022 Accepted: 15 August 2022 Published: 18 August 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

convergence and the occurrence of singularities by applying Sinc methods to the computation of these functions. Sinc methods initially introduced by Frank Stenger are a powerful numerical tool that allows representing nearly any calculus operation in an efficient and exponentially converging way [11]. For example, Sinc methods allow for computing of definite or indefinite integrals, convolution integrals, linear integral transforms and their inverse, to solve fractional differential and integral equations, and many other practical computations [12,13]. One essential characteristic of Sinc methods is the use of a small number of computing aids; i.e., small programs, a small number of discretization points, less memory, etc., in connection with a high precision output of numerical results [14]. We shall apply these approaches to the numerical computation of Mellin-Barnes integrals used in the presentation of special functions.

Next, we will introduce the definition of Fox, ℵ, and Saxena *I* functions. In Section 2, we shall introduce the approximation methods needed for this work. The application of these Sinc methods is demonstrated in Section 3. Section 4 summarizes the results and addresses open problems with the current approach.

#### *1.1. The Fox H Function*

The Fox H function was introduced by Charles Fox in connection with dual integral equations in 1965 [15]. As he stated at that time "These H functions contain Bessel functions as special cases and my aim is to show that, with the help of a suitable terminology, it is possible to write down a solution by inspection". Today we know that Fox H functions are a remarkably broad set of functions and include the elementary as well as special functions. The application of these functions is versatile and permits to derive solutions just by "inspection" as Fox noted. A collection of such applications are comprised in the book by Mathai et al. [4] which extends the classical text by Mathai and Saxena [16]. Both monographs concentrate on the part of getting solutions "by inspection". However, since then there exists a tremendous need and pressure to boil down the solutions to numbers; i.e., to represent the symbolic Fox *H* solutions as numerical estimations at least or as accurate numbers. Our aim here is to use the analytic and symbolic ideas of Fox and his successors to evaluate such kinds of solutions numerically. To this end let us introduce some notations for these functions.

A Fox function *Hm*,*<sup>n</sup> <sup>p</sup>*,*<sup>q</sup>* (*z*) is defined via a Mellin-Barnes type integral using integers *m*, *n*, *p*, *q* such that 0 ≤ *m* ≤ *q*, 0 ≤ *n* ≤ *p*, for *ai*, *bj* ∈ C with C, the set of complex numbers, and for *<sup>α</sup>i*, *<sup>β</sup><sup>j</sup>* <sup>∈</sup> <sup>R</sup><sup>+</sup> = (0, <sup>∞</sup>) (*<sup>i</sup>* <sup>=</sup> 1, 2, . . . , *<sup>p</sup>*; *<sup>j</sup>* <sup>=</sup> 1, 2, . . . , *<sup>q</sup>*) in the form

$$H\_{p,q}^{m,n} \left( z \Big| \begin{array}{c} (a\_i, \alpha\_i)\_{1,p} \\ (b\_j, \beta\_j)\_{1,q} \end{array} \right) = \frac{1}{2\pi i} \int\_{\mathcal{C}} \mathcal{H}\_{p,q}^{m,n}(s) \, z^{-s} ds \tag{1}$$

with

$$\mathcal{H}\_{p,q}^{m,n}\left(\begin{array}{c} \left(a\_i, a\_i\right)\_{1,p} \\ \left(b\_j, \beta\_j\right)\_{1,q} \end{array}\bigg| s\right) = \frac{\prod\_{j=1}^m \Gamma\left(b\_j + \beta\_j s\right) \prod\_{j=1}^n \Gamma\left(1 - a\_j - a\_j s\right)}{\prod\_{j=m+1}^q \Gamma\left(1 - b\_j - \beta\_j s\right) \prod\_{j=n+1}^p \Gamma\left(a\_j + a\_j s\right)}. \tag{2}$$

Here

$$z^{-s} = \exp\left[-s\{\log|z| + i\arg z\}\right], \; z \neq 0, \; i = \sqrt{-1}, \tag{3}$$

where log |*z*| represents the natural logarithm of |*z*| and arg *z* is not necessarily the principal value. An empty product in (2), if it occurs, is taken to be one, and the poles

$$b\_{j,l} = \frac{-b\_{\bar{j}} - l}{\beta\_{\bar{j}}} \left( j = 1, \dots, m; l = 0, 1, 2, \dots \right) \tag{4}$$

of the gamma function Γ *bj* + *βjs* and the poles

$$a\_{i,k} = \frac{1 - a\_i + k}{a\_i} \ (i = 1, \dots, n; k = 0, 1, 2, \dots) \tag{5}$$

of the gamma function Γ(1 − *ai* − *αis*) do not coincide:

$$a\_i(b\_j + l) \ne \beta\_j(a\_i - k - 1) \ (i = 1, \dots, n; j = 1, \dots, m; l, k = 0, 1, 2, \dots). \tag{6}$$

The contour C in (1) is the infinite contour which separates all the poles *bj*,*<sup>l</sup>* in (4) to the left and all the poles *ai*,*<sup>k</sup>* in (5) to the right of C. In fact there are many ways to define C in the complex plane. However, we will concentrate on the cases where C is parallel to the imaginary axis in the complex plane. For this reason let *s* = *γ* + *iσ*, where *γ* and *σ* are real; then the contour C along which the integral of (1) is taken is the straight line whose equation is *γ* = *γ*0, where *γ*<sup>0</sup> is a constant. This line is parallel to the imaginary axis in the complex *s* plane and separates the poles.

For numeric integration we take C as a contour starting at the point *γ* − *i*∞ and terminating at the point *γ* + *i*∞, where *γ* ∈ R = (−∞, ∞). To simplify the integration we use the substitutions *s* = *γ* + *iσ* and *ds* = *idσ* which delivers

$$\begin{split} \mathcal{H}^{m,n}\_{p,\emptyset} \left( z \Big| \begin{array}{c} (a\_i, a\_i)\_{1,p} \\ (b\_j, \beta\_j)\_{1,q} \end{array} \right)\_{1,q} &= \ \frac{1}{2\pi i} \int\_{\mathcal{C}} \mathcal{H}^{m,n}\_{p,\emptyset}(s) \, z^{-s} ds = \frac{1}{2\pi i} \int\_{\gamma - i\infty}^{\gamma + i\infty} \mathcal{H}^{m,n}\_{p,\emptyset}(s) \, z^{-s} ds \\ &= \ \ \_{\mathbb{Z}\pi} \int\_{-\infty}^{\infty} \mathcal{H}^{m,n}\_{p,\emptyset}(\gamma + i\sigma) \, z^{-\gamma - i\sigma} d\sigma \end{array} \tag{7}$$

allowing a direct integration to represent the Fox *H* function at points *z* ∈ C. Note that the rightmost integral in (7) is a highly oscillating integral. Such kinds of integrals need special care if treated by standard quadrature methods. We will deal with this problem by using the Sinc quadrature discussed in Section 2. The absolute convergence of the integral can be guaranteed under certain conditions on the parameters of the Fox *H* function; for details see [3].
