On some geometric properties of the Le Roy-type Mittag-Leffler function

Mehrez, Khaled; Das, Sourav

doi:10.15672/hujms.989236

Cited by 5 publications

(6 citation statements)

References 20 publications

(37 reference statements)

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“…Both relations ( 59) and ( 60) are exactly the expected analogues of those for the multi-index M-L functions, (51) and (50). Now, we aim to find an integral representation of the G-L integral (58), as an analogue of the generalized fractional integration operator (48) in Theorem 4. Instead of the Hfunction H m,0 m,m , we have now an I-function as a kernel.…”

Section: Theoremmentioning

confidence: 79%

“…It was introduced by Gerhold [52] and Garra and Polito [53], and studied further by Garrappa-Rogosin-Mainardi [54] and Garra-Orsingher-Polito [55], and then also by Gorska-Horzela [56], Simon [57], Mehrez-Das [58] and Mehrez [59]. In [55], it is mentioned that the Le Roy-type functions (21) are used in probability in the context of the studies of COM-Poisson distributions (in the sense of Conway and Maxwell), which are special classes of weighted Poisson distributions.…”

Section: Multi-index M-l-p Functions Of Le Roy Type As I I Iand H H H...mentioning

confidence: 99%

“…We also have to emphasize in terms of the other class of special functions the G m,0 m,m and H m,0 m,m -functions of type (ii), as discussed in survey [9], because here in Section 4, the classical FC operators of integration (such as R-L and E-K, (40)) are mentioned as having such kernels for m = 1, and also the generalized fractional integrals (42) have these kernels for arbitrary order m ≥ 1. In addition, in Section 4.4 (Theorem 6), their I m,0 m,m -analogue appears as a kernel function of the integral operator (61), being an alternative representation of the series (58) for the introduced Gelfond-Leontiev generalized integration generated by the Le Roy-type functions (22). Thus, the need for the analogy is evident.…”

Section: Classes Of G-and H-functions As Kernels Of Laplace-type Inte...mentioning

confidence: 99%

“…As mentioned in Kiryakova [64], even for the simplest case m = 1 and γ = 1, the problem stays open if τ ̸ = 1. In the particular case of Prabhakar parameters ∀τ i = 1, we proposed such operators in Section 4 here, see (58) as well as (61). 8.2.…”

Section: 1mentioning

confidence: 99%

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Going Next after “A Guide to Special Functions in Fractional Calculus”: A Discussion Survey

Kiryakova,

Paneva-Konovska

2024

Mathematics

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In the survey Kiryakova: “A Guide to Special Functions in Fractional Calculus” (published in this same journal in 2021) we proposed an overview of this huge class of special functions, including the Fox H-functions, the Fox–Wright generalized hypergeometric functions pΨq and a large number of their representatives. Among these, the Mittag-Leffler-type functions are the most popular and frequently used in fractional calculus. Naturally, these also include all “Classical Special Functions” of the class of the Meijer’s G- and pFq-functions, orthogonal polynomials and many elementary functions. However, it so happened that almost simultaneously with the appearance of the Mittag-Leffler function, another “fractionalized” variant of the exponential function was introduced by Le Roy, and in recent years, several authors have extended this special function and mentioned its applications. Then, we introduced a general class of so-called (multi-index) Le Roy-type functions, and observed that they fall in an “Extended Class of SF of FC”. This includes the I-functions of Rathie and, in particular, the H¯-functions of Inayat-Hussain, studied also by Buschman and Srivastava and by other authors. These functions initially arose in the theory of the Feynman integrals in statistical physics, but also include some important special functions that are well known in math, like the polylogarithms, Riemann Zeta functions, some famous polynomials and number sequences, etc. The I- and H¯-functions are introduced by Mellin–Barnes-type integral representations involving multi-valued fractional order powers of Γ-functions with a lot of singularities that are branch points. Here, we present briefly some preliminaries on the theory of these functions, and then our ideas and results as to how the considered Le Roy-type functions can be presented in their terms. Next, we also introduce Gelfond–Leontiev generalized operators of differentiation and integration for which the Le Roy-type functions are eigenfunctions. As shown, these “generalized integrations” can be extended as kinds of generalized operators of fractional integration, and are also compositions of “Le Roy type” Erdélyi–Kober integrals. A close analogy appears with the Generalized Fractional Calculus with H- and G-kernel functions, thus leading the way to its further development. Since the theory of the I- and H¯-functions still needs clarification of some details, we consider this work as a “Discussion Survey” and also provide a list of open problems.

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Section: Theoremmentioning

confidence: 79%

Section: Multi-index M-l-p Functions Of Le Roy Type As I I Iand H H H...mentioning

confidence: 99%

Section: Classes Of G-and H-functions As Kernels Of Laplace-type Inte...mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

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Going Next after “A Guide to Special Functions in Fractional Calculus”: A Discussion Survey

Kiryakova,

Paneva-Konovska

2024

Mathematics

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“…Moreover, in the proof of Theorem 3.1 [30], under the relation (26), the authors proved that the sequence (ξ n ) n≥1 defined by…”

Section: Example 1 the Function Ementioning

confidence: 99%

Further Geometric Properties of the Barnes–Mittag-Leffler Function

Alenazi,

Mehrez

2023

Axioms

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In this paper, we find sufficient conditions to be imposed on the parameters of a class of functions related to the Barnes–Mittag-Leffler function that allow us to conclude that it possesses certain geometric properties (such as starlikeness, uniformly starlike (convex), strongly starlike (convex), convexity, and close-to-convexity) in the unit disk. The key tools in some of our proofs are the monotonicity properties of a certain class of functions related to the gamma function.

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Normalized generalized Bessel function and its geometric properties

Zayed

Bulboacă

2022

J Inequal Appl

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The normalization of the generalized Bessel functions $\mathrm{U}_{\sigma,r}$ U σ , r $(\sigma,r\in \mathbb{C}\mathbbm{)}$ ( σ , r ∈ C ) defined by $$\begin{aligned} \mathrm{U}_{\sigma,r}(z)=z+\sum_{j=1}^{\infty} \frac{(-r)^{j}}{4^{j} (1)_{j}(\sigma )_{j}}z^{j+1} \end{aligned}$$ U σ , r ( z ) = z + ∑ j = 1 ∞ ( − r ) j 4 j ( 1 ) j ( σ ) j z j + 1 was introduced, and some of its geometric properties have been presented previously. The main purpose of the present paper is to complete the results given in the literature by employing a new procedure. We first used an identity for the logarithmic of the gamma function as well as an inequality for the digamma function to establish sufficient conditions on the parameters so that $\mathrm{U}_{\sigma,r}$ U σ , r is starlike or convex of order α$(0\leq \alpha \leq 1)$ ( 0 ≤ α ≤ 1 ) in the open unit disk. Moreover, the starlikeness and convexity of $\mathrm{U}_{\sigma,r}$ U σ , r have been considered where the leading concept of the proofs comes from the starlikeness of the power series $f(z)=\sum_{j=1}^{\infty}A_{j}z^{j}$ f ( z ) = ∑ j = 1 ∞ A j z j and the classical Alexander theorem between the classes of starlike and convex functions. We gave a simple proof to show that our conditions are not contradictory. Ultimately, the close-to-convexity of $(z\cos \sqrt{z} ) \ast \mathrm{U}_{\sigma,r}$ ( z cos z ) ∗ U σ , r and $(\sin z ) \ast \frac {\mathrm{U}_{\sigma,r}(z^{2})}{z}$ ( sin z ) ∗ U σ , r ( z 2 ) z have been determined, where “∗” stands for the convolution between the power series.

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On some geometric properties of the Le Roy-type Mittag-Leffler function

Cited by 5 publications

References 20 publications

Going Next after “A Guide to Special Functions in Fractional Calculus”: A Discussion Survey

Going Next after “A Guide to Special Functions in Fractional Calculus”: A Discussion Survey

Further Geometric Properties of the Barnes–Mittag-Leffler Function

Normalized generalized Bessel function and its geometric properties

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