A Survey of Some Recent Developments on Higher Transcendental Functions of Analytic Number Theory and Applied Mathematics

Symmetry, Special Functions and Number Theory

This editorial is a short review of papers accepted in Symmetry in 2020–2022 about the topic of Number Theory [...]

Modified Fractional Difference Operators Defined Using Mittag-Leffler Kernels

The discrete fractional operators of Riemann–Liouville and Liouville–Caputo are omnipresent due to the singularity of the kernels. Therefore, convexity analysis of discrete fractional differences of these types plays a vital role in maintaining the safe operation of kernels and symmetry of discrete delta and nabla distribution. In their discrete version, the generalized or modified forms of various operators of fractional calculus are becoming increasingly important from the viewpoints of both pure and applied mathematical sciences. In this paper, we present the discrete version of the recently modified fractional calculus operator with the Mittag-Leffler-type kernel. Here, in this article, the expressions of both the discrete nabla derivative and its counterpart nabla integral are obtained. Some applications and illustrative examples are given to support the theoretical results.

New Results Involving Riemann Zeta Function Using Its Distributional Representation

The relation of special functions with fractional integral transforms has a great influence on modern science and research. For example, an old special function, namely, the Mittag–Leffler function, became the queen of fractional calculus because its image under the Laplace transform is known to a large audience only in this century. By taking motivation from these facts, we use distributional representation of the Riemann zeta function to compute its Laplace transform, which has played a fundamental role in applying the operators of generalized fractional calculus to this well-studied function. Hence, similar new images under various other popular fractional transforms can be obtained as special cases. A new fractional kinetic equation involving the Riemann zeta function is formulated and solved. Thereafter, a new relation involving the Laplace transform of the Riemann zeta function and the Fox–Wright function is explored, which proved to significantly simplify the results. Various new distributional properties are also derived.

Some Integral Inequalities in 𝒱-Fractional Calculus and Their Applications

We consider the Steffensen–Hayashi inequality and remainder identity for V-fractional differentiable functions involving the six parameters truncated Mittag–Leffler function and the Gamma function. In view of these, we obtain some integral inequalities of Steffensen, Hermite–Hadamard, Chebyshev, Ostrowski, and Grüss type to the V-fractional calculus.

Geometric Properties of a Certain Class of Mittag–Leffler-Type Functions

The main objective of this paper is to establish some sufficient conditions so that a class of normalized Mittag–Leffler-type functions satisfies several geometric properties such as starlikeness, convexity, close-to-convexity, and uniform convexity inside the unit disk. Moreover, pre-starlikeness and k-uniform convexity are discussed for these functions. Some sufficient conditions are also derived so that these functions belong to the Hardy spaces Hp and H∞. Furthermore, the inclusion properties of some modified Mittag–Leffler-type functions are discussed. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples. Some potential directions for analogous further research on the subject of the present investigation are indicated in the concluding section.