Fractional calculus, zeta functions and Shannon entropy

Abstract

This paper deals with the fractional calculus of zeta functions. In particular, the study is focused on the Hurwitz ζ \zeta function. All the results are based on the complex generalization of the Grünwald-Letnikov fractional derivative. We state and prove the functional equation together with an integral representation by Bernoulli numbers. Moreover, we treat an application in terms of Shannon entropy.

Existence and uniqueness of distributional solution for semilinear fractional elliptic equation involving new operator and some numerical results

“…For some applications, we refer to other studies, [4][5][6][7][8][9][10][11][12][13] and the references therein. For applied science, we refer to other studies [14][15][16][17][18][19] as an application in the theory of special functions.…”

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

“…In fact, having an overview of the on-going contributions to the theory and applications of fractional calculus, which are continually appearing in some of the leading journals devoted to mathematical and physical sciences, biological sciences, statistical sciences, engineering sciences, and so on, the subject-matter, which we have dealt with in this review article, is remarkably important and potentially useful. Moreover, the interested future researchers will surely benefit from the listing of references to some of the other applications of various fractional-calculus operators in the mathematical and other sciences, which we have not considered in the preceding sections (see, for example, [94][95][96][97][98][99][100][101][102][103][104][105][106][107][108][109][110][111]).…”

On some inequalities involving the function and its fractional derivative

“…Namely, it was shown that many real-world phenomena can be better modeled using fractional operators; see, e.g., previous studies. [14][15][16][17][18][19][20] This fact motivated many researchers to study fractional analogues of known functional inequalities; see, e.g., previous studies [21][22][23][24][25][26][27][28][29][30] and the references therein. In particular, in Anastassiou, 25 some fractional Sobolev-and Hilbert-Pachpatte-type inequalities were obtained with respect to ψ-fractional derivatives (see previous studies 16,31 ).…”