scholarly journals Fractional calculus, zeta functions and Shannon entropy

2021 ◽  
Vol 19 (1) ◽  
pp. 87-100
Author(s):  
Emanuel Guariglia
Keyword(s):  
Functional Equation ◽  
Fractional Calculus ◽  
Integral Representation ◽  
Fractional Derivative ◽  
Shannon Entropy ◽  
Zeta Functions ◽  
Bernoulli Numbers

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.

THE ERROR ZETA FUNCTION

2007 ◽  
Vol 03 (03) ◽  
pp. 439-453 ◽  
Author(s):  
ABDUL HASSEN ◽  
HIEU D. NGUYEN
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Special Function ◽  
Zeta Functions ◽  
Bernoulli Numbers ◽  
Intimate Connection

This paper investigates a new special function referred to as the error zeta function. Derived as a fractional generalization of hypergeometric zeta functions, the error zeta function is shown to exhibit many properties analogous to its hypergeometric counterpart, including its intimate connection to Bernoulli numbers. These new properties are treated in detail and used to demonstrate a pre-functional equation satisfied by this special function.

scholarly journals HYPERGEOMETRIC ZETA FUNCTIONS

2010 ◽  
Vol 06 (01) ◽  
pp. 99-126 ◽  
Author(s):  
ABDUL HASSEN ◽  
HIEU D. NGUYEN
Keyword(s):  
Integral Representation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Special Functions ◽  
Zeta Functions ◽  
Functional Inequality ◽  
Bernoulli Numbers ◽  
Classical Counterpart ◽  
New Family ◽  
Riemann Zeta

This paper investigates a new family of special functions referred to as hypergeometric zeta functions. Derived from the integral representation of the classical Riemann zeta function, hypergeometric zeta functions exhibit many properties analogous to their classical counterpart, including the intimate connection to Bernoulli numbers. These new properties are treated in detail and are used to demonstrate a functional inequality satisfied by second-order hypergeometric zeta functions.

scholarly journals A remark on local fractional calculus and ordinary derivatives

2016 ◽  
Vol 14 (1) ◽  
pp. 1122-1124 ◽  
Author(s):  
Ricardo Almeida ◽  
Małgorzata Guzowska ◽  
Tatiana Odzijewicz
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Short Note ◽  
General Definition ◽  
Local Fractional Derivative ◽  
Fundamental Properties ◽  
Local Fractional Calculus ◽  
Definition Of

AbstractIn this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.

Half-lives of spherical proton emitters within the framework of fractional calculus

2014 ◽  
Vol 23 (09) ◽  
pp. 1450044 ◽  
Author(s):  
Abdullah Engin Çalik ◽  
Hüseyin Şirin ◽  
Hüseyin Ertik ◽  
Buket Öder ◽  
Mürsel Şen
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Nuclear Structure ◽  
Caputo Fractional Derivative ◽  
Half Life ◽  
Nuclear Decay ◽  
Life Values ◽  
Derivative Order

In this paper, the half-life values of spherical proton emitters such as Sb , Tm , Lu , Ta , Re , Ir , Au , Tl and Bi have been calculated within the framework of fractional calculus. Nuclear decay equation, related to this phenomenon, has been resolved by using Caputo fractional derivative. The order of fractional derivative μ being considered is 0 < μ ≤ 1, and characterizes the fractality of time. Half-life values have been calculated equivalent with empirical ones. The dependence of fractional derivative order μ on the nuclear structure has also been investigated.

The bouncing ball and the Grünwald-Letnikov definition of fractional derivative

2021 ◽  
Vol 24 (4) ◽  
pp. 1003-1014
Author(s):  
J. A. Tenreiro Machado
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Restitution Coefficient ◽  
Classical Physics ◽  
Bouncing Ball ◽  
Mechanical Experiment ◽  
Fractional Models ◽  
Definition Of

Abstract This paper proposes a conceptual experiment embedding the model of a bouncing ball and the Grünwald-Letnikov (GL) formulation for derivative of fractional order. The impacts of the ball with the surface are modeled by means of a restitution coefficient related to the coefficients of the GL fractional derivative. The results are straightforward to interpret under the light of the classical physics. The mechanical experiment leads to a physical perspective and allows a straightforward visualization. This strategy provides not only a motivational introduction to students of the fractional calculus, but also triggers possible discussion with regard to the use of fractional models in mechanics.

p-adic zeta functions and Bernoulli numbers

1982 ◽  
Vol 19 (2) ◽  
pp. 1186-1194 ◽  
Author(s):  
Yu. V. Osipov
Keyword(s):  
Zeta Functions ◽  
Bernoulli Numbers

scholarly journals Mathematical Models Arising in the Fractal Forest Gap via Local Fractional Calculus

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Chun-Ying Long ◽  
Yang Zhao ◽  
Hossein Jafari
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Mathematical Models ◽  
Cantor Sets ◽  
Forest Gap ◽  
Local Fractional Derivative ◽  
Local Fractional Calculus ◽  
Growth Equations ◽  
Gap Models ◽  
Individual Trees

The forest new gap models via local fractional calculus are investigated. The JABOWA and FORSKA models are extended to deal with the growth of individual trees defined on Cantor sets. The local fractional growth equations with local fractional derivative and difference are discussed. Our results are first attempted to show the key roles for the nondifferentiable growth of individual trees.

Bernoulli Numbers and Zeta Functions

2014 ◽  
Author(s):  
Tsuneo Arakawa ◽  
Tomoyoshi Ibukiyama ◽  
Masanobu Kaneko
Keyword(s):  
Zeta Functions ◽  
Bernoulli Numbers

scholarly journals Some Difference Equations for Srivastava’s λ-Generalized Hurwitz–Lerch Zeta Functions with Applications

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 311 ◽  
Author(s):  
Asifa Tassaddiq
Keyword(s):  
Integral Representation ◽  
Difference Equations ◽  
Zeta Functions ◽  
Numerical Computations ◽  
Direct Evaluation ◽  
Mellin Transforms ◽  
The Family

In this article, we establish some new difference equations for the family of λ-generalized Hurwitz–Lerch zeta functions. These difference equations proved worthwhile to study these newly defined functions in terms of simpler functions. Several authors investigated such functions and their analytic properties, but no work has been reported for an estimation of their values. We perform some numerical computations to evaluate these functions for different values of the involved parameters. It is shown that the direct evaluation of involved integrals is not possible for the large values of parameter s; nevertheless, using our new difference equations, we can evaluate these functions for the large values of s. It is worth mentioning that for the small values of this parameter, our results are 100% accurate with the directly computed results using their integral representation. Difference equations so obtained are also useful for the computation of some new integrals of products of λ-generalized Hurwitz–Lerch zeta functions and verified to be consistent with the existing results. A derivative property of Mellin transforms proved fundamental to present this investigation.

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