Fractional Sums Research Articles

On fractional sums of the divisor functions

In this paper, we consider the fractional sum of the divisor functions. We can improve previous results considered by [O. Bordellés, On certain sums of number theory, Int. J. Number Theory 18(9) (2022) 2053–2074; K. Liu, J. Wu and Z. S. Yang, On some sums involving the integral part function, preprint (2021); arXiv:2109.01382v1 [math.NT]]. Precisely, we can show that [Formula: see text] where [Formula: see text] is an arbitrary small positive constant and [Formula: see text] is the number of representations of [Formula: see text] as product of [Formula: see text] natural numbers.

A review of definitions of fractional differences and sums

<p style='text-indent:20px;'>Given the increasing importance of discrete fractional calculus in mathematics, science engineering and so on, many different concepts of fractional difference and sum operators have been defined. In this paper, we mainly reviews some definitions of fractional differences and sum operators that emerged in the fields of discrete calculus. Moreover, some properties of those operators are also analyzed and compared with each other, including commutation rules, linearity, Leibniz rules, etc.</p>

Hyperbolic summation for fractional sums

Hyperbolic summation for fractional sums

Some fractional Dirac systems

In this work, including $\alpha\epsilon(0,1)$; we examined the Dirac system in the frame which includes$\ \alpha$ order right and left Reimann-Liouville fractional integrals and derivatives with exponential kernels, and the Dirac system which includes $\alpha$ order right and left Caputo fractional integrals and derivatives with exponential kernels. Furthermore, we have given some definitions and properties for discrete exponential kernels and their associated fractional sums and fractional differences, and we have studied discrete fractional Dirac systems.

Explicit identities for infinite families of series involving squared binomial coefficients

Ramanujan famously evaluated the integer/half-integer moments of the complete elliptic integral of the first kind as finite sums, and, quite recently, innovative coefficient-extraction techniques based on Gauss' hypergeometric identity and a Watson–Whipple-like F23-series were introduced by Wang and Chu to generalize this hypergeometric identity due to Ramanujan, providing explicit evaluations, as finite sums, for infinite families of series involving harmonic numbers and squared central binomial coefficients. In this article, we make use of a modified Abel summation lemma to greatly build upon these results, solving two open problems concerning the recent work of Wang and Chu. We also consider how our results may be extended via fractional sums, and we introduce formulas involving the Landau constants.

NEW DEVELOPMENTS IN WEIGHTED n-FOLD TYPE INEQUALITIES VIA DISCRETE GENERALIZED ℏ̂-PROPORTIONAL FRACTIONAL OPERATORS

This study explores some significant consequences of discrete [Formula: see text]-proportional fractional sums [Formula: see text] having an exponential function as a nonlocal kernel. Certain novel weighted versions comprising a group of positive mappings via [Formula: see text] are given. A variety of refinements can be derived by taking into account the extraction of the new estimates and the nabla [Formula: see text]-fractional sums. The suggested technique is a revolutionary formulation of conventional operators that may be used to design efficient mechanism descriptions in short time spans by provoking certain noteworthy properties of chaos theory. Moreover, novel generalizations of the discrete [Formula: see text]-fractional sum can be generated by the specific value of the proportionality index. Derived outcomes and investigation confirm that the proposed plan will offer gains in many modeling and chaotic framework applications.

SOME RECENT DEVELOPMENTS ON DYNAMICAL ℏ-DISCRETE FRACTIONAL TYPE INEQUALITIES IN THE FRAME OF NONSINGULAR AND NONLOCAL KERNELS

Discrete fractional calculus ([Formula: see text]) is significant for neural networks, complex dynamic systems and frequency response analysis approaches. In contrast with the continuous-time frameworks, fewer outcomes are accessible for discrete fractional operators. This study investigates some major consequences of two sorts of inequalities by considering discrete Atangana–Baleanu [Formula: see text]-fractional operator having [Formula: see text]-discrete generalized Mittag-Leffler kernels in the sense of Riemann type ([Formula: see text]). Certain novel versions of reverse Minkowski and related Hölder-type inequalities via discrete [Formula: see text]-fractional operators having [Formula: see text]-discrete generalized Mittag-Leffler kernels are given. Moreover, several other generalizations can be generated for nabla [Formula: see text]-fractional sums. The proposing discretization is a novel form of the existing operators that can be provoked by some intriguing features of chaotic systems to design efficient dynamics description in short time domains. Furthermore, by combining two mechanisms, numerous new special cases are introduced.

Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums

This work presents an overview of the summability of divergent series and fractional finite sums, including their connections. Several summation methods listed, including the smoothed sum, permit obtaining an algebraic constant related to a divergent series. The first goal is to revisit the discussion about the existence of an algebraic constant related to a divergent series, which does not contradict the divergence of the series in the classical sense. The well-known Euler–Maclaurin summation formula is presented as an important tool. Throughout a systematic discussion, we seek to promote the Ramanujan summation method for divergent series and the methods recently developed for fractional finite sums.

Hermite–Hadamard Type Inequalities Involving k-Fractional Operator for (h¯,m)-Convex Functions

The principal motivation of this paper is to establish a new integral equality related to k-Riemann Liouville fractional operator. Employing this equality, we present several new inequalities for twice differentiable convex functions that are associated with Hermite–Hadamard integral inequality. Additionally, some novel cases of the established results for different kinds of convex functions are derived. This fractional integral sums up Riemann–Liouville and Hermite–Hadamard’s inequality, which have a symmetric property. Scientific inequalities of this nature and, particularly, the methods included have applications in different fields in which symmetry plays a notable role. Finally, applications of q-digamma and q-polygamma special functions are presented.

SOME FURTHER EXTENSIONS CONSIDERING DISCRETE PROPORTIONAL FRACTIONAL OPERATORS

In this paper, some attempts have been devoted to investigating the dynamic features of discrete fractional calculus (DFC). To date, discrete fractional systems with complex dynamics have attracted the most consideration. By considering discrete [Formula: see text]-proportional fractional operator with nonlocal kernel, this study contributes to the major consequences of the certain novel versions of reverse Minkowski and related Hölder-type inequalities via discrete [Formula: see text]-proportional fractional sums, as presented. The proposed system has an intriguing feature not investigated in the literature so far, it is characterized by the nabla [Formula: see text] fractional sums. Novel special cases are reported with the intention of assessing the dynamics of the system, as well as to highlighting the several existing outcomes. In terms of applications, we can employ the derived consequences to investigate the existence and uniqueness of fractional difference equations underlying worth problems. Finally, the projected method is efficient in analyzing the complexity of the system.

On Discrete Delta Caputo–Fabrizio Fractional Operators and Monotonicity Analysis

The discrete delta Caputo-Fabrizio fractional differences and sums are proposed to distinguish their monotonicity analysis from the sense of Riemann and Caputo operators on the time scale Z. Moreover, the action of Q− operator and discrete delta Laplace transform method are also reported. Furthermore, a relationship between the discrete delta Caputo-Fabrizio-Caputo and Caputo-Fabrizio-Riemann fractional differences is also studied in detail. To better understand the dynamic behavior of the obtained monotonicity results, the fractional difference mean value theorem is derived. The idea used in this article is readily applicable to obtain monotonicity analysis of other discrete fractional operators in discrete fractional calculus.

On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

Monotonicity analysis of delta fractional sums and differences of order υ∈(0,1] on the time scale hZ are presented in this study. For this analysis, two models of discrete fractional calculus, Riemann–Liouville and Caputo, are considered. There is a relationship between the delta Riemann–Liouville fractional h-difference and delta Caputo fractional h-differences, which we find in this study. Therefore, after we solve one, we can apply the same method to the other one due to their correlation. We show that y(z) is υ-increasing on Ma+υh,h, where the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and then, we can show that y(z) is υ-increasing on Ma+υh,h, where the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) for each z∈Ma+h,h. Conversely, if y(a+υh) is greater or equal to zero and y(z) is increasing on Ma+υh,h, we show that the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) on Ma,h. Furthermore, we consider some related results for strictly increasing, decreasing, and strictly decreasing cases. Finally, the fractional forward difference initial value problems and their solutions are investigated to test the mean value theorem on the time scale hZ utilizing the monotonicity results.

More efficient estimates viaℏ-discrete fractional calculus theory and applications

Discrete fractional calculus (DFC) is continuously spreading in the engineering practice, neural networks, chaotic maps, and image encryption, which is appropriately assumed for discrete-time modelling in continuum problems. First, we start with a novel discrete ℏ-proportional fractional sum defined on the time scale ℏZ so as to give the premise to the more broad and complex structures, for example, the suitably accustomed transformations conjuring the property of observing the new chaotic behaviors of the logistic map. Here, we aim to present the novel discrete versions of Grüss and certain other associated variants by employing discrete ℏ-proportional fractional sums are established. Moreover, several novel consequences are recaptured by the ℏ-discrete fractional sums. The present study deals with the modification of Young, weighted-arithmetic and geometric mean formula by taking into account changes in the exponential function in the kernel represented by the parameters of the operator, varying delivery noted outcomes. In addition, two illustrative examples are apprehended to demonstrate the applicability and efficiency of the proposed technique.

Discrete weighted fractional calculus and applications

We introduce fractional (nabla) sums and differences with a weight function and prove some of their properties. We give particular emphasis to the tempered fractional case. In the end, we postulate a novel entropic functional and discuss some of its properties.

More Properties of Fractional Proportional Differences

The main aim of this paper is to clarify the action of the discrete Laplace transform on the fractional proportional operators. First of all, we recall the nabla fractional sums and differences and the discrete Laplace transform on a time scale equivalent to $h\mathbb{Z}$. The discrete $h-$Laplace transform and its convolution theorem are then used to study the introduced discrete fractional operators.

On a new type of fractional difference operators on h-step isolated time scales

In this article, a new type of fractional sums and differences called the discrete weighted fractionaloperators are presented. The weighted backward and forward difference operators are defined on anisolated time scale with arbitrary step size and they obey the power law.

New discrete inequalities of Hermite\u2013Hadamard type for convex functions

We introduce new time scales on mathbb{Z}. Based on this, we investigate the discrete inequality of Hermite–Hadamard type for discrete convex functions. Finally, we improve our result to investigate the discrete fractional inequality of Hermite–Hadamard type for the discrete convex functions involving the left nabla and right delta fractional sums.

A unifying computational framework for novel estimates involving discrete fractional calculus approaches

The aim of this paper is to evaluate the potential improvement of classification results in the frame of discrete proportional fractional operator. The nonlocal kernel of the generalized proportional fractional sum depending on ĥ-discrete exponential functions defined on time scale ĥZ. This paper deals novel discrete versions of the Pólya-Szegö and ČebyšeV type inequalities via discrete ĥ-proportional fractional sums. These generalizations have potential utilities in the study of finite difference equations and statistical analysis. Taking into account the discrete ĥ-proportional fractional sums, the main consequences concerns a quite general form of the Pólya-Szegö and ČebyšeV variants. In addition, the present investigation is a discrete analogue of integral inequalities established in the relative literature and also expands several discrete variants for nabla ĥ-fractional sums in particular.

New formulation for discrete dynamical type inequalities via $ h $-discrete fractional operator pertaining to nonsingular kernel.

Discrete fractional calculus (DFC) use to analyse nonlocal behaviour of models has acquired great importance in recent years. The aim of this paper is to address the discrete fractional operator underlying discrete Atangana-Baleanu (AB)-fractional operator having $\hbar$-discrete generalized Mittag-Leffler kernels in the sense of Riemann type (ABR). In this strategy, we use the $\hbar$-discrete AB-fractional sums in order to obtain the Gr\"{u}ss type and certain other related variants having discrete generalized $\hbar$-Mittag-Leffler function in the kernel. Meanwhile, several other variants found by means of Young, weighted-arithmetic-geometric mean techniques with a discretization are formulated in the time domain $\hbar\mathbb{Z}$. At first, the proposed technique is compared to discrete AB-fractional sums that uses classical approach to derive the numerous inequalities, showing how the parameters used in the proposed discrete $\hbar$-fractional sums can be estimated. Moreover, the numerical meaning of the suggested study is assessed by two examples. The obtained results show that the proposed technique can be used efficiently to estimate the response of the neural networks and dynamic loads.

Discrete generalized fractional operators defined using h‐discrete Mittag‐Leffler kernels and applications to AB fractional difference systems

This study investigates the h‐fractional difference operators with h‐discrete generalized Mittag‐Leffler kernels ( in the sense of Riemann type (namely, the ABR) and Caputo type (namely, the ABC). For which, we will discuss the region of convergent. Then, we study the h‐discrete Laplace transforms to formulate their corresponding AB‐fractional sums. Also, it is useful in obtaining the semi‐group properties. We will prove the action of fractional sums on the ABC type h‐fractional differences and then it can be used to solve the system of ABC h‐fractional difference. By using the h‐discrete Laplace transforms and the Picard successive approximation technique, we will solve the nonhomogeneous linear ABC h‐fractional difference equation with constant coefficient, and also we will remark the h‐discrete Laplace transform method for the continuous counterpart. Meanwhile, we will obtain a nontrivial solution for the homogeneous linear ABC h‐fractional difference initial value problem with constant coefficient for the case δ ≠ 1. We will formulate the relation between the ABC and ABR h‐fractional differences by using the h‐discrete Laplace transform. By iterating the fractional sums of order −(ϕ, δ, 1), we will generate the h‐fractional sum‐differences, and in view of this, a semigroup property will be proved. Due to these new powerful techniques, we can calculate the nabla h‐discrete transforms for the AB h‐fractional sums and the AB iterated h‐fractional sum‐differences. Furthermore, we will obtain some particular cases that can be found in examples and remarks. Finally, we will discuss the higher order case of the h‐discrete fractional differences and sums.