Related papers: Dual identities in fractional difference calculus …

We Investigate two types of dual identities for Caputo fractional differences. The first type relates nabla and delta type fractional sums and differences. The second type represented by the Q-operator relates left and right fractional sums…

Recently, some authors have proved monotonicity results for delta and nabla fractional differences separately. In this article, we use dual identities relating delta and nabla fractional difference operators to prove shortly the…

In fractional calculus there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and…

A discrete version of the symmetric duality of Caputo-Torres, to relate left and right Riemann-Liouville and Caputo fractional differences, is considered. As a corollary, we provide an evidence to the fact that in case of right fractional…

This article investigates a class of discrete nabla fractional operators by using the discrete nabla convolution theorem. Inspired by this, we define the discrete generalized nabla fractional sum and differences of Riemann-Liouville and…

Different fractional difference types of Euler-Lagrange equations are obtained within Riemann and Caputo by making use of different versions of integration by part forumlas in fractional difference calculus. An example is presented to…

We introduce a more general discrete fractional operator, given by convex linear combination of the delta and nabla fractional sums. Fundamental properties of the new fractional operator are proved. As particular cases, results on delta and…

We prove duality between the left and right fractional derivatives, independently on the type of fractional operator. Main result asserts that the right derivative of a function is the dual of the left derivative of the dual function or,…

This paper intends to develop a $q$-difference operator $\nabla^{(\gamma)}_q$ of fractional order $\gamma$, and give several intriguing properties of this new difference operator. Our main focus remains on the construction of sequence…

We introduce a nabla, a delta, and a symmetric fractional calculus on arbitrary nonempty closed subsets of the real numbers. These fractional calculi provide a study of differentiation and integration of noninteger order on discrete,…

Based on the fractional $q$-integral with the parametric lower limit of integration, we define fractional $q$-derivative of Riemann-Liouville and Caputo type. The properties are studied separately as well as relations between them. Also, we…

In this paper, we introduce the nabla fractional derivative and fractional integral on time scales in the Riemann-Liouville sense. We also introduce the nabla fractional derivative in Gr\"unwald-Letnikov sense. Some of the basic properties…

We introduce a natural definition for sums of the form \[ \sum_{\nu=1}^x f(\nu) \] when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the…

In 2003, Zudilin presented a $q$-analogue of Euler's identity for one of the variants of $q$-double zeta function. This article focuses on exploring identities related to another variant of $q$-double zeta function and its star variant.…

We formulate fractional difference equations of Riemann-Liouville and Caputo type in a functional analytical framework. Main results are existence of solutions on Hilbert space-valued weighted sequence spaces and a condition for stability…

There are several approaches to the fractional differential operator. Generalized q-fractional difference operator was defined in the aid of q-iterated Cauchy integral and q-calculus techniques. We introduce Caputo type derivative related…

We introduce a fractional calculus on time scales using the theory of delta (or nabla) dynamic equations. The basic notions of fractional order integral and fractional order derivative on an arbitrary time scale are proposed, using the…

While the definition of a fractional integral may be codified by Riemann and Liouville, an agreed-upon fractional derivative has eluded discovery for many years. This is likely a result of integral definitions including numerous constants…

In this paper we derive a sufficient condition for the existence of a unique solution of a Cauchy type q-fractional problem (involving the fractional q-derivative of Riemann-Liouville type) for some nonlinear differential equations. The key…

We formulate a new class of fractional difference and sum operators, study their fundamental properties, and find their discrete Laplace transforms. The method depends on iterating the fractional sum operators corresponding to fractional…