Related papers: Analytic computations of digamma function using so…
We derive new identities involving zeros of the Bessel function $J_{\nu}$ and some related functions. These are special cases of more general identities obtained in this note, which might also be of interest.
The theory of fractional calculus has developed in a number of directions over the years, including: the formulation of multiple different definitions of fractional differintegration; the extension of various properties of standard calculus…
We introduce and study the properties of a new family of fractional differential and integral operators which are based directly on an iteration process and therefore satisfy a semigroup property. We also solve some ODEs in this new model…
Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We…
We investigate the possibilities to calculate vector partition functions by means of iterated partial fraction decomposition, as suggested by Beck (2004). Particularly, for an important type of families of rational functions, we describe an…
Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite…
A new mathematical notation is proposed for the iteration of functions. It facilitates the application of the iteration of functions in mathematical and logical expressions, definitions of sets, and formulations of algorithms. Illustrations…
Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel…
We obtain two new algorithms for partial fraction decompositions; the first is over algebraically closed fields, and the second is over general fields. These algorithms takes $O(M^2)$ time, where $M$ is the degree of the denominator of the…
We present an algorithm to compute values L(s) and derivatives of L-functions of motivic origin numerically to required accuracy. Specifically, the method applies to any L-series whose Gamma-factor is a product of any number of…
This paper continues a series of investigations on converging representations for the Riemann Zeta function. We generalize some identities which involve Riemann's zeta function, and moreover we give new series and integrals for the zeta…
The function spaces of continuously differentiable functions are extensively studied and appear in various mathematical settings. In this context, we investigate the spaces of continuously fractional differentiable functions of order…
We derive simple new expressions, in various dimensions, for the functional determinant of a radially separable partial differential operator, thereby generalizing the one-dimensional result of Gel'fand and Yaglom to higher dimensions. We…
In this article, we illustrate the asymptotic behaviour of exponential functions of nabla fractional calculus. For this purpose, we propose a novel matrix technique to compute these functions numerically.
In this paper we propose an algorithm for the numerical solution of arbitrary differential equations of fractional order. The algorithm is obtained by using the following decomposition of the differential equation into a system of…
A new definition of a fractional derivative has recently been developed, making use of a fractional Dirac delta function as its integral kernel. This derivative allows for the definition of a distributional fractional derivative, and as…
It is well known that using high-order numerical algorithms to solve fractional differential equations leads to almost the same computational cost with low-order ones but the accuracy (or convergence order) is greatly improved, due to the…
We give a new proof of Chan's identity involving the cubic partition function and we also give a new identity for the cubic partition function which is analogues to the Zuckerman's identity for the ordinary partition function.
In this paper, we present the definitions and some properties of the general fractional integrals (GFIs) and general fractional derivatives (GFDs) of a function f(x) with respect to another function g(x). Examples of special cases of…
We present a new discretization for the Hadamard fractional derivative, that simplifies the computations. We then apply the method to solve a fractional differential equation and a fractional variational problem with dependence on the…