Related papers: On formulas and fractional exponents for umbral op…
We give a proper fractional extension of the classical calculus of variations by considering variational functionals with a Lagrangian depending on a combined Caputo fractional derivative and the classical derivative. Euler-Lagrange…
The primary goal of this paper is to introduce and investigate generalized incomplete exponential functions with matrix parameters. Integral representation, differential formula, addition formula, multiplication formula, and recurrence…
The purpose of this paper is to derive some applications of umbral calculus by using extended fermionic p-adic q-integral on Zp. From those applications, we derive some new interesting properties on the new family of Euler numbers and…
We introduce a suitable notion of integral operators (comprising the fractional Laplacian as a particular case) acting on functions with minimal requirements at infinity. For these functions, the classical definition would lead to divergent…
The objective of this paper is to present an approximation formula for the Katugampola fractional integral, that allows us to solve fractional problems with dependence on this type of fractional operator. The formula only depends on…
`Umbral calculus' deals with representations of the canonical commutation relations. We present a short exposition of it and discuss how this calculus can be used to discretize continuum models and to construct representations of Lie…
In this paper, we revisit foundations of umbral calculus using a straightforward approach based on an explicit matrix realization of binomial convolution. We construct an umbral duality of Wronskian type for rational curves in echelon form,…
The paper provides a coherent presentation of an operator scheme, which is used in an approach to inverse problems of mathematical physics (the boundary control method). The scheme is based on the triangular factorization of operators. It…
A very simple closed-form formula for Sheppard's corrections is recovered by means of the classical umbral calculus. By means of this symbolic method, a more general closed-form formula for discrete parent distributions is provided and the…
This paper focuses on the equivalent expression of fractional integrals/derivatives with an infinite series. A universal framework for fractional Taylor series is developed by expanding an analytic function at the initial instant or the…
An algebraic interpretation of the $q$-Meixner polynomials is obtained. It is based on representations of $\mathcal{U}_q(\mathfrak{su}(1,1))$ on $q$-oscillator states with the polynomials appearing as matrix elements of unitary…
We investigate the properties of some recently developed variable-order differential operators involving order transition functions of exponential type. Since the characterisation of such operators is performed in the Laplace domain it is…
The main aim of this present paper is to present a new extension of the fractional derivative operator by using the extension of Beta function recently defined by Shadab et al.[19]. Moreover, we establish some results related to the newly…
In this paper we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann-Liouville fractional integral and derivative operators on a compact of the real axis.This approach has some advantages and allows us to…
We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and $q$-difference equations for these polynomials. A general functional equation is found which allows one to relate…
We prove the decomposition of arbitrary diagonal operators into tensor and matrix products of smaller matrices, focusing on the analytic structure of the resulting formulas and their inherent symmetries. Diagrammatic representations are…
Recently, fractional differential equations have been investigated via the famous variational iteration method. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to…
In this paper we obtain a representation as martingale transform operators for the rearrangement and shift operators introduced by T. Figiel. The martingale transforms and the underlying sigma algebras are obtained explicitly by…
In this paper we provide a definition of fractional gradient operators, related to directional derivatives. We develop a fractional vector calculus, providing a probabilistic interpretation and mathematical tools to treat multidimensional…
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…