Related papers: Fourth-order Bessel-type special functions: a surv…

We observe that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In addition, we describe numerical experiments which illustrate…

In a recent paper we investigated the internal space of Bessel functions associated with their orders. We found a formula (new) unifying Bessel functions of integer and of real orders. In this paper we study the deformed exterior derivative…

The order derivatives of the modified Bessel function of the second kind at s = .5 are obtained as finite expressions of integrals that generalize the exponential integral appearing in the first derivative (Theorem 1.) The derivatives arise…

We evaluate definite integrals involving the product of four modified Bessel functions of the first and second kind and a power function. We provide general formulas expressed in terms of the Meijer $G$-function and generalized…

We present a straightforward discretization of the Bessel functions $J_n(x)$ to discrete counterparts $B^{(N)}_n(x_m)$, of $N$ integer orders $n$ on $N$ integer points $x_m \equiv m$, that we call discrete Bessel functions. These are built…

Discrete analogs of the index transforms, involving Bessel and Lommel functions are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.

Discrete analogs of the index transforms, involving Bessel and the modified Bessel functions are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.

This paper offers a new and complete description of subnormal solutions of certain non-homogeneous second order periodic linear differential equations first studied by Gundersen and Steinbart in 1994. We have established a previously…

Differential subordination and superordination preserving properties for univalent functions in the open unit disk with an operator involving generalized Bessel functions are derived. Some particular cases involving trigonometric functions…

In this paper we study fractional powers of the Bessel differential operator defined on a semiaxis. Some important properties of such fractional powers of the Bessel differential operator are proved. They include connections with Legendre…

We use the operator method to evaluate a class of integrals involving Bessel or Bessel-type functions. The technique we propose is based on the formal reduction of these family of functions to Gaussians.

We obtain integral representations of the $n$-th derivatives of the Bessel functions with respect to the order. The numerical evaluation of these expressions is very efficient using a double exponential integration strategy. Also, from the…

The goal of this paper is to extend the classical and multiplicative fractional derivatives. For this purpose, it is introduced the new extended modified Bessel function and also given an important relation between this new function…

For integral representations of associated Legendre functions in terms of modified Bessel functions, we establish justification for differentiation under the integral sign with respect to parameters. With this justification, derivatives for…

Using the theory of orthogonal polynomials, their associated recursion relations and differential formulas we develop a method for evaluating new integrals. The method is illustrated by obtaining a closed-form expression for the value of an…

Series involving hypergeometric functions are used to derive, extend and evaluate integrals involving the product of two Bessel functions of the first kind $J_{u}(a z)$ $J_{v}(b z)$ with order $u,v$, studied by Landau et al. The method used…

In this paper, sums represented in (3) are studied. The expressions are derived in terms of Bessel functions of the first and second kinds and their integrals. Further, we point out the integrals can be written as a Meijer G function.

Functions like the exponential, Chebyshev polynomials, and monomial symmetric polynomials are preeminent among all special functions. They have simple definitions and can be expressed using easily specified integers like n!. Families of…

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a…

A new method is presented for obtaining indefinite integrals of common special functions. The approach is based on a Lagrangian formulation of the general homogeneous linear ordinary differential equation of second order. A general integral…