Related papers: Reflection formulas for order derivatives of Besse…
In this work, we consider the Dunkl complex reflection operators related to the group $G(m,1,N)$ in the complex plane \begin{align*} T_i=\frac{\partial}{\partial z_i}+k_0\sum_{j\neq i}\sum_{r=0}^{m-1}\frac{1-s_i^{-r}(i,j)s_i^r}…
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 contribution proposes a new formulation to efficiently compute directional derivatives of order one to fourth. The formulation is based on automatic differentiation implemented with dual numbers. Directional derivatives are particular…
In this work, series expansions in terms of Bessel functions of the first kind are given for the sine and cosine integrals. These representations differ from many of the known Neumann-type series expansions for the sine and cosine…
We find an explicit integral formula for the eigenfunctions of a fourth order differential operator against the kernel involving two Bessel functions. Our formula establishes the relation between K-types in two different realizations of the…
The $\nu$-zeros of the Bessel functions of purely imaginary order are examined for fixed argument $x>0$. In the case of the modified Bessel function of the second kind $K_{i\nu}(x)$, it is known that it possesses a countably infinite…
In this note we give a simplified ordinal analysis of first-order reflection. An ordinal notation system $OT$ is introduced based on $\psi$-functions. Provable $\Sigma_{1}$-sentences on $L_{\omega_{1}^{CK}}$ are bounded through…
We give an algorithm to compute the series expansion for the inverse of a given function. The algorithm is extremely easy to implement and gives the first $N$ terms of the series. We show several examples of its application in calculating…
Using Jacobi's identity we derive a simple expression for the Bessel functions of integer order in terms of combinations of powers and hyperbolic functions of the same argument.
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…
This paper derives new integral representations for products of two parabolic cylinder functions. In particular, expressions are obtained for D_{nu}(x)D_{mu}(y), with x>0 and y>0, that allow for different orders and arguments in the two…
Based on known definite integrals of Bessel functions of the first kind, we obtain exact solutions to unknown definite integrals using the method of integral transforms from Hankel's transform.
The purpose of this note is to compare various approximation methods as applied to the inverse of the Bessel function of the first kind, in a given domain of the complex plane.
The Gr\"unwald and shifted Gr\"unwald formulas for the function $y(x)-y(b)$ are first order approximations for the Caputo fractional derivative of the function $y(x)$ with lower limit at the point $b$. We obtain second and third order…
Generalized integral formulas involving the generalized modified k-Bessel function $J_{k,\nu }^{c,\gamma ,\lambda }\left( z\right) $ of first kind are expressed in terms generalized $k-$Wright functions. Some interesting special cases of…
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.
We derive two distinct asymptotic expansions for the zeros $j_{\nu,k}^{(n)}$ of the $n$-th derivative of Bessel function $J_\nu^{(n)}(x)$. The first is a McMahon-type expansion for the case when $k \to \infty$ with fixed $\nu$, for which we…
Given a right-non-degenerate set-theoretic solution $(X,r)$ to the Yang-Baxter equation, we construct a whole family of YBE solutions $r^{(k)}$ on $X$ indexed by its reflections $k$ (i.e., solutions to the reflection equation for $r$). This…
We introduce the notion of reflections for selfinjective algebras from the point of view of torsion theories induced by two-term tilting complexes. As an application, we determine the transformations of Brauer trees associated with…
It is known that Struve function $\mathbf H_\nu$ and modified Struve function $\mathbf L_\nu$ are closely connected to Bessel function of the first kind $J_\nu$ and to modified Bessel function of the first kind $I_\nu$ and possess…