The initial roots of the Bessel function can be calculated using relation ςk=ςk−1+π, where ςk denotes kth positive roots of Bessel function [26,41]. After obtaining the initial roots, further roots can be calculated accurately in an iterative manner using Newton-Raphson method [...
The derivative of the Bessel function is given by the recursion relation (see Ref. [3], 10.6.2 and Refs. [2], [6])(1)Jn′(z)=12(Jn−1(z)−Jn+1(z)).The roots of this expression are then found using standard built in root finding methods of Mathematica to an accuracy of ...
The recursion relation for where is the McDonald function, is extended to the case of together with the same for the Bessel functions of the first and second kind. An application to the Bessel-type kernels is indicated which is of importance for inversion problems for the Riesz potential ...
recursion relationsSubroutine BESYN implements the three-term recursion relation for Y/sub nu /(x) in forward recursion starting with Bessel function values for two consecutive integers. If nu is less than 100, Y sub 0 (x) and Y sub 1 (x) are generated for the first members of the ...
Bessel functionsum rulerecursion relationA four-term recurrence relation for squared spherical Bessel functions is shown to yield closed-form expressions for several types of finite weighted sums of these functions. The resulting sum rules, which may contain an arbitrarily large number of terms, are ...
In this contribution, we first present a new recursion relation fulfilled by the linearization coefficients of Bessel polynomials (LCBPs), which is different than the one presented by Berg and Vignat in 2008. We will explain why this new recursion formula is as important as Berg ...
. We also show that the residues and values of z((nu),s) at negative integers are simply expressed in terms of a sequence of polynomials (in (nu) with rational coefficients) that satisfy a quadratic recursion relation.;We also prove that z((nu),s) has an analytic continuation in (nu...
Show that Iν(x) and Kν(x) satisfy the Wronskian relation Iν(x)Kν′(x)−Iν′(x)Kν(x)=−1x. 14.5.12 Verify that the coefficient in the axial Green's function of Eq. (14.117) is −1. 14.5.13 If r=(x2+y2)1∕2, prove that 1r=2π∫0∞cos(xt)K0(yt)dt.Th...
For real ν the Bessel function Jν has infinitely many real zeros, and whenν > −1, then all the zeros are real. All the zeros are simple (except possibly at the origin). Each of the functions Jν(z),Yν(z),Hν(1)(z), or Hν(2)(z) satisfies the recurrence relation za...
Bessel functionassociated Laguerre polynomialGegenbauer polynomialrecursion relationfunction spectral decompositionSCATTERINGUsing the theory of orthogonal polynomials, their associated recursion relations and differential formulas we develop a new method for evaluating integrals that include orthogonal polynomials. The...