On a Class of Integrals Involving a Bessel Function times Gegenbauer Polynomials

Recommended by Feng Qi We provide information and explicit formulae for a class of integrals involving Bessel functions and Gegenbauer polynomials. We present a simple proof of an old formula of Gegenbauer. Some interesting special cases and applications of this result are obtained. In particular, we give a short proof of a recent result of A. A. R. Neves et al. regarding the analytical evaluation of an integral of a Bessel function times associated Legendre functions. These integrals arise in problems of vector diffraction in electromagnetic theory.


Introduction
In the recent article [1], Neves et  where P m n (cosθ) is the associated Legendre function and J m (z) is the Bessel function of the first kind and order m.
The authors of [1] encountered this integral in their work [2] dealing with the calculation of the optical force of the optical tweezers in a complete electromagnetic treatment for any beam shape focused on an arbitrary position.The integral (1.1) appears also in fields related to vector diffraction theory where computationally intensive methods or approximations are employed.
As the authors of [1] pointed out, an explicit formula for the integral (1.1) would be a useful result, making unnecessary any numerical approximations of it.More specifically, 2 International Journal of Mathematics and Mathematical Sciences Neves et al. showed in [1] that for all integers n, m such that n ≥ 0 and −n ≤ m ≤ n, one has where j n (R) is the spherical Bessel function of order n, that is, 3) The authors mention in [1] that the integral (1.1) has not been reported in a closed form and it is not shown in any integral tables.In [3, page 379, formula (1)], however, the following closely related formula is given: where C ν n (x) is the Gegenbauer (or ultraspherical) polynomial of degree n and order ν defined by the generating function The formula (1.4) is due to Gegenbauer and holds for all real numbers ν such that ν An equivalent form of (1.4) can also be found in the integral tables [4, pages 838-839, formula 7.333 (1) and ( 2)].
In this note we show that (1.2) follows easily from (1.4) using properties of the associated Legendre functions.
A proof of (1.4) is given in [3, pages 378-379], using a method which is based on integration over a unit sphere.Since formula (1.4) is important in applications in many different fields in physics, especially for those that require partial wave decomposition, we give in the next section a new proof of (1.4) which is simpler than the one given in [3].We will also present some other consequences of (1.4).

Proofs and additional comments
We first show how to obtain (1.2) from (1.4).Suppose first that m, n are nonnegative integers such that m ≤ n.It is well known that the relation between associated Legendre functions P m n (cosθ) and Gegenbauer polynomials is given by the formula is negative such that −n ≤ m < 0, we cannot apply (1.4) as above, because the condition ν = m + 1/2 > −1/2 is not fulfilled.This case can be handled using the formulae see [3, page 15, formula (2)], and (2.4) Since n ≥ −m > 0, applying (1.2) established in the previous case and using (2.3) and (2.4), we get which completes the proof of (1.2).We next give a simple proof of (1.4).Let ν > −1/2.The Gegenbauer polynomials C ν n (cosθ) satisfy the following orthogonality property: (2.6) We will also use the formula whence (1.4) follows at once.It is a further confirmation of the importance of formula (1.4) that some other results are obtained by it as special cases.Indeed, taking the limiting case of (1.4) when α → 0 using the dominated convergence theorem and the fact that , for λ > −1,  (2.10) we obtain sin 2ν θ dθ.
(2.11) Formula (2.11) is Gegenbauer's generalization of the Poisson integral representation of Bessel functions.For n = 0, it reduces to Poisson's formula .12) see [4, page 48, formula (6)].Since = cos nθ, (2.13) taking the limit in (2.11) as ν → 0 applying once more the dominated convergence theorem, we deduce that for all nonnegative integers n, we have (2.14) In view of (2.2), these equalities hold also for all negative integers n, hence (2.14) gives all the Fourier coefficients of the function exp(iRcos θ).
Finally, we note that the Gouesbet and Lock result [6] for the integral where n, m are integers such that |m| ≤ n, can be derived from (2.11) in exactly the same way as (1.2) is obtained from (1.4).

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: al. gave an analytical evaluation of the integral I m n := π 0 sinθ exp(iRcos αcos θ)P m n (cosθ)J m (Rsinαsinθ)dθ, (1.1)