Integral Representations for Products of Two Bessel or Modified Bessel Functions

Abstract

The first part of the article contains integral expressions for products of two Bessel functions of the first kind having either different integer orders or different arguments. A similar question for a product of modified Bessel functions of the first kind is solved next, when the input functions are of different integer orders and have different arguments. Keywords: Bessel function of the first kind; modified Bessel function of the first kind; integral representation of product functions; Chebyshev polynomial of the first kind MSC: Primary: 33C10; Secondary: 33C05; 33C20 √ X 2 e it + x 2 e −it , while Nicholson proved that ([5], p. 236, Equation (39)) J n (z)J −n (z) = 1 π sin(nπ) π 0 sin(2nx) J 0 (2z sin x) dx, where z ∈ R and n / ∈ Z. This result implies for n non-negative integer the formula ([5], p. 236, Equation (40)) J 2 n (z) = 1 π π 0 sin(2nx) J 0 (2z sin x) dx .Nicholson also derived integral representation for the mixed product J n (z)Y n (z) of the Bessel function of the first kind J n (z) and of the Bessel function of the second kind Y n (z) when n ∈ Z; similar results can be found also in the already cited papers by Dixon and Ferrar [4,6] and by Görlich et al. [7].