In boundary value problems involving the diffusion equation the following infinite sums sometimes appear,
(1) |
most often for , where are the positions of the zeros located away from the origin of the regular cylindrical Bessel function , with real and integer . The sums are convergent for . As we will show in what follows, all these sums have the property that they are given by the ratio of two polynomials on with integer coefficients. The simplest and most common example is
(2) |
In a few cases the exact expression of these polynomials are available in the literature [1]. The known cases are those obtained by Rayleigh, extending investigations by Euler, for through , and one discovered by Cayley, for . The cases and seem not to be generally known, and will be given explicitly further on. The known cases were obtained in a case-by-case fashion, using the expression of the Bessel functions as infinite products involving its zeros.
In this paper we will provide a simple, independent proof of all the known formulas, and will present a general formula from which the specific formulas can be derived, for any given strictly positive integer value of , by purely algebraic means. The proof will rely entirely on the general properties of analytical functions and on the well-known properties of the functions , which are generally available in the literature, for example in [2].