A METHOD FOR SUMMING BESSEL SERIES AND A COUPLE OF ILLUSTRATIVE EXAMPLES

For mu, nu > -1, we consider the Bessel series U-mu,nu(a)(x) = 2 mu Gamma(mu+1)/x(mu) Sigma(m >= 1jm= 1) are the positive zeros of J(nu) and a = (a(m))(m >= 1) is a sequence of real numbers satisfying Sigma(m >= 1) vertical bar a(m)vertical bar/j(m,nu)(mu+1/2) < +infinity. We propose a method for computing in a closed form the sum of the Bessel series U-mu,nu(a) assuming that for a particular value eta of the parameter mu a closed expression for U-eta,nu(a) as a power series of x (not necessarily with integer exponents) is known. We illustrate the method with some examples. One of them is related to the sine coefficients of the function 1-x(s), s > -1. The closed form of the sum is then given in terms of a generalization of the Bernoulli numbers.