On the properties of zeros of Bessel series in the real line

For a given sequence of real numbers a = (a(m))(m >= 1), we define the function U-mu,v(a) (x) = 2(mu)Gamma(mu + 1)/x(mu) Sigma(m >= 1) a(m)/j(m,v)(mu) J(mu)(j(m,v)x), x is an element of (0,+infinity), where mu, v > -1, J(mu) denotes the Bessel function of order mu, and (j(m,v))m >= 1 are the positive zeros of J(v). In this paper, we study the function U-mu,v(a) and some outstanding instances of it on the whole real line and propose a number of conjectures about its positive zeros.