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

For a given sequence of real numbers a=(am)m≥1, we define the function Uμ,νa(x)=2μΓ(μ+1)xμ∑m≥1amjm,νμJμ(jm,νx),x∈(0,+∞), where μ,ν>−1, Jμ denotes the Bessel function of order μ, and (jm,ν)m≥ 1 are the positive zeros of Jν. In this paper, we study the function Uμ,νa and some outstanding instances of it on the whole real line and propose a number of conjectures about its positive zeros.