Partial Fraction Expansions and Zeros of Hankel Transforms

It is proved by the method of partial fraction expansion and Sturm’s oscillation theory that the zeros of certain Hankel transforms are all real, simple and distributed one by one between consecutive zeros of Bessel functions. As an application, we obtain a list of sufficient conditions as well as necessary conditions on parameters for which $${}_1F_2$$ hypergeometric functions belong to the Laguerre-Pólya class.