A sampling theorem of hypergeometric $$ _1F_2$$ functions and the Laguerre-Pólya class

We investigate the pattern and the range of parameters for which hypergeometric $$ _1F_2$$ functions belong to the Laguerre-Pólya class with infinitely many real zeros. Our approach is based on the sampling theorem which provides sufficient conditions in terms of the sign of samples. In the case where the Lommel function comes into play, we use the differential equation method to determine the sign change, thereby specifying those parameters. As for the reality of all zeros of the Lommel function, our results give an extensive range of parameters, which turns out to be the best possible for the Struve function.