The Sampling Theorem and the Lerche–Newberger Formula

ABSTRACT In this paper, we prove that the sampling theorem for the product of two entire functions, which grows no faster than reciprocal gamma functions, implies the generalization of the Lerche–Newberger formula for the series involving product of Bessel functions of the first kind. As a consequence of the generalized Lerche–Newberger formula, we present a new addition formula for the Bessel function, a particular case of which extend the well‐known Graf 's addition formula.