Generalized series of Bessel functions

Known series of Bessel functions, currently available in handbooks, and many of Neumann type, are generalized to arbitrary order. The underlying result is a Poisson formula due to Titchmarsh. This formula gives rise to a Neumann series involving modified Bessel functions of integral order. The latter is the basis of many of the generalized series that follow. Included are examples of generalized trigonometric identities. The paper concludes by indicating the wide range of results that can be obtained.