Some novel infinite series of spherical Bessel functions

By a double application of the translational additional theorem for spherical wave functions, whereby one shifts an $n$ th order axisymmetric wave function from some origin to another and then in turn back to the first, one obtains a mathematical identity in the form of the $n$th order spherical wave function equated to an infinite series containing every order spherical wave function. The coefficients of the terms in this infinite series are themselves infinite series of spherical Bessel functions of arbitrary argument. These latter series must either sum to zero or unity to satisfy the mathematical identity. Following this reasoning, a collection of infinite series involving spherical Bessel functions has been generated. Some of the low mode order results are presented.