Generalization of Schlefli formulas of Bessel functions and some of its applications

The work is dedicated to the series of Neumann. This is one of the four varieties of Bessel function series and has the form [1, p. 574] Σn cnJn + q (z), (−1 ≤ q <1) (1) where Jn + q (z) is the Bessel function of the first kind. A new basic formula is derived - an integral representation of the product of a Bessel function by its index in degree. With the help of the new formula are produced below: 1. Integral representations of the Neumann series. 2. Summation of the Neumann series.