Integral representations and asymptotic expansions for the second type Neumann series of Bessel functions of the first kind

In this paper we study the following Bessel series ${\sum }_{l=1}^{\mathrm{\infty }}{J}_{l+m^{\mathrm{\prime }}}(r)J_{l+m}(r)(l+\beta {)}^{\alpha }$ for any $m,m^{\mathrm{\prime }}\in \mathbb{Z}$, $\alpha \in \mathbb{R}$ and $\beta >-1$. They are a particular case of the second type Neumann series of Bessel functions of the first kind. More specifically, we derive fully explicit integral representations and study the asymptotic behavior with explicit terms. As a corollary, the asymptotic behavior of series of the derivatives of Bessel functions can be understood.