In this paper we derive formulas for summation of series involving J.~Bourget's generalization of Bessel functions of integer order, as well as the analogous generalizations by H.~M.~Srivastava. These series are expressed in terms of the Riemann $\z$ function and Dirichlet functions $\eta$, $\la$, $\b$, and can be brought into closed form in certain cases, which means that the infinite series are represented by finite sums.

Keywords:

Riemann zeta function, Bessel functions, Bourget functions, Dirichlet functions Riemann zeta function, Bessel functions, Bourget functions, Dirichlet functions

MSC Classifications:

33C10, 11M06, 65B10 show english descriptions Bessel and Airy functions, cylinder functions, ${}_0F_1$
$\zeta (s)$ and $L(s, \chi)$
Summation of series 33C10 - Bessel and Airy functions, cylinder functions, ${}_0F_1$
11M06 - $\zeta (s)$ and $L(s, \chi)$
65B10 - Summation of series

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