On one-sided series of Bessel functions of the first kind: $\sum_{m=0}^{\infty} (\pm 1)^m J_{Nm+p}(x)$

We present generic expressions for one-sided series of Bessel functions of the first kind: $\sum_{m=0}^{\infty} J_{Nm+p}(x)$ and $\sum_{m=0}^{\infty} (-1)^m J_{Nm+p}(x)$. The result is expressed in terms of incomplete Lipschitz-Hankel integrals of Bessel functions of the first kind, which can be rep