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

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