Integrability of series with respect to multiplicative systems and generalized derivatives

We give some necessary and sufficient conditions for the convergence of generalized derivatives of sums of series with respect to multiplicative systems and the corresponding Fourier series. These conditions are counterparts of trigonometric results of S. Sheng, W.O. Bray and $\check{C}$.V. Stanojević and extend some results of F. Móricz proved for Walsh–Fourier series.