We report on some generalizations of the square root functions. We correspond the classical results for recurrence of vector chain with results obtained from the ergodic theory, to prove the main results of this paper. We will prove that the series $S(\frac x4;y)$ converges in the region \[ 0<xeq4,\quad0<yeq\bigg(\frac{2-qrt x}{qrt x}\bigg)^2. \] Also we prove that the following holds \[ t^2+(t^2-2t^3+t^4)+(t^2-4t^3+7t^4-6t^5+2t^6)+\dots=t,\quad0<t<1. \]