Novel solutions of RG equations for alpha(s) and beta(alpha) in the large N_c limit

General solution of RG equations in the framework of background perturbation theory is written down in the large N_c limit. A simplified (model) approximation to the general solution is suggested which allows to write beta(alpha) and alpha(beta) to any loop order. The resulting alpha_B(Q^2) coincides asymptotically at large $\mathrm{\mid }{Q}^2\mathrm{\mid }$ with standard (free) alpha_s, saturates at small Q^2 and has poles at time-like Q^2 in agreement with analytic properties of physical amplitudes in the large N_c limit.