Recurrent relations for the solutions of an infinite system of linear algebraic equations

We obtain recurrent relations for bounded solutions of the system of equations $$X_k - sumlimits_{n = 0}^infty {frac{{(k + n)!}}{{k!n!}}} alpha ^{k + n + 1} x_n = f_{k,} k = 0,1,..., alpha in (0,1/2),$$ with right-hand sides {f _k } _k=0 ^∞ ={δ _kj } _k=0 ^∞ ,j=0,1,..., where δ _kj is the Kronecker symbol.