Nonhomogeneous recursions and Generalised Continued Fractions

Consider the (n+1)st order nonhomogeneous recursionX_k+n+1=b_kX_k+n+a_k⁽ⁿ⁾X_k+n-1+...+a_k⁽¹⁾X_k+X_k.Leth be a particular solution, andf⁽¹⁾,...,f⁽ⁿ⁾,g independent solutions of the associated homogeneous equation. It is supposed thatg dominatesf⁽¹⁾,...,f⁽ⁿ⁾ andh. If we want to calculate a solutiony which is dominated byg, but dominatesf⁽¹⁾,...,f⁽ⁿ⁾, then forward and backward recursion are numerically unstable. A stable algorithm is derived if we use results constituting a link between Generalised Continued Fractions and Recursion Relations.