Asymptotic expansions for second-order linear difference equations

Formal series solutions are obtained for the difference equation y(n+2)+a(n)y(n+1)+b(n)y(n) = 0, where a(n) and b(n) have asymptotic expansions of the form a(n)∼∑^∞ _s=0 a_s n^sand b(n)∼∑^∞ _s=0 b_s n^s, for large values of n, and b₀ ≠ 0. These solutions are characterized by the roots of the characteristic equation ρ²+a₀ρ+b₀ = 0. Our discussion is divided into three cases, according to whether the roots are distinct, or equal and do not satisfy the auxiliary equation a₁ρ+b₁ = 0, or equal and do satisfy the auxiliary equation. The last case is further divided into three subcases, according to whether the roots of the indicial equation α(α-1)ρ²+(a₁α+a₂)ρ+b₂ = 0 do not differ by a nonnegative integer, or differ by a positive integer, or are equal. In all cases, the formal series solutions will be shown to be asymptotic. Our approach is based on the method of successive approximations. © 1992.