User:Jmokland/Poincaré-Perron theorem

The theorem concerns homogeneous linear recurrence relations with variable coefficients.

Statement of the Poincaré-Perron theorem
If the coefficients $$α_{i,n}, i = 1,...,k$$ of a linear homogeneous difference equation $$u_{n+k} + α_{1,n}u_{n+k−1} + α_{2,n}u_{n+k−2} + ... + α_{k,n}u_n = 0$$ have limits $$\lim_{n→∞} α_{i,n} = α_i, i = 1, ..., k$$ and if the roots $$λ_1, ..., λ_k$$ of the characteristic equation $$t^k + α_1t^{k−1} + ... + α_k = 0$$ have distinct absolute values then (i) for any solution u either u(n) = 0 for all sufficiently large n or $$\lim_{n→∞} \frac{u(n+1)}{u(n)}$$ for n → ∞ equals one of the roots of the characteristic equation. (ii) if additionally $$α_{k,n}\neq 0$$ for all n then for every $$λ_i$$ there exists a solution u with $$\lim_{n→∞} \frac{u(n+1)}{u(n)} = λ_i$$.

Original papers