User:Jmokland/Poincaré-Perron theorem
The theorem concerns homogeneous linear recurrence relations with variable coefficients.
Statement of the Poincaré-Perron theorem[edit]
If the coefficients Failed to parse (syntax error): {\displaystyle α_{i,n}, i = 1,...,k} of a linear homogeneous difference equation Failed to parse (syntax error): {\displaystyle 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 Failed to parse (syntax error): {\displaystyle \lim_{n→∞} α_{i,n} = α_i, i = 1, ..., k} and if the roots Failed to parse (syntax error): {\displaystyle λ_1, ..., λ_k} of the characteristic equation Failed to parse (syntax error): {\displaystyle 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 Failed to parse (syntax error): {\displaystyle \lim_{n→∞} \frac{u(n+1)}{u(n)}} for n → ∞ equals one of the roots of the characteristic equation. (ii) if additionally Failed to parse (syntax error): {\displaystyle α_{k,n}\neq 0} for all n then for every Failed to parse (syntax error): {\displaystyle λ_i} there exists a solution u with Failed to parse (syntax error): {\displaystyle \lim_{n→∞} \frac{u(n+1)}{u(n)} = λ_i} .
References[edit]
Original papers[edit]
- Perron, Oskar (1921), "Über Summengleichungen und Poincarésche Differenzengleichungen", Mathematische Annalen, 84: 1–15, doi:10.1007/BF01458689, S2CID 120429963
- Poincaré, Henri (1885), "Sur les Equations Lineaires aux Differentielles Ordinaires et aux Differences Finies.", Amer. J. Math., 7 (3): 203–258, doi:10.2307/2369270, JSTOR 2369270
Further reading[edit]
- Borcea, Julius; Friedland, Schmuel; Shapiro, Boris (2011), "Parametric Poincaré-Perron theorem with applications", Journal d'Analyse Mathématique, 113 (1): 197–225, doi:10.1007/s11854-011-0004-0, S2CID 3298201
- Saber Elaydi, "An Introduction to Difference Equations."