Limit of ratio of consecutive terms for general order-k linear homogeneous recurrences with constant coefficients

For complex linear homogeneous recursive sequences with constant coefficients we find a necessary and sufficient condition for the existence of the limit of the ratio of consecutive terms. The result can be applied even if the characteristic polynomial has not necessarily roots with modulus pairwise distinct, as in the celebrated Poincaré’s theorem. In case of existence, we characterize the limit as a particular root of the characteristic polynomial, which depends on the initial conditions and that is not necessarily the unique root with maximum modulus and multiplicity. The result extends to a quite general context the way used to find the Golden mean as limit of ratio of consecutive terms of the classical Fibonacci sequence.

Research highlights► We prove a result true for all linear homogeneous recurrences with constant coefficients. ► As a corollary of our results we immediately get the celebrated Poincare’ theorem. ► The limit of the ratio of adjacent terms is characterized as the unique leading root of the characteristic polynomial. ► The Golden Ratio, Kepler limit of the classical Fibonacci sequence, is the unique leading root. ► The Kepler limit may differ from the unique root of maximum modulus and multiplicity.