recurrencevaries

Recurrencevaries is a term used to describe a class of recurrence relations in which the coefficients or the forcing term depend on the index, rather than remaining fixed. In other words, the rule that generates each term from previous terms changes with n, producing non-stationary sequences. Formally, a recurrencevaries relation defines a sequence {x_n} by x_n = sum_{k=1}^m c_k(n) x_{n-k} + f(n) for n ≥ n0, where c_k(n) are coefficient functions of n and f(n) is a prescribed input.

A simple example is x_0 = x0, and x_n = (1 + 1/n) x_{n-1} for n ≥ 1. More generally, one may have a first-order equation x_n = a_n x_{n-1} + b_n, with a_n and b_n varying with n. Another common form is a linear recurrence with two or more lag terms, x_n = a_n x_{n-1} + b_n x_{n-2} + …, where each a_n, b_n, etc., depends on n.

Properties and analysis of recurrencevaries differ from constant-coefficient recurrences. The lack of a fixed characteristic equation means closed-form solutions are rare, and stability or long-term behavior depends on the behavior of the coefficient functions and the input term. Solutions are often studied via numerical methods, state-space representations, or asymptotic techniques, such as analyzing products of variable matrices.

Applications arise in areas with changing dynamics, including population models with time-varying rates, economic or financial models, adaptive signal processing, and numerical schemes for differential equations with variable step sizes. See also recurrence relations, difference equations, and linear time-varying systems.