An equation that defines each term of a sequence or discrete array in terms of preceding term(s) and possibly the index, together with initial condition(s) that uniquely determine the sequence.

Principle

Specify a base case and a deterministic rule linking later terms to earlier ones so the entire sequence is generated stepwise; linear recurrences admit algebraic solution methods while nonlinear ones may require other techniques.

Demonstration

The Fibonacci recurrence F_n = F_{n-1} + F_{n-2} with base values F_0 = 0, F_1 = 1 generates the Fibonacci sequence; its linearity enables closed form via characteristic polynomial (Binet's formula).

Misapplication

Using a recurrence without providing sufficient initial conditions or applying solution techniques for linear recurrences to nonlinear recurrences, yielding ambiguous or incorrect sequences.

Consequence

Recurrences provide constructive definitions amenable to iterative computation, complexity analysis, and asymptotic estimation; linear recurrences map to polynomial characteristic equations and generating functions.

Reversal

An explicit closed‑form formula for the nth term that does not require previous terms for evaluation and provides direct computation and analytic asymptotics.

Boundary

Pertains to discrete indexed collections (sequences, arrays) and excludes continuous differential equations, though analogies with difference equations exist; requires well‑posed initial/boundary data for uniqueness.

Semantic Tension

Tension between 'difference equation' used broadly (including relations involving shifts and inhomogeneities) and 'recurrence relation' often reserved for sequences with a fixed forward generation rule.

Synthesis

A recurrence relation is a rule plus base data that recursively generates a discrete sequence by expressing each term in terms of earlier terms, enabling stepwise computation and, for linear cases, algebraic solution and asymptotic analysis.