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Difference equations, also known as recurrence relations, are equations that define a sequence recursively. Instead of providing an explicit formula for each term, a difference equation describes how a term in a sequence relates to previous terms. This relationship allows for the generation of the entire sequence once a set of initial conditions is provided.

The general form of a difference equation is often expressed as $y_{n+k} = f(y_n, y_{n+1}, ..., y_{n+k-1}, n)$, where $y_n$ represents the $n$-th term of the sequence, $k$ is the order of the difference equation, and $f$ is a function that depends on the previous $k$ terms and possibly the index $n$. The order of the equation is determined by the largest difference in the indices of the terms involved.

Simple examples include arithmetic progressions, where each term is the previous term plus a constant ($y_{n+1} = y_n + d$), and geometric progressions, where each term is the previous term multiplied by a constant ($y_{n+1} = r y_n$). More complex difference equations can model phenomena in various fields such as economics, biology, computer science, and physics. Solving difference equations involves finding an explicit formula for $y_n$ or understanding the behavior of the sequence. Techniques for solving them vary depending on the type and order of the equation, often employing methods similar to those used for differential equations.