Difference and functional equations

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• ConceptRecurrence relation

  A recurrence relation defines each term of a sequence from earlier terms. Recurrences are a main tool to describe counting sequences, and solving one means finding a closed formula for its terms. The node recurrence relation gives the precise definition.

• ConceptOrder of a difference equation

  The order of a difference equation is the gap between the largest and the smallest index of the unknown sequence that occur in it. When the equation can be solved for its highest term, the order is the number of initial values needed to fix a solution. It plays the role that the order of a differential equation plays for derivatives.

• ConceptInitial value problem for a recurrence

  An initial value problem for a recurrence asks for a sequence that satisfies a given recurrence and has prescribed first terms. For a recurrence of order $k$ in explicit form, prescribing $k$ initial values gives exactly one solution. It is the discrete counterpart of the initial value problem for ordinary differential equations.

• ConceptFinite difference

  A finite difference of a function is the change of its value between two points at a fixed distance, such as $f(x + h) - f(x)$. Divided by the step $h$, it becomes a difference quotient, which approximates the derivative when $h$ is small. Finite differences are the basic operation of difference equations, of interpolation and of numerical methods for differential equations.

• ConceptForward difference operator

  The forward difference operator $\Delta_h$ sends a function $f$ to the function $x \mapsto f(x + h) - f(x)$. It is the discrete analogue of differentiation, and difference equations are often written with it. It equals $E_h - I$, where $E_h$ is the shift operator and $I$ is the identity.

• ConceptBackward difference operator

  The backward difference operator $\nabla_h$ sends a function $f$ to the function $x \mapsto f(x) - f(x - h)$. It looks one step back instead of one step ahead. It is natural in time-stepping and interpolation schemes that use only values already computed.