Recurrence relation

In sequence of numbers is constitute to some combination of a previous terms. Often, only previous terms of the sequencein the equation, for a parameter that is independent of ; this number is called the order of the relation. whether the values of the number one numbers in the sequence create been given, the rest of the sequence can be calculated by repeatedly applying the equation.

In linear recurrences, the nth term is equated to a Fibonacci numbers, $${\displaystyle F_{n}=F_{n-1}+F_{n-2}}$$ where the layout is two in addition to the linear function merely adds the two preceding terms. This example is a closed-form expression of . As well, special functions clear a Taylor series whose coefficients satisfy such(a) a recurrence representation see holonomic function.

The concept of a recurrence description can be extended to multidimensional arrays, that is, indexed families that are indexed by tuples of natural numbers.

Solving

Moreover, for the general first-order non-homogeneous linear recurrence relation with variable coefficients:

there is also a nice method to solve it:

Let

Then

If we apply the formula to together with take the limit , we receive the formula for first order linear differential equations with variable coefficients; the written becomes an integral, and the product becomes the exponential function of an integral.

Many homogeneous linear recurrence relations may be solved by means of the generalized hypergeometric series. Special cases of these lead to recurrence relations for the orthogonal polynomials, and numerous special functions. For example, the sum to

is assumption by

the Bessel function, while

is solved by

the rational or hypergeometric solutions.

A first positioning rational difference equation has the form . such(a) an equation can be solved by writing as a nonlinear transformation of another variable which itself evolves linearly. Then specifications methods can be used to solve the linear difference equation in .