stylistic:style_transfer, linguistic:paraphrasing
Definition

A recurrence relation is an equation that expresses each element of a sequence as a function of the preceding ones. More precisely, in the case where only the immediately preceding element is involved, a recurrence relation has the form

where

is a function, where X is a set to which the elements of a sequence must belong. For any u 0 ∈ X {\displaystyle u_{0}\in X}, this defines a unique sequence with u 0 {\displaystyle u_{0}} as its first element, called the initial value.

It is easy to modify the definition for getting sequences starting from the term of index 1 or higher.

This defines recurrence relation of first order. A recurrence relation of order k has the form

where φ: N × X k → X {\displaystyle \varphi :\mathbb {N} \times X^{k}\to X} is a function that involves k consecutive elements of the sequence.In this case, k initial values are needed for defining a sequence.