Difference and functional equations Definition

Order of a difference equation

Definition

Let $X$ be a set and let $(u_n)_{n \ge 0}$ denote an unknown sequence in $X$ . Consider a difference equation of the form $F(n, u_n, u_{n+1}, \dots, u_{n+k}) = 0 \qquad (n \ge 0),$ where $k \ge 0$ is an integer and $F$ is a given function. The equation has order $k$ if $F$ depends on both its second argument $u_n$ and its last argument $u_{n+k}$ . For a recurrence relation $u_{n+k} = f(n, u_n, \dots, u_{n+k-1})$ in which $f$ depends on $u_n$ , the order is $k$ .

Remarks

Example: $u_{n+3} = u_{n+1} + 2u_n$ has order $3$ . The indices that occur are $n$ , $n+1$ and $n+3$ , and the largest minus the smallest is $3$ . Non-example: $u_{n+3} = 2u_{n+2}$ looks like order $3$ , but $u_n$ does not occur. After the substitution $v_n = u_{n+2}$ it reads $v_{n+1} = 2v_n$ , which has order $1$ . For a linear equation $a_k(n) u_{n+k} + \dots + a_0(n) u_n = b(n)$ the order is $k$ when neither $a_k$ nor $a_0$ is the zero sequence.

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Recurrence relation definition
Order of a difference equation definition

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Initial value problem for a recurrence

Tag	`J00G7MTBFB`
URI	`m2://J00G7MTBFB`
Kind	definition
Writing	W0 stub
MSC	39A05
Defines	Order of a difference equation
Uses	Recurrence relation

Order of a difference equation

Definition

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