Difference and functional equations Definition

Linear recurrence with constant coefficients

Definition

Let $R$ be a commutative ring and let $d$ be a natural number. A linear recurrence of order $d$ with constant coefficients over $R$ is given by coefficients $c_0, c_1, \dots, c_{d-1} \in R$ . It is the recurrence relation $u_{n+d} = c_0 u_n + c_1 u_{n+1} + \dots + c_{d-1} u_{n+d-1} \qquad (n \ge 0)$ for sequences $(u_n)_{n \ge 0}$ in $R$ .

Remarks

This form is homogeneous. A non-homogeneous recurrence adds a term $g(n)$ that does not depend on $u$ . The Fibonacci sequence satisfies the recurrence of order $2$ with $c_0 = c_1 = 1$ . The sequences that satisfy it are its solutions. Its characteristic polynomial is $X^d - c_{d-1} X^{d-1} - \dots - c_1 X - c_0$ .

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Recurrence relation definition
Linear recurrence with constant coefficients definition

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Tag	`081THG13JZ`
URI	`m2://081THG13JZ`
Kind	definition
Writing	W1 stated
MSC	39A06
Defines	Linear recurrence with constant coefficients
Uses	Recurrence relation

Linear recurrence with constant coefficients

Definition

Remarks

Reading path

Where this leads

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