3rd Order Recurrence Relation

**jsbach** · April 26th 2011, 12:43 AM

Hi,

I have a 3rd Order Homogenous Recurrence Relation, with a characteristic polynomial of

x^3 - r1x^2 - r2x - r3. This polynomial has three roots, \lambda1, \lambda2, \lambda3, which are the same.

I know for a 2nd Order Recurrence Relation with equal roots, the general solution of the relation is C1\lambda^n + C2n\lambda^n, for some constants C1, C2, but what is the general solution for a 3rd Order with equal roots?

Thankyou.

**emakarov** · April 26th 2011, 03:03 AM

See Linear homogeneous recurrence relations with constant coefficients in Wikipedia and the main theorem on the same page.

**Soroban** · April 26th 2011, 03:44 AM

Hello, jsbach!

I have a 3rd Order Homogenous Recurrence Relation,
. . with a characteristic polynomial of: . $x^3$ - $r_1$ $x^2$ - $r_2$ $x$ - $r_3$

This polynomial has three roots: $\lambda_1$ , $\lambda_2$ , $\lambda_3$ , which are equal.

I know for a 2nd Order Recurrence Relation with equal roots, the general
solution is: . $C_1$ $\lambda^n$ $+$ $C_2$ $n$ $\lambda^n$ , for some constants $C_1$ , $C_2$ .

But what is the general solution for a 3rd Order with equal roots?

It is: . $C_1$ $\lambda^n$ + $C_2$ $n$ $\lambda^n$ $+$ $C_3$ $n^2$ $\lambda^n$

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