TODO

Linear recurrences

Homogeneous

$a_n = 4a_{n−1} − 5a_{n−2} + 2a_{n−3}$
https://math.stackexchange.com/questions/518088/solving-a-recurrence-relation-with-the-characteristic-polynomial/518092#518092
https://www.probabilitycourse.com/chapter14/Chapter_14.pdf
https://artofproblemsolving.com/wiki/index.php/Characteristic_polynomial#Linear_recurrences

When multiplicity of all $\lambda_i$ is one
$a_n = \alpha_1 \lambda_1^n + \ldots \alpha_k \lambda_k^n$

When multiplicity of $\lambda_1$ is two:
$a_n = (\alpha_1 n + \beta_1) \lambda_1^n + \ldots$

Inhomogeneous with constant coefficients

$a_n = 4a_{n−1} − 5a_{n−2} + 2a_{n−3} + 3^n$
$a_n = 4a_{n−1} − 5a_{n−2} + 2a_{n−3} + 6$
https://www.math.kth.se/math/student/courses/5B1203/F/200304/linrek.pdf

Descartes’ rule of signs

https://en.wikipedia.org/wiki/Descartes'_rule_of_signs

Number of positive roots is number of sign differences between consecutive nonzero coefficients, or is less than it by an even number.
Number of negative roots is counted in the same way by looking at $f(-x)$.

Non-real roots: if no root at 0, then minimum number is $n - (p + q)$.

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