Problem 2, Part 2: Linear recurrence sequences with constant coefficients We consider a sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ of complex numbers defined by the data of $u _ { 0 } , \ldots , u _ { d }$ and the linear recurrence relation $$u _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } a _ { i } u _ { n + i } + b ,$$ where the $a _ { i }$ and $b$ are complex numbers. We define $P \in \mathbb { C } [ X ]$ by $P ( X ) = X ^ { d } - \sum _ { i = 0 } ^ { d - 1 } a _ { i } X ^ { i }$ and we assume that all complex roots of $P$ have modulus strictly less than 1. The matrix $A \in \mathrm{M}_d(\mathbb{C})$ is as defined in question 7. Calculate the characteristic polynomial of the matrix $A$ (one may reason by induction on $d$).
\textbf{Problem 2, Part 2: Linear recurrence sequences with constant coefficients}
We consider a sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ of complex numbers defined by the data of $u _ { 0 } , \ldots , u _ { d }$ and the linear recurrence relation
$$u _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } a _ { i } u _ { n + i } + b ,$$
where the $a _ { i }$ and $b$ are complex numbers. We define $P \in \mathbb { C } [ X ]$ by $P ( X ) = X ^ { d } - \sum _ { i = 0 } ^ { d - 1 } a _ { i } X ^ { i }$ and we assume that all complex roots of $P$ have modulus strictly less than 1. The matrix $A \in \mathrm{M}_d(\mathbb{C})$ is as defined in question 7.
Calculate the characteristic polynomial of the matrix $A$ (one may reason by induction on $d$).