We assume that $p$ is an integer greater than or equal to 2, that $\left( a _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( b _ { k } \right) _ { k \in \mathbb { N } }$ are two sequences of real numbers that are $p$-periodic and that $\forall k \in \mathbb { N } , b _ { k } \neq 0$. We denote by Sol(II.2) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ that satisfy the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ Justify that the application $\left. \Psi : \left\lvert \, \begin{array} { l l l } \operatorname { Sol } ( \mathrm { II } .2 ) & \rightarrow & \mathbb { C } ^ { 2 } \\ \left( z _ { k } \right) _ { k \in \mathbb { N } } & \mapsto & \left( z _ { 0 } \right) \\ z _ { 1 } \end{array} \right. \right)$ is an isomorphism of $\mathbb { C }$-vector spaces.
We assume that $p$ is an integer greater than or equal to 2, that $\left( a _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( b _ { k } \right) _ { k \in \mathbb { N } }$ are two sequences of real numbers that are $p$-periodic and that $\forall k \in \mathbb { N } , b _ { k } \neq 0$. We denote by Sol(II.2) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ that satisfy the recurrence relation
$$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$
Justify that the application $\left. \Psi : \left\lvert \, \begin{array} { l l l } \operatorname { Sol } ( \mathrm { II } .2 ) & \rightarrow & \mathbb { C } ^ { 2 } \\ \left( z _ { k } \right) _ { k \in \mathbb { N } } & \mapsto & \left( z _ { 0 } \right) \\ z _ { 1 } \end{array} \right. \right)$ is an isomorphism of $\mathbb { C }$-vector spaces.