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 $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ To any complex sequence $\left( z _ { k } \right) _ { k \in \mathbb { N } }$, we associate the sequence $\left( Z _ { k } \right) _ { k \in \mathbb { N } }$ of elements of $\mathbb { C } ^ { 2 }$ defined by $$\forall k \in \mathbb { N } , \quad Z _ { k } = \binom { z _ { k } } { z _ { k + 1 } }$$ Prove that the sequence $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ is a solution of (II.2) if and only if the sequence $\left( Z _ { k } \right) _ { k \in \mathbb { N } }$ is a solution of a system (II.3) of the form $$\forall k \in \mathbb { N } , \quad Z _ { k + 1 } = A _ { k } Z _ { k }$$ Specify the matrix $A _ { k } \in \mathcal { M } _ { 2 } ( \mathbb { C } )$.
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
$$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$
To any complex sequence $\left( z _ { k } \right) _ { k \in \mathbb { N } }$, we associate the sequence $\left( Z _ { k } \right) _ { k \in \mathbb { N } }$ of elements of $\mathbb { C } ^ { 2 }$ defined by
$$\forall k \in \mathbb { N } , \quad Z _ { k } = \binom { z _ { k } } { z _ { k + 1 } }$$
Prove that the sequence $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ is a solution of (II.2) if and only if the sequence $\left( Z _ { k } \right) _ { k \in \mathbb { N } }$ is a solution of a system (II.3) of the form
$$\forall k \in \mathbb { N } , \quad Z _ { k + 1 } = A _ { k } Z _ { k }$$
Specify the matrix $A _ { k } \in \mathcal { M } _ { 2 } ( \mathbb { C } )$.