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 $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ We fix $\left( y _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( z _ { k } \right) _ { k \in \mathbb { N } }$, two solution sequences of (II.2), and set for all $k \in \mathbb { N } , W _ { k } = b _ { k } \left( y _ { k } z _ { k + 1 } - z _ { k } y _ { k + 1 } \right)$. Show that the two sequences $\left( y _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ form a basis of $\operatorname { Sol } ($ II.2 $)$ if and only if $W _ { 0 } \neq 0$.
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
$$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$
We fix $\left( y _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( z _ { k } \right) _ { k \in \mathbb { N } }$, two solution sequences of (II.2), and set for all $k \in \mathbb { N } , W _ { k } = b _ { k } \left( y _ { k } z _ { k + 1 } - z _ { k } y _ { k + 1 } \right)$. Show that the two sequences $\left( y _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ form a basis of $\operatorname { Sol } ($ II.2 $)$ if and only if $W _ { 0 } \neq 0$.