Let $a _ { 0 }$ and $a _ { 1 }$ be complex numbers and define $a _ { n } = 2 a _ { n - 1 } + a _ { n - 2 }$ for $n \geq 2$. (A) Show that there are polynomials $p ( z ) , q ( z ) \in \mathbb { C } [ z ]$ such that $q ( 0 ) \neq 0$ and $\sum _ { n \geq 0 } a _ { n } z ^ { n }$ is the Taylor series expansion (around 0) of $\frac { p ( z ) } { q ( z ) }$. (B) Let $a _ { 0 } = 1$ and $a _ { 1 } = 2$. Show that there exist complex numbers $\beta _ { 1 } , \beta _ { 2 } , \gamma _ { 1 } , \gamma _ { 2 }$ such that $$a _ { n } = \beta _ { 1 } \gamma _ { 1 } ^ { n + 1 } + \beta _ { 2 } \gamma _ { 2 } ^ { n + 1 }$$ for all $n$.
Let $a _ { 0 }$ and $a _ { 1 }$ be complex numbers and define $a _ { n } = 2 a _ { n - 1 } + a _ { n - 2 }$ for $n \geq 2$.\\
(A) Show that there are polynomials $p ( z ) , q ( z ) \in \mathbb { C } [ z ]$ such that $q ( 0 ) \neq 0$ and $\sum _ { n \geq 0 } a _ { n } z ^ { n }$ is the Taylor series expansion (around 0) of $\frac { p ( z ) } { q ( z ) }$.\\
(B) Let $a _ { 0 } = 1$ and $a _ { 1 } = 2$. Show that there exist complex numbers $\beta _ { 1 } , \beta _ { 2 } , \gamma _ { 1 } , \gamma _ { 2 }$ such that
$$a _ { n } = \beta _ { 1 } \gamma _ { 1 } ^ { n + 1 } + \beta _ { 2 } \gamma _ { 2 } ^ { n + 1 }$$
for all $n$.