Show that there exists a real sequence $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ having the following property: for every integer $p \in \mathbb { N } ^ { * }$, for every non-degenerate interval $I$ and for every complex function $f$ of class $C ^ { \infty }$ on $I$, the function $g$ defined on $I$ by $g = a _ { 0 } f + a _ { 1 } f ^ { \prime } + \cdots + a _ { p - 1 } f ^ { ( p - 1 ) }$ satisfies $$g ^ { \prime } + \frac { 1 } { 2 ! } g ^ { \prime \prime } + \frac { 1 } { 3 ! } g ^ { ( 3 ) } + \cdots + \frac { 1 } { p ! } g ^ { ( p ) } = f ^ { \prime } + \sum _ { l = 1 } ^ { p - 1 } b _ { l , p } f ^ { ( p + l ) }$$ where the $b _ { l , p }$ are coefficients independent of $f$ which we do not seek to calculate.
Show that there exists a real sequence $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ having the following property: for every integer $p \in \mathbb { N } ^ { * }$, for every non-degenerate interval $I$ and for every complex function $f$ of class $C ^ { \infty }$ on $I$, the function $g$ defined on $I$ by $g = a _ { 0 } f + a _ { 1 } f ^ { \prime } + \cdots + a _ { p - 1 } f ^ { ( p - 1 ) }$ satisfies
$$g ^ { \prime } + \frac { 1 } { 2 ! } g ^ { \prime \prime } + \frac { 1 } { 3 ! } g ^ { ( 3 ) } + \cdots + \frac { 1 } { p ! } g ^ { ( p ) } = f ^ { \prime } + \sum _ { l = 1 } ^ { p - 1 } b _ { l , p } f ^ { ( p + l ) }$$
where the $b _ { l , p }$ are coefficients independent of $f$ which we do not seek to calculate.