In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$. Show that there exists a sequence $\left( P _ { k } \right) _ { k \in \mathbf { N } }$ of polynomials with coefficients in $\mathbf { C }$, independent of $p$, such that $$\forall \theta \in \mathbf { R } , \forall k \in \mathbf { N } , \Phi _ { X } ^ { ( k ) } ( \theta ) = p i ^ { k } e ^ { i \theta } \frac { P _ { k } \left( q e ^ { i \theta } \right) } { \left( 1 - q e ^ { i \theta } \right) ^ { k + 1 } } \quad \text { and } \quad P _ { k } ( 0 ) = 1$$
In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Show that there exists a sequence $\left( P _ { k } \right) _ { k \in \mathbf { N } }$ of polynomials with coefficients in $\mathbf { C }$, independent of $p$, such that
$$\forall \theta \in \mathbf { R } , \forall k \in \mathbf { N } , \Phi _ { X } ^ { ( k ) } ( \theta ) = p i ^ { k } e ^ { i \theta } \frac { P _ { k } \left( q e ^ { i \theta } \right) } { \left( 1 - q e ^ { i \theta } \right) ^ { k + 1 } } \quad \text { and } \quad P _ { k } ( 0 ) = 1$$