We define $\varphi$ the function defined on $] - 1 , + \infty [$ by $\varphi ( x ) = ( \psi ( 1 + x ) - \psi ( 1 ) ) ^ { 2 } + \left( \psi ^ { \prime } ( 1 ) - \psi ^ { \prime } ( 1 + x ) \right)$.
Show that $\varphi$ is $\mathcal { C } ^ { \infty }$ on its domain of definition and give for every natural integer $n \geqslant 2$ the value of $\varphi ^ { ( n ) } ( 0 )$ as a function of the successive derivatives of $\psi$ at the point 1.