Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k }$. We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$ on $\mathbb{R}^{+*}$, satisfying $\psi ( x + 1 ) - \psi ( x ) = \frac{1}{x}$ for all $x > 0$. Show that for every real $x > - 1$ and for every integer $n \geqslant 1$ $$\psi ( 1 + x ) - \psi ( 1 ) = \psi ( n + x + 1 ) - \psi ( n + 1 ) + \sum _ { k = 1 } ^ { n } \left( \frac { 1 } { k } - \frac { 1 } { k + x } \right)$$
Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k }$. We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$ on $\mathbb{R}^{+*}$, satisfying $\psi ( x + 1 ) - \psi ( x ) = \frac{1}{x}$ for all $x > 0$.
Show that for every real $x > - 1$ and for every integer $n \geqslant 1$
$$\psi ( 1 + x ) - \psi ( 1 ) = \psi ( n + x + 1 ) - \psi ( n + 1 ) + \sum _ { k = 1 } ^ { n } \left( \frac { 1 } { k } - \frac { 1 } { k + x } \right)$$