I am trying to prove the following relation $$\begin{align*} \frac{2j+1}{z} \int_{0}^{\infty} \frac{dt}{(1+t)^{2j+2}}[\sum_{n=0}^{2j} [\frac{(2j)!}{(2j- n)! n!}]^{\frac{1}{2}}f_n(\frac{t}{z^*})^n]^* \\ = \sum_{n=0}^{2j} [\frac{((2j-n)!n!)}{(2j!)}]^{\frac{1}{2}}f^*_n z^{-n-1} \end{align*}$$ That means I have to prove that
$$\begin{align*} \int_{0}^{\infty} \frac{(t^*)^n}{(1+t)^{2j+2}}dt = \sum_{n=0}^{2j} [\frac{(2j- n)! n!}{(2j)!}] \frac {1}{2j+1} \end{align*}$$ where $j= 1/2,1,3/2, 2,\dots $we can choose j equals any value from these values I go through the proof but I could not find the last relation between the infinite integral and this sum
I will appreciate any help.