Pfaff-Saalschütz Theorem/Historical Note

Historical Note on Pfaff-Saalschütz Theorem
According to by,  and , the Pfaff-Saalschütz Theorem was discovered by  in $1797$, and rediscovered by  in $1890$.

The proof shown here is a more detailed version of a proof presented in chapter $2$ of by, as follows.

In the formula:

obtained in $\S 1.2$, equate the coefficients of $z^n$ to obtain:

Hence:

As can be seen, vaulted over several intermediate steps between steps $(2)$ and $(3)$, but his assertions were correct.