Pfaff-Saalschütz Theorem/Historical Note

Historical Note on Saalschütz's Hypergeometric Theorem
Some sources refer to this theorem as the Pfaff-Saalschütz Theorem after mathematicians Johann Friedrich Pfaff and Louis Saalschütz.

According to Special Functions by, and , Pfaff discovered the theorem in 1797 and Saalschütz rediscovered the theorem in 1890.

The proof shown above is a more detailed version of a proof shown on page $9$ of Generalized Hypergeometric Series by Wilfrid Norman Bailey.

On page $9$, the proof goes as follows:

In the formula

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

Hence:

As can be seen, Bailey vaulted over several intermediate steps between steps $\paren {2}$ and $\paren {3}$, but his assertions were correct.