Pfaff-Saalschütz Theorem/Historical Note

From ProofWiki
Jump to navigation Jump to search

Historical Note on Pfaff-Saalschütz Theorem

According to Special Functions by George E. Andrews, Richard Askey and Ranjan Roy, the Pfaff-Saalschütz Theorem was discovered by Johann Friedrich Pfaff in $1797$, and rediscovered by Louis Saalschütz in $1890$.

The proof shown here is a more detailed version of a proof presented in chapter $2$ of Generalized Hypergeometric Series by Wilfrid Norman Bailey, as follows.


In the formula:

\(\text {(1)}: \quad\) \(\ds \paren {1 - z}^{a + b - c} \map F {a, b; c; z}\) \(=\) \(\ds \map F {c - a, c - b; c; z}\)

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

\(\text {(2)}: \quad\) \(\ds \sum_{r \mathop = 0}^n \dfrac {a^{\overline r} b^{\overline r} } {r! c^{\overline r} } \dfrac {\paren {c - a - b}^{\overline {n - r} } } {\paren {n - r}!}\) \(=\) \(\ds \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } {n! c^{\overline n} }\)

Hence:

\(\text {(3)}: \quad\) \(\ds \sum_{r \mathop = 0}^n \dfrac { a^{\overline r} b^{\overline r} \paren {c - a - b}^{\overline n} \paren {-n}^{\overline r} } {r! c^{\overline r} \paren {1 + a + b - c - n}^{\overline r} n!}\) \(=\) \(\ds \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } {n! c^{\overline n} }\)

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


Sources