Pfaff-Saalschütz Theorem/Historical Note
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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
- 1999: George E. Andrews, Richard Askey and Ranjan Roy: Special Functions: Chapter $\text {2}$. The Hypergeometric Functions
- 1935: W.N. Bailey: Generalized Hypergeometric Series Chapter $\text {2}$. The hypergeometric series