Pfaff-Saalschütz Theorem

Theorem
Let $a, b, c \in \C$.

Let $n \in Z_{\ge 0}$.

Then:
 * $\ds \map { {}_3 \operatorname F_2} { { {a, b, -n} \atop {c, 1 + a + b - c - n} } \, \middle \vert \, 1} = \dfrac {\paren {c - a}^{\overline n} \paren {c - b}^{\overline n} } { c^{\overline n} \paren {c - a - b}^{\overline n} }$

where:
 * ${}_3 \operatorname F_2$ is the generalized hypergeometric function
 * $x^{\overline k}$ denotes the $k$th rising factorial power of $x$
 * $\Gamma$ denotes the Gamma function.

Proof
By definition of the generalized hypergeometric function:
 * $\ds \map { {}_3 \operatorname F_2} { { {a, b, -n} \atop {c, 1 + a + b - c - n} } \, \middle \vert \, 1} = \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} \paren {-n}^{\overline k} } { c^{\overline k} \paren {1 + a + b - c - n}^{\overline k} } \dfrac {1^k} {k!}$

From Euler's Transformation, we have:

Next, we observe that:

Equating coefficients of $x^n$ in $(1)$ above, we obtain:

Also known as
Some sources refer to the Pfaff-Saalschütz Theorem as Saalschütz's Hypergeometric Theorem.

Also see

 * Dixon's Hypergeometric Theorem
 * Dougall's Hypergeometric Theorem
 * Gauss's Hypergeometric Theorem
 * Kummer's Hypergeometric Theorem