---
tag: 9X6KG770GV
kind: statement
role: theorem
name: Pfaff–Saalschütz theorem
msc: [33C20, 33C05]
uses: [VDBAW9PN88, YZHM3HSM0F]
about: [VDBAW9PN88]
sources:
  - url: https://en.wikipedia.org/wiki/Generalized_hypergeometric_function
---

## Statement

Let $n$ be a nonnegative integer and let $a, b, c$ be complex numbers.
Assume that neither $c$ nor $1 + a + b - c - n$ lies in $\set{0, -1, \dots, 1 - n}$.
Write $(x)_k = x (x + 1) \cdots (x + k - 1)$ for the [[Pochhammer symbol|YZHM3HSM0F]].
Then the terminating [[generalized hypergeometric series|VDBAW9PN88]] ${}_3F_2$ at $1$ is
$${}_3F_2\left( \begin{matrix} -n,\ a,\ b \\ c,\ 1 + a + b - c - n \end{matrix} ; 1 \right) = \frac{(c - a)_n\, (c - b)_n}{(c)_n\, (c - a - b)_n}.$$

## Remarks

The series is *balanced* (Saalschützian): the sum of the lower parameters exceeds the sum of the upper parameters by $1$.
A standard proof starts from Euler's transformation $(1 - x)^{a + b - c}\, {}_2F_1(a, b; c; x) = {}_2F_1(c - a, c - b; c; x)$.
Comparing the coefficients of $x^n$ on both sides gives the identity after a rearrangement of Pochhammer symbols.
For $n = 1$ the left side is $1 - \frac{ab}{c(a + b - c)}$.
This equals $\frac{c(a + b - c) - ab}{c(a + b - c)} = \frac{(c - a)(c - b)}{c(c - a - b)}$, which is the right side.
The hypothesis on $c$ and $1 + a + b - c - n$ keeps the denominators of the terms of the series nonzero; it also gives $(c - a - b)_n \ne 0$, since $(c - a - b)_n = (-1)^n (1 + a + b - c - n)_n$.

## Explanations

### L1 human

A hypergeometric series with a negative integer $-n$ on top stops after $n + 1$ terms, so it is a finite sum.
In general such a sum at $1$ has no simple closed form.
The theorem says that when the parameters are balanced, the sum collapses to a single product of rising factorials.

### L2 human

The theorem is the ${}_3F_2$ analogue of the Chu–Vandermonde identity for terminating ${}_2F_1$ sums at $1$.
It is one of the basic summation formulas used to evaluate sums of products of binomial coefficients.
It has a $q$-analogue in the theory of basic hypergeometric series.