Special functions Hypergeometric functions Theorem

Pfaff–Saalschütz theorem

Builds on Generalized hypergeometric function Pochhammer symbol

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. Then the terminating generalized hypergeometric series ${}_3F_2$ 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 $(c - a - b)_n = (-1)^n (1 + a + b - c - n)_n$ .

Going deeper

For a student

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.

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Generalized hypergeometric function concept
Pochhammer symbol concept
Pfaff–Saalschütz theorem statement

Tag	`9X6KG770GV`
URI	`m2://9X6KG770GV`
Kind	statement, theorem
Writing	W0 stub
MSC	33C20, 33C05
Status	This copy of the site has no build data. Run `m2 status 9X6KG770GV`.
About	Generalized hypergeometric function
Uses	Generalized hypergeometric function, Pochhammer symbol