Special functions Hypergeometric functions Theorem

Dougall's formula

Builds on Generalized hypergeometric function Pochhammer symbol

Statement

Let $m$ be a nonnegative integer and let $a, b, c, d, e$ be complex numbers with $1 + 2a = b + c + d + e - m.$ Assume that none of the lower parameters $\frac{a}{2}, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a + m$ lies in $\set{0, -1, \dots, 1 - m}$ , and that $(1 + a - b - c - d)_m \ne 0$ . Write $(x)_k = x (x + 1) \cdots (x + k - 1)$ for the Pochhammer symbol. Then the terminating generalized hypergeometric series ${}_7F_6$ at $1$ is ${}_7F_6\left( \begin{matrix} a,\ 1 + \frac{a}{2},\ b,\ c,\ d,\ e,\ -m \\ \frac{a}{2},\ 1 + a - b,\ 1 + a - c,\ 1 + a - d,\ 1 + a - e,\ 1 + a + m \end{matrix} ; 1 \right) = \frac{(1 + a)_m\, (1 + a - b - c)_m\, (1 + a - b - d)_m\, (1 + a - c - d)_m}{(1 + a - b)_m\, (1 + a - c)_m\, (1 + a - d)_m\, (1 + a - b - c - d)_m}.$

Remarks

The series is very well-poised: each upper parameter plus the matching lower parameter gives $1 + a$ , and the pair $1 + \frac{a}{2}, \frac{a}{2}$ is present. The condition $1 + 2a = b + c + d + e - m$ says that the series is $2$ -balanced: the lower parameters sum to $2$ more than the upper ones. One proof applies Whipple's transformation, which turns the ${}_7F_6$ into a multiple of ${}_4F_3(-m, d, e, 1 + a - b - c; 1 + a - b, 1 + a - c, d + e - a - m; 1)$ . Under the condition $1 + 2a = b + c + d + e - m$ one has $d + e - a - m = 1 + a - b - c$ , so this ${}_4F_3$ reduces to a balanced ${}_3F_2$ . The Pfaff–Saalschütz theorem then sums it, and the product on the right comes out. For $m = 0$ both sides equal $1$ . Letting $m \to \infty$ term by term gives the summation of the well-poised ${}_5F_4$ with upper parameters $a, 1 + \frac{a}{2}, b, c, d$ , which in turn contains part 3 of Dixon's identity.

Going deeper

For a student

Dougall's formula sits near the top of the classical summation formulas for hypergeometric series at $1$ . Specialising parameters gives many other summations, which is why it is a hub in tables of identities. Its $q$ -analogue is Jackson's summation for a very well-poised ${}_8\phi_7$ .

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Generalized hypergeometric function concept
Pochhammer symbol concept
Dougall's formula statement

Tag	`CSCWM5ZKV4`
URI	`m2://CSCWM5ZKV4`
Kind	statement, theorem
Writing	W0 stub
MSC	33C20
Status	This copy of the site has no build data. Run `m2 status CSCWM5ZKV4`.
About	Generalized hypergeometric function
Uses	Generalized hypergeometric function, Pochhammer symbol