Special functions Hypergeometric functions Theorem

Whipple's transformation

Builds on Generalized hypergeometric function Pochhammer symbol

Statement

Let $n$ be a nonnegative integer and let $a, b, c, d, e$ be complex numbers. Assume that none of $\frac{a}{2}, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a + n$ and $d + e - a - n$ 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 satisfy ${}_7F_6\left( \begin{matrix} a,\ 1 + \frac{a}{2},\ b,\ c,\ d,\ e,\ -n \\ \frac{a}{2},\ 1 + a - b,\ 1 + a - c,\ 1 + a - d,\ 1 + a - e,\ 1 + a + n \end{matrix} ; 1 \right)$ $= \frac{(1 + a)_n\, (1 + a - d - e)_n}{(1 + a - d)_n\, (1 + a - e)_n}\; {}_4F_3\left( \begin{matrix} -n,\ d,\ e,\ 1 + a - b - c \\ 1 + a - b,\ 1 + a - c,\ d + e - a - n \end{matrix} ; 1 \right).$

Remarks

The left side is a very well-poised ${}_7F_6$ : each upper parameter plus the matching lower parameter gives $1 + a$ . The ${}_4F_3$ on the right is balanced: its lower parameters sum to $1$ more than its upper parameters. So the transformation trades well-poised structure for balanced structure, and the number of free parameters stays the same. A standard proof uses the Pfaff–Saalschütz summation together with an exchange of the order of summation. For $n = 0$ both sides equal $1$ . If moreover $1 + 2a = b + c + d + e - n$ , then $d + e - a - n = 1 + a - b - c$ , the ${}_4F_3$ reduces to a balanced ${}_3F_2$ , and the Pfaff–Saalschütz theorem sums it; this proves Dougall's formula.

Going deeper

For a student

The theorem connects the two main families of hypergeometric series at $1$ : the well-poised ones and the balanced ones. Combined with the Pfaff–Saalschütz theorem it gives Dougall's ${}_7F_6$ summation. Its $q$ -analogue is Watson's transformation between a very well-poised ${}_8\phi_7$ and a balanced ${}_4\phi_3$ . The name Whipple is attached to several other transformations of ${}_3F_2$ and ${}_7F_6$ series as well.

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Generalized hypergeometric function concept
Pochhammer symbol concept
Whipple's transformation statement

Tag	`QNGGCTXK9S`
URI	`m2://QNGGCTXK9S`
Kind	statement, theorem
Writing	W0 stub
MSC	33C20
Status	This copy of the site has no build data. Run `m2 status QNGGCTXK9S`.
About	Generalized hypergeometric function
Uses	Generalized hypergeometric function, Pochhammer symbol