Watson's Hypergeometric Theorem

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

Then:
 * $\ds \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {\dfrac 1 2 \paren {a + b + 1}, 2 c } } \, \middle \vert \, 1} = \dfrac {\map \Gamma {\dfrac 1 2} \map \Gamma {\dfrac 1 2 + c} \map \Gamma {\dfrac 1 2 \paren {1 + a + b} } \map \Gamma {\dfrac 1 2 \paren {1 - a - b} + c } } {\map \Gamma {\dfrac 1 2 \paren {1 + a} } \map \Gamma {\dfrac 1 2 \paren {1 + b} } \map \Gamma {\dfrac 1 2 \paren {1 - a} + c } \map \Gamma {\dfrac 1 2 \paren {1 - b} + c } }$

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

Proof
From Thomae's Transformation, we have:

The generalized hypergeometric function on the can be summed using Dixon's Hypergeometric Theorem.

From Dixon's Hypergeometric Theorem, we have:
 * $\ds \map { {}_3 \operatorname F_2} { { {n, -x, -y} \atop {x + n + 1, y + n + 1} } \, \middle \vert \, 1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {\dfrac n 2 + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {y + \dfrac n 2 + 1} } $

Let:

Then:

Making these substitutions to the generalized hypergeometric function on the, we have:

We now have:

From Legendre's Duplication Formula, we have:

Therefore:

Also known as
Some sources refer to Watson's Hypergeometric Theorem as Watson's Theorem.

Also see

 * Dixon's Hypergeometric Theorem
 * Gauss's Hypergeometric Theorem
 * Kummer's Hypergeometric Theorem
 * Properties of Generalized Hypergeometric Function