Dixon's Hypergeometric Theorem/Proof 1

Theorem

$\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} } $

Proof

From Dougall's Hypergeometric Theorem, we have:

$\ds \map { {}_5 \operatorname F_4} { { {\dfrac n 2 + 1, n, -x, -y, -z} \atop {\dfrac n 2, x + n + 1, y + n + 1, z + n + 1} } \, \middle \vert \, 1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {z + n + 1} \map \Gamma {x + y + z + n + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {y + z + n + 1} \map \Gamma {x + z + n + 1} } $

where:

$\ds \map { {}_5 \operatorname F_4} { { {\dfrac n 2 + 1, n, -x, -y, -z} \atop {\dfrac n 2, x + n + 1, y + n + 1, z + n + 1} } \, \middle \vert \, 1}$ is the generalized hypergeometric function of $1$: $\ds \sum_{k \mathop = 0}^\infty \dfrac { \paren {\dfrac n 2 + 1}^{\overline k} n^{\overline k} \paren {-x}^{\overline k} \paren {-y}^{\overline k} \paren {-z}^{\overline k} } { \paren {\dfrac n 2}^{\overline k} \paren {x + n + 1}^{\overline k} \paren {y + n + 1}^{\overline k} \paren {z + n + 1}^{\overline k} } \dfrac {1^k} {k!}$

$x^{\overline k}$ denotes the $k$th rising factorial power of $x$

$\map \Gamma {n + 1} = n!$ is the Gamma function.

Setting $z = -\dfrac n 2$, we obtain:

\(\ds \sum_{k \mathop = 0}^\infty \dfrac { \paren {\dfrac n 2 + 1}^{\overline k} n^{\overline k} \paren {-x}^{\overline k} \paren {-y}^{\overline k} \paren {-z}^{\overline k} } { \paren {\dfrac n 2}^{\overline k} \paren {x + n + 1}^{\overline k} \paren {y + n + 1}^{\overline k} \paren {z + n + 1}^{\overline k} } \dfrac {1^k} {k!}\)

\(=\)

\(\ds \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {z + n + 1} \map \Gamma {x + y + z + n + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {y + z + n + 1} \map \Gamma {x + z + n + 1} }\)

before substitution

\(\ds \sum_{k \mathop = 0}^\infty \dfrac { \paren {\dfrac n 2 + 1}^{\overline k} n^{\overline k} \paren {-x}^{\overline k} \paren {-y}^{\overline k} \paren {\dfrac n 2}^{\overline k} } { \paren {\dfrac n 2}^{\overline k} \paren {x + n + 1}^{\overline k} \paren {y + n + 1}^{\overline k} \paren {-\dfrac n 2 + n + 1}^{\overline k} } \dfrac {1^k} {k!}\)

\(=\)

\(\ds \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {-\dfrac n 2 + n + 1} \map \Gamma {x + y - \dfrac n 2 + n + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {y - \dfrac n 2 + n + 1} \map \Gamma {x - \dfrac n 2 + n + 1} }\)

after substitution

\(\ds \sum_{k \mathop = 0}^\infty \dfrac { \paren {\dfrac n 2}^{\overline k} \paren {\dfrac n 2 + 1}^{\overline k} n^{\overline k} \paren {-x}^{\overline k} \paren {-y}^{\overline k} } { \paren {\dfrac n 2}^{\overline k} \paren {\dfrac n 2 + 1}^{\overline k} \paren {x + n + 1}^{\overline k} \paren {y + n + 1}^{\overline k} } \dfrac {1^k} {k!}\)

\(=\)

\(\ds \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {\dfrac n 2 + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {y + \dfrac n 2 + 1} \map \Gamma {x + \dfrac n 2 + 1} }\)

reorganizing

\(\ds \sum_{k \mathop = 0}^\infty \dfrac { n^{\overline k} \paren {-x}^{\overline k} \paren {-y}^{\overline k} } { \paren {x + n + 1}^{\overline k} \paren {y + n + 1}^{\overline k} } \dfrac {1^k} {k!}\)

\(=\)

\(\ds \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {\dfrac n 2 + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {y + \dfrac n 2 + 1} \map \Gamma {x + \dfrac n 2 + 1} }\)

$\paren {\dfrac n 2}^{\overline k}$ and $\paren {\dfrac n 2 + 1}^{\overline k}$ cancel

Therefore:

$\ds \map { {}_3 \operatorname F_2} { { {n, -x, -y} \atop {x + n + 1, y + n + 1} } \, \middle \vert \, 1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {\dfrac n 2 + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } {\map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {y + \dfrac n 2 + 1} }$

$\blacksquare$

Source of Name

This entry was named for Alfred Cardew Dixon.

Sources

• 1989: Bruce C. Berndt: Ramanujan's Notebooks: Part II: Chapter $\text {10}$. Hypergeometric Series: $\text I$