Dixon's Hypergeometric Theorem

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This proof is about Dixon's Theorem in the context of Combinatorics. For other uses, see Dixon's Theorem.

Theorem

Let $x, y, n \in \C$.

Let $n \notin \Z_{\lt 0}$.

Let $\map \Re {x + y + \dfrac n 2 + 1} > 0$.


Then:

$\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} } $

where:

$\ds \map { {}_3 \operatorname F_2} { { {n, -x, -y} \atop {x + n + 1, y + n + 1} } \, \middle \vert \, 1}$ is the generalized hypergeometric function of $1$: $\ds \sum_{k \mathop = 0}^\infty \dfrac { n^{\overline k} \paren {-x}^{\overline k} \paren {-y}^{\overline k} } { \paren {x + n + 1}^{\overline k} \paren {y + n + 1}^{\overline k} } \dfrac {1^k} {k!}$
$x^{\overline k}$ denotes the $k$th rising factorial power of $x$
$\map \Gamma {n + 1} = n!$ is the Gamma function.


Corollary 1

Let $\map \Re n < \dfrac 2 3$.

Then:

$\ds 1 + \paren {\dfrac n {1!} }^3 + \paren {\dfrac {n \paren {n + 1} } {2!} }^3 + \paren {\dfrac {n \paren {n + 1} \paren {n + 2} } {3!} }^3 + \cdots = \dfrac {6 \map \sin {\dfrac {\pi n} 2} \map \sin {\pi n} \map {\Gamma^3} {\dfrac n 2 + 1} } {\pi^2 n^2 \paren {1 + 2 \map \cos {\pi n} } \map \Gamma {\dfrac {3 n} 2 + 1} } $


Proof 1

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:

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

$\blacksquare$


Proof 2

From Gauss's Hypergeometric Theorem, we have:

$\map F {a, b; c; 1} = \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }$

Therefore:

\(\ds \map F {b + n, c + n; 1 + a + 2 n; 1}\) \(=\) \(\ds \dfrac {\map \Gamma {1 + a + 2 n} \map \Gamma {\paren {1 + a + 2 n} - \paren {b + n} - \paren {c + n} } } {\map \Gamma {\paren {1 + a + 2 n} - \paren {b + n} } \map \Gamma {\paren {1 + a + 2 n} - \paren {c + n} } }\) Gauss's Hypergeometric Theorem
\(\text {(1)}: \quad\) \(\ds \) \(=\) \(\ds \dfrac {\map \Gamma {1 + a + 2 n} \map \Gamma {- b - c + a + 1} } {\map \Gamma {1 + a + n - b } \map \Gamma {1 + a + n - c } }\) simplifying

Therefore:

\(\ds \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {1 + a - b} \map \Gamma {1 + a - c} } \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {1 + a - b, 1 + a - c} } \, \middle \vert \, 1}\) \(=\) \(\ds \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {1 + a - b} \map \Gamma {1 + a - c} } \sum_{n \mathop = 0}^\infty \dfrac {\dfrac {\map \Gamma {a + n} } {\map \Gamma a} \dfrac {\map \Gamma {b + n} } {\map \Gamma b} \dfrac {\map \Gamma {c + n} } {\map \Gamma c} } {\dfrac {\map \Gamma {1 + a - b + n} } {\map \Gamma {1 + a - b} } \dfrac {\map \Gamma {1 + a - c + n} } {\map \Gamma {1 + a - c} } } \dfrac {1^n} {n!}\) Definition of Hypergeometric Function and Rising Factorial as Quotient of Factorials
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {1 + a - b + n} \map \Gamma {1 + a - c + n} } \times \dfrac {\map \Gamma {1 + a + 2 n} } {\map \Gamma {1 + a + 2 n} } \times \dfrac {\map \Gamma {- b - c + a + 1} } {\map \Gamma {1 + a - b - c} }\) multiplying by $1$ and $1^n = 1$
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {1 + a + 2 n} \map \Gamma {- b - c + a + 1} } \map F {b + n, c + n; 1 + a + 2 n; 1}\) from $(1)$ above
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \dfrac {\map \Gamma {a + n} \map \Gamma {b + n} \map \Gamma {c + n} } {n! \map \Gamma {1 + a + 2 n} \map \Gamma {- b - c + a + 1} } \sum_{m \mathop = 0}^\infty \dfrac {\dfrac {\map \Gamma {b + n + m} } {\map \Gamma {b + n} } \dfrac {\map \Gamma {c + n + m} } {\map \Gamma {c + n} } } {\dfrac {\map \Gamma {1 + a + 2 n + m} } {\map \Gamma {1 + a + 2 n} } } \dfrac {1^m} {m!}\) Definition of Hypergeometric Function and Rising Factorial as Quotient of Factorials
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \sum_{m \mathop = 0}^\infty \dfrac {\map \Gamma {a + n} \map \Gamma {b + n + m} \map \Gamma {c + n + m} } {n! m! \map \Gamma {1 + a + 2 n + m} \map \Gamma {- b - c + a + 1} }\) simplifying
\(\ds \) \(=\) \(\ds \sum_{p \mathop = 0}^\infty \sum_{n \mathop = 0}^p \dfrac {\map \Gamma {a + n} \map \Gamma {b + p} \map \Gamma {c + p} } {n! \paren {p - n}! \map \Gamma {1 + a + n + p} \map \Gamma {- b - c + a + 1} }\) Letting $p = m + n$, so $m = p - n$ and Product of Absolutely Convergent Series
\(\ds \) \(=\) \(\ds \sum_{p \mathop = 0}^\infty \sum_{n \mathop = 0}^p \dfrac {\map \Gamma {a + n} \map \Gamma {b + p} \map \Gamma {c + p} } {n! \paren {p - n}! \map \Gamma {1 + a + n + p} \map \Gamma {- b - c + a + 1} } \times \dfrac {\map \Gamma {1 + a + p} } {\map \Gamma {1 + a + p} } \times \dfrac {\map \Gamma a } {\map \Gamma a } \times \dfrac {p!} {p!} \times \paren {-1}^n \times

\paren {-1}^n\)

multiplying by $1$
\(\ds \) \(=\) \(\ds \sum_{p \mathop = 0}^\infty \dfrac {\map \Gamma a \map \Gamma {b + p} \map \Gamma {c + p} } {p! \map \Gamma {1 + a + p} \map \Gamma {- b - c + a + 1} } \sum_{n \mathop = 0}^p \dfrac {\paren {\dfrac {\map \Gamma {a + n} } {\map \Gamma a} } \paren {\dfrac {\paren {-1}^n p!} {\paren {p - n}!} } } {\dfrac {\map \Gamma {1 + a + n + p} } {\map \Gamma {1 + a + p} } } \dfrac {\paren {-1}^n} {n!}\) rearranging terms
\(\ds \) \(=\) \(\ds \sum_{p \mathop = 0}^\infty \dfrac {\map \Gamma a \map \Gamma {b + p} \map \Gamma {c + p} } {p! \map \Gamma {1 + a + p} \map \Gamma {- b - c + a + 1} } \map F {a, -p; 1 + a + p; -1}\) Definition of Hypergeometric Function, Rising Factorial in terms of Falling Factorial of Negative and Rising Factorial as Quotient of Factorials
\(\ds \) \(=\) \(\ds \sum_{p \mathop = 0}^\infty \dfrac {\map \Gamma a \map \Gamma {b + p} \map \Gamma {c + p} } {p! \map \Gamma {1 + a + p} \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {p + a + 1} \map \Gamma {\dfrac a 2 + 1} } {\map \Gamma {p + \dfrac a 2 + 1} \map \Gamma {a + 1} }\) Kummer's Hypergeometric Theorem
\(\ds \) \(=\) \(\ds \sum_{p \mathop = 0}^\infty \dfrac {\map \Gamma a \map \Gamma {b + p} \map \Gamma {c + p} } {p! \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} } {\map \Gamma {p + \dfrac a 2 + 1} \map \Gamma {a + 1} }\) canceling $\map \Gamma {1 + a + p}$
\(\ds \) \(=\) \(\ds \sum_{p \mathop = 0}^\infty \dfrac {\map \Gamma a \map \Gamma {b + p} \map \Gamma {c + p} } {p! \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} } {\map \Gamma {p + \dfrac a 2 + 1} \map \Gamma {a + 1} } \times \dfrac {\map \Gamma b} {\map \Gamma b} \times \dfrac {\map \Gamma c} {\map \Gamma c}\) multiplying by $1$
\(\ds \) \(=\) \(\ds \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \sum_{p \mathop = 0}^\infty \dfrac {\dfrac {\map \Gamma {b + p} } {\map \Gamma b } \dfrac {\map \Gamma {c + p} } {\map \Gamma c } } {\dfrac {\map \Gamma {p + \dfrac a 2 + 1} } {\map \Gamma {\dfrac a 2 + 1} } } \dfrac 1 {p!}\)
\(\ds \) \(=\) \(\ds \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \map F {b, c; \dfrac a 2 + 1; 1}\) Definition of Hypergeometric Function and Rising Factorial as Quotient of Factorials
\(\ds \) \(=\) \(\ds \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} \map \Gamma {- b - c + \dfrac a 2 + 1} } {\map \Gamma {- b+ \dfrac a 2 + 1} \map \Gamma {- c + \dfrac a 2 + 1} }\) Gauss's Hypergeometric Theorem

We now have:

\(\ds \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {- b + a + 1} \map \Gamma {- c + a + 1} } \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {- b + a + 1, - c+ a + 1} } \, \middle \vert \, 1}\) \(=\) \(\ds \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} \map \Gamma {- b - c + \dfrac a 2 + 1} } {\map \Gamma {- b+ \dfrac a 2 + 1} \map \Gamma {- c + \dfrac a 2 + 1} }\)
\(\ds \leadsto \ \ \) \(\ds \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {- b + a + 1, - c + a + 1} } \, \middle \vert \, 1}\) \(=\) \(\ds \dfrac {\map \Gamma {- b + a + 1} \map \Gamma {- c + a + 1} } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} \map \Gamma {- b - c + \dfrac a 2 + 1} } {\map \Gamma {- b+ \dfrac a 2 + 1} \map \Gamma {- c + \dfrac a 2 + 1} }\)


Substituting $a = n$, $b = -x$ and $c = -y$, we obtain:

\(\ds \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {1 + a - b, 1 + a - c} } \, \middle \vert \, 1}\) \(=\) \(\ds \dfrac {\map \Gamma {- b + a + 1} \map \Gamma {- c + a + 1} } {\map \Gamma {a + 1} \map \Gamma {- b - c + a + 1} } \dfrac {\map \Gamma {\dfrac a 2 + 1} \map \Gamma {- b - c + \dfrac a 2 + 1} } {\map \Gamma {- b + \dfrac a 2 + 1} \map \Gamma {-c + \dfrac a 2 + 1} }\) before substitution
\(\ds \leadsto \ \ \) \(\ds \map { {}_3 \operatorname F_2} { { {n, -x, -y} \atop {x + n + 1, y + n + 1} } \, \middle \vert \, 1}\) \(=\) \(\ds \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {\dfrac n 2 + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {y + \dfrac n 2 + 1} }\) after substitution


$\blacksquare$


Also known as

Dixon's Hypergeometric Theorem is usually known as Dixon's Theorem, but there are a number of such and similarly named theorems.

Some sources refer to it as Dixon's Summation Theorem.


The term Dixon's Hypergeometric Theorem was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$ in order to identify this theorem uniquely..

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.


Examples

Example: $\map { {}_3 \operatorname F_2} {\dfrac 1 2, \dfrac 1 2, \dfrac 1 4; 1, \dfrac 5 4; 1}$

$1 + \dfrac 1 5 \paren {\dfrac 1 2}^2 + \dfrac 1 9 \paren {\dfrac {1 \times 3} {2 \times 4} }^2 + \dfrac 1 {13} \paren {\dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} }^2 + \cdots = \dfrac {\pi^2} {4 \paren {\map \Gamma {\dfrac 3 4} }^4}$


Example: $\map { {}_3 \operatorname F_2} {\dfrac 1 2, \dfrac 1 4, \dfrac 1 4; \dfrac 5 4, \dfrac 5 4; 1}$

$1 + \dfrac 1 {5^2} \paren {\dfrac 1 2} + \dfrac 1 {9^2} \paren {\dfrac {1 \times 3} {2 \times 4} } + \dfrac 1 {13^2} \paren {\dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} } + \cdots = \dfrac {\pi^{\frac 5 2} } {8 \sqrt 2 \paren {\map \Gamma {\dfrac 3 4} }^2}$


Example: $\map { {}_3 \operatorname F_2} {\dfrac 1 2, \dfrac 1 2, \dfrac 1 2; 1, 1; 1}$

$1 + \paren {\dfrac 1 2}^3 + \paren {\dfrac {1 \times 3} {2 \times 4} }^3 + \paren {\dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} }^3 + \cdots = \dfrac \pi {\paren {\map \Gamma {\dfrac 3 4} }^4 }$


Also see


Source of Name

This entry was named for Alfred Cardew Dixon.


Sources

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