Kummer's Hypergeometric Theorem/Proof 1

From ProofWiki
Jump to navigation Jump to search

Theorem

$\map F {n, -x; x + n + 1; -1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } {\map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {n + 1} }$


Proof

First we note the definition of Gaussian hypergeometric function:

$\map F {n, -x; x + n + 1; -1} = \ds \sum_{k \mathop = 0}^\infty \dfrac { n^{\overline k} \paren {-x}^{\overline k} } {\paren {x + n + 1}^{\overline k} } \dfrac {\paren {-1}^k} {k!}$

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


Two lemmata:

Lemma 1

$\ds \lim_{y \mathop \to \infty} \dfrac {y^{\underline k} } {\paren {y + n + 1}^{\overline k} } = 1$

$\Box$


Lemma 2

$\ds \lim_{y \mathop \to \infty} \dfrac {\paren {y + \dfrac n 2 + 1}^{\overline x} } {\paren {y + n + 1}^{\overline x} } = 1$

$\Box$


We use Dixon's Hypergeometric Theorem:

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

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 $1$.


So:

\(\ds \map { {}_3 \operatorname F_2} { { {n, -x, -y} \atop {x + n + 1, y + n + 1} } \, \middle \vert \, 1}\) \(=\) \(\ds \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {\dfrac n 2 + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {y + \dfrac n 2 + 1} }\) Dixon's Hypergeometric Theorem
\(\ds \leadsto \ \ \) \(\ds \sum_{k \mathop = 0}^\infty \dfrac { n^{\overline k} \paren {-x}^{\overline k} \paren {-y}^{\overline k} } { \paren {x + n + 1}^{\overline k} \paren {y + n + 1}^{\overline k} } \dfrac {1^k} {k!}\) \(=\) \(\ds \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {\dfrac n 2 + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {y + \dfrac n 2 + 1} }\) Definition of Generalized Hypergeometric Function
\(\ds \leadsto \ \ \) \(\ds \sum_{k \mathop = 0}^\infty \dfrac { n^{\overline k} \paren {-x}^{\overline k} } { \paren {x + n + 1}^{\overline k} } \dfrac {\paren {-y}^{\overline k} } {\paren {y + n + 1}^{\overline k} } \dfrac {1^k} {k!}\) \(=\) \(\ds \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + \dfrac n 2 + 1} } \times \dfrac {\map \Gamma {y + n + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {x + y + n + 1} \map \Gamma {y + \dfrac n 2 + 1} }\) reorganizing both sides: isolating $y$
\(\ds \leadsto \ \ \) \(\ds \sum_{k \mathop = 0}^\infty \dfrac { n^{\overline k} \paren{-x}^{\overline k} } { \paren {x + n + 1}^{\overline k} } \dfrac {y^{\underline k} } {\paren {y + n + 1}^{\overline k} } \dfrac {\paren {-1}^k} {k!}\) \(=\) \(\ds \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + \dfrac n 2 + 1} } \times \dfrac {\map \Gamma {y + n + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {x + y + n + 1} \map \Gamma {y + \dfrac n 2 + 1} }\) on the left hand side: Rising Factorial in terms of Falling Factorial of Negative
\(\ds \leadsto \ \ \) \(\ds \sum_{k \mathop = 0}^\infty \dfrac { n^{\overline k} \paren{-x}^{\overline k} } { \paren {x + n + 1}^{\overline k} } \dfrac {y^{\underline k} } {\paren {y + n + 1}^{\overline k} } \dfrac {\paren {-1}^k} {k!}\) \(=\) \(\ds \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + \dfrac n 2 + 1} } \times \dfrac {\paren {y + \dfrac n 2 + 1}^{\overline x} } {\paren {y + n + 1}^{\overline x} }\) on the right hand side: Rising Factorial as Quotient of Factorials
\(\ds \leadsto \ \ \) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {n^{\overline k} \paren{-x}^{\overline k} } {\paren {x + n + 1}^{\overline k} } \paren 1 \dfrac {\paren {-1}^k} {k!}\) \(=\) \(\ds \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } {\map \Gamma {n + 1} \map \Gamma {x + \dfrac n 2 + 1} } \times 1\) Lemma 1 and Lemma 2: letting $y \to \infty$
\(\ds \leadsto \ \ \) \(\ds \map F {n, -x; x + n + 1; -1}\) \(=\) \(\ds \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } {\map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {n + 1} }\) Definition of Gaussian Hypergeometric Function

$\blacksquare$


Source of Name

This entry was named for Ernst Eduard Kummer.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Kummer%27s_Hypergeometric_Theorem/Proof_1&oldid=609514"