Gauss's Hypergeometric Theorem/Corollary 2

Corollary to Gauss's Hypergeometric Theorem

Let $\map \Re {a - 1} < 0$.

Then:

$\ds \dfrac 1 a + \dfrac {a } {\paren {a + 1} 1!} + \dfrac {a \paren {a + 1} } {\paren {a + 2} 2!} + \dfrac {a \paren {a + 1} \paren {a + 2} } {\paren {a + 3} 3!} + \cdots = \dfrac {\pi} {\map \sin {\pi a } } $

Proof

Set $c - 1 = a$ in Corollary 1:

Before substitution:

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

\(=\)

\(\ds \dfrac {\map \Gamma {c - 1} \map \Gamma {1 - a} } {\map \Gamma {c - a} }\)

Corollary 1

After substitution:

\(\ds \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} } { \paren {a + k} k!}\)

\(=\)

\(\ds \dfrac {\map \Gamma a \map \Gamma {1 - a} } {\map \Gamma {\paren {a + 1} - a} }\)

\(\ds \)

\(=\)

\(\ds \map \Gamma a \map \Gamma {1 - a}\)

Definition of Gamma Function and $\map \Gamma 1 = 1$

\(\ds \)

\(=\)

\(\ds \dfrac {\pi} {\map \sin {\pi a } }\)

Euler's Reflection Formula

$\blacksquare$

Sources

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