Morley's Formula

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Theorem

Let $n \in \C$ be a complex number.

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

Then:

\(\ds \sum_{k \mathop = 0}^\infty \paren {\dfrac {n^{\overline k} } {k!} }^3\) \(=\) \(\ds \dfrac {\map \Gamma {1 - \dfrac {3 n} 2} } {\map {\Gamma^3} {1 - \dfrac n 2} } \map \cos {\dfrac {\pi n} 2}\)


Proof

\(\ds \sum_{k \mathop = 0}^\infty \paren {\dfrac {n^{\overline k} } {k!} }^3\) \(=\) \(\ds \dfrac {6 \map \sin {\dfrac {\pi n} 2} \map \sin {\pi n} } {\pi^2 n^2 \paren {1 + 2 \map \cos {\pi n} } } \times \dfrac {\map {\Gamma^3} {\dfrac n 2 + 1} } {\map \Gamma {\dfrac {3 n} 2 + 1} }\) Dixon's Hypergeometric Theorem: Corollary 1
\(\ds \) \(=\) \(\ds \dfrac {6 \map \sin {\dfrac {\pi n} 2} \map \sin {\pi n} } {\pi^2 n^2 \paren {1 + 2 \map \cos {\pi n} } } \times \dfrac {\paren {\dfrac n 2}^3 \map {\Gamma^3} {\dfrac n 2} } {\paren {\dfrac {3 n} 2} \map \Gamma {\dfrac {3 n} 2} }\) Definition of Gamma Function
\(\ds \) \(=\) \(\ds \dfrac {\map \sin {\dfrac {\pi n} 2} \map \sin {\pi n} } {2 \pi^2 \paren {1 + 2 \map \cos {\pi n} } } \times \dfrac {\map {\Gamma^3} {\dfrac n 2} } {\map \Gamma {\dfrac {3 n} 2} } \times \dfrac {\map {\Gamma^3} {1 - \dfrac n 2} } {\map {\Gamma^3} {1 - \dfrac n 2} } \times \dfrac {\map \Gamma {1 - \dfrac {3 n} 2} } {\map \Gamma {1 - \dfrac {3 n} 2} }\) multiplying by $1$ and simplifying
\(\ds \) \(=\) \(\ds \dfrac {\map \sin {\dfrac {\pi n} 2} \map \sin {\pi n} \map \Gamma {1 - \dfrac {3 n} 2} } {2 \pi^2 \paren {1 + 2 \map \cos {\pi n} } \map {\Gamma^3} {1 - \dfrac n 2} } \times \dfrac {\pi^3 } {\paren {\map \sin {\dfrac {\pi n} 2}^3 } } \times \dfrac {\map \sin {\dfrac {3\pi n} 2} } \pi\) Euler's Reflection Formula
\(\ds \) \(=\) \(\ds \dfrac {\map \sin {\dfrac {\pi n} 2} \map \sin {\pi n} \map \Gamma {1 - \dfrac {3 n} 2} } {2 \map \sin {\dfrac {3\pi n} 2} \map {\Gamma^3} {1 - \dfrac n 2} } \times \dfrac 1 {\paren {\map \sin {\dfrac {\pi n} 2}^2} } \times \map \sin {\dfrac {3\pi n} 2}\) Sine of Integer Multiple of Argument: Formulation 6 and simplifying
\(\ds \) \(=\) \(\ds \dfrac {\map \sin {\pi n} \map \Gamma {1 - \dfrac {3 n} 2} } {2 \map \sin {\dfrac {\pi n} 2} \map {\Gamma^3} {1 - \dfrac n 2} }\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {2 \map \sin {\dfrac {\pi n} 2} \map \cos {\dfrac {\pi n} 2} \map \Gamma {1 - \dfrac {3 n} 2} } {2 \map \sin {\dfrac {\pi n} 2} \map {\Gamma^3} {1 - \dfrac n 2} }\) Double Angle Formula for Sine
\(\ds \) \(=\) \(\ds \dfrac {\map \Gamma {1 - \dfrac {3 n} 2} } {\map {\Gamma^3} {1 - \dfrac n 2} } \map \cos {\dfrac {\pi n} 2}\) rearranging

$\blacksquare$


Source of Name

This entry was named for Frank Morley.


Sources

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