Difference of Even Powers of z + a and z - a

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Theorem

Let $m \in \Z$ be an integer such that $m > 1$.

Then for all complex number $z$:

$\paren {z + a}^{2 m} - \paren {z - a}^{2 m} = 4 m a z \ds \prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \dfrac {k \pi} {2 m} }$


Proof

From Factors of Difference of Two Even Powers:

$x^{2 n} - y^{2 n} = \paren {x - y} \paren {x + y} \ds \prod_{k \mathop = 1}^{n - 1} \paren {x^2 - 2 x y \cos \dfrac {k \pi} n + y^2}$

Substituting $z + a$ for $x$, $z - a$ for $y$, and $m$ for $n$ we get:

\(\ds \) \(\) \(\ds \paren {z + a}^{2 m} - \paren {z - a}^{2 m}\)
\(\ds \) \(=\) \(\ds \paren {\paren {z + a} - \paren {z - a} } \paren {\paren {z + a} + \paren {z - a} } \prod_{k \mathop = 1}^{m - 1} \paren {\paren {z + a}^2 - 2 \paren {z + a} \paren {z - a} \cos \frac {k \pi} m + \paren {z - a}^2}\)
\(\ds \) \(=\) \(\ds \paren {2 a} \paren {2 z} \prod_{k \mathop = 1}^{m - 1} \paren {z^2 + 2 a z + a^2 - 2 \paren {z^2 - a^2} \cos \frac {k \pi} m + \paren {z^2 - 2 a z + a^2} }\)
\(\ds \) \(=\) \(\ds 4 a z \prod_{k \mathop = 1}^{m - 1} \paren {2 z^2 + 2 a^2 - 2 \paren {z^2 - a^2} \cos \frac {k \pi} m}\)
\(\ds \) \(=\) \(\ds 4 a z \prod_{k \mathop = 1}^{m - 1} \paren {2 z^2 \paren {1 - \cos \frac {k \pi} m} + 2 a^2 \paren {1 + \cos \frac {k \pi} m} }\)
\(\ds \) \(=\) \(\ds 4 a z \prod_{k \mathop = 1}^{m - 1} \paren {2 z^2 \paren {2 \sin^2 \frac {k \pi} {2 m} } + 2 a^2 \paren {2 \cos^2 \frac {k \pi} {2 m} } }\) Double Angle Formula for Cosine: Corollary $1$ and Double Angle Formula for Cosine: Corollary $2$
\(\ds \) \(=\) \(\ds 4 a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\prod_{k \mathop = 1}^{m - 1} 4 \sin^2 \frac {k \pi} {2 m} }\) Definition of Cotangent
\(\ds \) \(=\) \(\ds 4^m a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\prod_{k \mathop = 1}^{m - 1} \sin \frac {k \pi} {2 m} } \paren {\prod_{k \mathop = 1}^{m - 1} \map \sin {\pi - \frac {k \pi} {2 m} } }\) Sine of Supplementary Angle
\(\ds \) \(=\) \(\ds 4^m a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\prod_{k \mathop = 1}^{m - 1} \sin \frac {k \pi} {2 m} } \paren {\prod_{k \mathop = 1}^{m - 1} \sin \frac {\paren {2 m - k} \pi} {2 m} } \sin \frac {m \pi} {2 m}\) Sine of Right Angle
\(\ds \) \(=\) \(\ds 4^m a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\prod_{k \mathop = 1}^m \sin \frac {k \pi} {2 m} } \paren {\prod_{k \mathop = m + 1}^{2 m - 1} \sin \frac {k \pi} {2 m} }\) Translation of Index Variable of Product
\(\ds \) \(=\) \(\ds 4^m a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\prod_{k \mathop = 1}^{2 m - 1} \sin \frac {k \pi} {2 m} }\)
\(\ds \) \(=\) \(\ds 4^m a z \paren {\prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} } } \paren {\frac {2 m} {2^{2 m - 1} } }\) Product of Sines of Fractions of Pi
\(\ds \) \(=\) \(\ds 4 m a z \prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \frac {k \pi} {2 m} }\)

$\blacksquare$


Sources

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