Z coth(z) = sum of 2^(2n)B (2n) z^(2n)/(2n)!

Theorem

The following formula holds for $|z| < \pi$: $$z \mathrm{coth}(z) = \displaystyle\sum_{k=0}^{\infty} 2^{2k} B_{2k} \dfrac{z^{2k}}{(2k)!},$$ where $\mathrm{coth}$ denotes hyperbolic cotangent, $B_{2k}$ denotes Bernoulli numbers, and $(2k)!$ denotes factorial.

References

• 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.20 (2)$