Treffer: Some polynomials associated with Williams' limit formula for $\zeta (2n)$: Some polynomials associated with Williams' limit formula for \(\zeta(2n)\)

In 1971 K. S. Williams proved that \[ \zeta(2n)= \lim_{q\to\infty} \Biggl({\pi\over 2q}\Biggr)^{2n}\,\sum^q_{p=1} \text{cot}^{2n}\Biggl(-{p\pi\over 2q+1}\Biggr). \] The authors show that Williams' limit formula, and three other analogous limit formulas proven here, involve polynomials of degree \(2n\). These polynomials are explicitly determined, and as a consequence, Euler's formula on \(\zeta(2n)\), involving Bernoulli numbers, is deduced. Each of these closed-form summation formulas, expressing a finite trigonometric sum in terms of higher-order Bernoulli polynomials is capable of yielding many special cases and consequences.