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

Abstract

An interesting limit formula for the Riemann Zeta function $\zeta (n) (n\in \mathbb{N}\backslash \{1\})$ was contained implicitly in a paper by K. S. Williams [17]. In the case of $\zeta(2n)\ (n\in \mathbb{N})$, we show that Williams' limit formula, and three other analogous limit formulas proven here, involve polynomials of degree $2n$. We also determine these polynomials explicitly and deduce, as an immediate consequence, Euler's celebrated relation between $\zeta(2n)$ and the familiar Bernoulli numbers $B_{2n}$. Each of our closed-form summation formulas, expressing a finite trigonometric sum in terms of higher-order Bernoulli polynomials, is capable of yielding many (new or known) special cases and consequences.