In this note, we calculate the Green’s function for the linear operator (D )n, whereD is the one-dimensional Dirichlet Laplacian in L2 ((0, 1); dx) defined byD f = − f′′ with (Dirichlet) boundary conditions f (0) = f (1) = 0. As a consequence of this computation, we obtain Euler’s formula ζ (2n) =∑k∈N k−2n =(−1)n−1 22n−1 π2n B2n, n ∈ N, (2n)! where ζ (·) denotes the Riemann zeta function and Bn is the nth Bernoulli number. This generalizes the example given by Grieser [29] for n = 1. In addition, we derive its z-dependent generalization for z ∈ C\{(kπ)2n}k∈N, ∑ k∈N [ (kπ)2n − z] [ −1 1= n − 2nz ∑n−1 j =0 ω1/2 j z1/(2n) cot(ω1/2 j z1/(2n))], n ∈ N, where ωj = e2πi j /n, 0 ≤ j ≤ n − 1, represent the nth roots of unity. In this context we also derive the Green’s function of((D )n − z I)−1, n ∈ N.
