A summation method based on the Fourier series of periodic distributions and an example arising in the Casimir effect

A generalised summation method is considered based on the Fourier series of periodic distributions. It is shown that [Formula presented], where [Formula presented] is the 2π-periodic distribution given by [Formula presented] for φ∈D(R) with support supp(φ)⊂(−δ,2π+δ), where δ∈(0,π). Applying the generalised summation method, the sum of the divergent series 1+2+3+⋯ (arising in a quantum field theory calculation in the Casimir effect) is determined, and more generally also the sum 1 k+2 k+3 k+⋯, for k∈N, is determined.