Periodic Bernoulli function (${\tilde{B}}_n(x)=B_n(\{x\})$)

The periodic Bernoulli function is the periodic extension ${\tilde{B}}_n(x)=B_n(\{x\})$ of the Bernoulli polynomial, where $\{x\}=x-\lfloor x\rfloor$ is the fractional part. It has period one. For example, ${\tilde{B}}_2(x)=\{x{\}}^2-\{x\}+1/6$.

For $n\ge 2$, its Fourier expansion isThe series is absolutely convergent. For $n=1$ the same formula holds away from the integers and describes a sawtooth. At an integer the symmetric Fourier value is zero, whereas the literal fractional-part convention gives $B_1(0)=-1/2$; endpoint conventions must therefore be distinguished.^[ref]

Periodic Bernoulli functions occur in the remainder of the Euler-Maclaurin formula, connecting errors in discrete summation with periodic oscillation between successive integers.