I wonder, can anyone describe an expression or formula of a transform that converts


$$\sum\_{k=0}^\infty \frac{a\_k x^k}{k!}$$


into


$$\sum\_{k=0}^\infty \frac{a\_k B\_k(x)}{k!}$$


where $B\_k(x)$ are Bernoulli polynomials.


For instance,


$$x^2\, \to\, x^2-x+1/6$$


$$e^x \,\to \, \frac{e^x}{e-1}$$


etc.



Transformation converting power series to Bernoulli polynomial series
An operator performing the mapping is $$O= D/(e^D-1)=e^{B.(0)D},$$


where $D=d/dx $ and $(B.(0))^n=B\_n(x)|\_{x=0}$, since the Bernoulli polynomials are an Appell sequence.


Edit (6/20/2017):


This operator is essentially the Todd operator. See the discussions on pg. 30 and Appendix B of "[Permutohedra, associahedra, and beyond](https://arxiv.org/abs/math/0507163)" by Postnikov of the Todd operator as a transform of the homogeneous volume polynomials for classes of polytopes into a generalized Ehrhart polynomial coding the number of lattice points in the polytopes.


(Edit 8/2018)


For some idea of the importance of this Todd operator in modern mathematics and physics, see [New Models for Veneziano Amplitudes: Combinatorial, Symplectic and Supersymmetric Aspects](https://arxiv.org/abs/hep-th/0503232) by Kholodenko.


