Multiplication Theorem - Bernoulli Polynomials

Bernoulli Polynomials

For the Bernoulli polynomials, the multiplication theorems were given by Joseph Ludwig Raabe in 1851:

and for the Euler polynomials,

k^{-m} E_m(kx)= \sum_{n=0}^{k-1} (-1)^n E_m \left(x+\frac{n}{k}\right) \quad \mbox{ for } k=1,3,\dots

and

k^{-m} E_m(kx)= \frac{-2}{m+1} \sum_{n=0}^{k-1} (-1)^n B_{m+1} \left(x+\frac{n}{k}\right) \quad \mbox{ for } k=2,4,\dots.

The Bernoulli polynomials may be obtained as a special case of the Hurwitz zeta function, and thus the identities follow from there.

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