Bernoulli Polynomials - Multiplication Theorems

Multiplication Theorems

The multiplication theorems were given by Joseph Ludwig Raabe in 1851:

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

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

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