Bourbaki

Exercise 3, § 2

Functions of a Real Variable, VI, p. 23

Prove, for every integer p>1p > 1, the formula

Bn(xp)+Bn(x+1p)+⋯+Bn(x+p−1p)=1pn−1Bn(X). B_n\left(\frac{x}{p}\right) + B_n\left(\frac{x+1}{p}\right) + \cdots + B_n\left(\frac{x+p-1}{p}\right) = \frac{1}{p^{n-1}} B_n(X).
Solution

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Machine translation of the English, by gpt-5-6-mini. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.