\(4 . ^ { * }\) The functions \(Q _ { k }\) in the previous problem are related to the Bernoulli polynomials \(B _ { k } ( x )\) by the formula
\[ Q _ {k} (x) = \frac {B _ {k + 1} (x)}{(k + 1) !} \quad \text {for} \quad 0 \leq x \leq 1. \]
Also, if k is a positive integer, then
\[ 2 \zeta (2 k) = (- 1) ^ {k + 1} \frac {(2 \pi) ^ {2 k}}{(2 k) !} B _ {2 k}, \]
where \(B _ { k } = B _ { k } ( 0 )\) are the Bernoulli numbers. For the definition of \(B _ { k } ( x )\) and \(B _ { k }\) see Chapter 3 in Book I.