#hm-research

TAOCP 1.2.4 Exercise 40

Section 1.2.4: Integer Functions and Elementary Number Theory Exercise 40. [ HM46 ] Study the class of replicative functions; determine all replicative functions of a special type. For example, is the function in (a) of exercise 39 the only continuous replicative function? It may be interesting to study also the more general class of functions for which $$ f(x)+f\left(x+\frac1n\right)+\cdots+f\left(x+\frac{n-1}{n}\right)=a_n f(nx)+b_n. $$ Here $a_n$ and $b_n$ are numbers that depend on...