4 May
2020
4 May
'20
4:38 p.m.
The Rademacher function r_n : [0,1] —> {-1,1} is your basic square wave of wavelength 1/2^n on [0,1], defined conveniently as r_n(x) = sign(sin(2^n * 2πx)) I noticed that in Mark Kac's great 1959 book "Statistical Independence in Probability, Analysis, and Number Theory" he states that Integral_{0 <= t <= 1} ((r_1(t) + ... + r_n(t))^2 dt = n It's easy to see that if the exponent 2 is replaced by an odd number, this integral is 0, by oddness. But my limited experiments have not suggested what the pattern is for even exponents, other than that it's always an integer. Question: What is an explicit formula for f(n, k) = Integral_{0 <= t <= 1} ((r_1(t) + ... + r_n(t))^(2k) dt ??? —Dan