M?bius inversion formulae for apostol-bernoulli type polynomials and numbers

摘要

In this paper, we establish M?bius inversion formulae for the Fourier expansions of the Apostol-Bernoulli, Apostol-Euler and Apostol- Genocchi polynomials. As an application, by specializing our formulae at some special values we obtain interesting number-theoritical relations. We derive explicit formulae for Apostol-Bernoulli numbers. These formulae involve Stirling numbers of the second kind and powers of cotangent. Our proofs are very simple.