Möbius inversion formulae for Apostol-Bernoulli type polynomials and numbers. (English) Zbl 1276.11029
Summary: In this paper, we establish Möbius 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.
MSC:
11B68 | Bernoulli and Euler numbers and polynomials |
11A25 | Arithmetic functions; related numbers; inversion formulas |
11B73 | Bell and Stirling numbers |
42A16 | Fourier coefficients, Fourier series of functions with special properties, special Fourier series |
41A58 | Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) |
Keywords:
Apostol-Bernoulli numbers and polynomials; Apostol-Euler polynomials; Apostol-Genocchi polynomials; Stirling numbers; Fourier series; Möbius function; Möbius inversionReferences:
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