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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| [3] | Manuel Benito, Luis M. Navas, and Juan L. Varona, Möbius inversion formulas for flows of arithmetic semigroups, J. Number Theory 128 (2008), no. 2, 390 – 412. · Zbl 1147.11006 · doi:10.1016/j.jnt.2007.04.009 |
| [4] | Qiu-Ming Luo, Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials, Math. Comp. 78 (2009), no. 268, 2193 – 2208. · Zbl 1214.11032 |
| [5] | Luis M. Navas, Francisco J. Ruiz, and Juan L. Varona, The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials, J. Approx. Theory 163 (2011), no. 1, 22 – 40. · Zbl 1253.11028 · doi:10.1016/j.jat.2010.02.005 |
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