The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials

Abstract

Hurwitz found the Fourier expansion of the Bernoulli polynomials over a century ago. In general, Fourier analysis can be fruitfully employed to obtain properties of the Bernoulli polynomials and related functions in a simple manner. In addition, applying the technique of Möbius inversion from analytic number theory to Fourier expansions, we derive identities involving Bernoulli polynomials, Bernoulli numbers, and the Möbius function; this includes formulas for the Bernoulli polynomials at rational arguments. Finally, we show some asymptotic properties concerning the Bernoulli and Euler polynomials.

Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials

“…A similar phenomenon was studied by the authors for Bernoulli polynomials in [5], namely, when the first conjugate term in the Fourier series fails to provide information because it vanishes, one turns to the next, which in that case cannot vanish simultaneously with the first. Proof.…”

“…The purpose of this article is to obtain analogous asymptotic estimates for B n (z; λ), valid for any z ∈ C. In short, the central result of this paper is that the Fourier series (4) of B n (x; λ) for x ∈ [0, 1], which a priori represents it only on this interval, is actually valid on the entire complex plane as an asymptotic series representing B n (z; λ) for z ∈ C. From this we deduce explicit asymptotic estimates for the Apostol-Bernoulli polynomials which include the pattern mentioned above, namely, geometric order of decrease for the differences between B n (z; λ) and its asymptotic approximations, with implicit constants that are exponential in |z|, as well as estimates for succesive quotients of these differences. The behavior of the approximations varies considerably depending on λ, with λ = 1, studied in [5], turning out to be the exception rather than the rule.…”

The Lerch transcendent from the point of view of Fourier analysis

“…Namely, one can obtain asymptotic series with simple explicit bounds on the error term, and use them to understand the growth of the functions and also certain limiting oscillatory phenomena brought to light by the Fourier expansions. We have done this for the classical Bernoulli and Euler polynomials in [8], for the Apostol-Bernoulli and Apostol-Euler polynomials in [10], and for the Lerch transcendent in [11]. Clearly one will have analogous results for the conjugate functions in each case.…”

“…2 A simple consequence of this, which also shows an alternative way of expressing the convolution, is as follows: The next application involves Möbius inversion of Fourier series having an arithmetically simple form, to which the general framework developed in [3] can be applied in a straightforward manner. This includes the Fourier series of the Bernoulli polynomials, studied in [8], and more generally, of the Apostol-Bernoulli polynomials. For example, in [2] and [12], it is shown that…”

A note on Appell sequences, Mellin transforms and Fourier series

“…This implies that their Fourier series are uniformly convergent for n ≥ 2. In addition, the Fourier coefficients are completely multiplicative arithmetic functions, a fact which can be used to find interesting Möbius inversion formulas (see [10]). The Apostol-Bernoulli polynomials B(x; λ), after introducing the factor λ x , have similar properties.…”