Luis M. Navas

Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials

We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials

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

\mathcal{B}_{n}(x;\lambda)

A Simple Computation of ζ (2 k )

in detail. The starting point is their Fourier series on

Old and New Identities for Bernoulli Polynomials via Fourier Series

[0,1]

Algebraic properties and Fourier expansions of two-dimensional Apostol-Bernoulli and Apostol-Euler polynomials

which, it is shown, remains valid as an asymptotic expansion over compact subsets of the complex plane. This is used to determine explicit estimates on the constants in the approximation, and also to analyze oscillatory phenomena which arise in certain cases. These results are transferred to the Apostol-Euler polynomials

Some functional relations derived from the Lindelöf-Wirtinger expansion of the Lerch transcendent function

\mathcal{E}_{n}(x;\lambda)

ANALYTIC CONTINUATION OF THE FIBONACCI DIRICHLET SERIES

via a simple relation linking them to the Apostol-Bernoulli polynomials.

Full length article: Asymptotic behavior of the Lerch transcendent function

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 Mobius inversion from analytic number theory to Fourier expansions, we derive identities involving Bernoulli polynomials, Bernoulli numbers, and the Mobius function; this includes formulas for the Bernoulli polynomials at rational arguments. Finally, we show some asymptotic properties concerning the Bernoulli and Euler polynomials.

Möbius inversion formulas for flows of arithmetic semigroups

Abstract We present a new simple proof of Euler’s formulas for ζ(2k), where k = 1, 2, 3,…. The computation is done using only the defining properties of the Bernoulli polynomials and summing a telescoping series. The same method also yields integral formulas for ζ(2k + 1).

THE LERCH TRANSCENDENT FROM THE POINT OF VIEW OF FOURIER ANALYSIS

The Bernoulli polynomials restricted to and extended by periodicity have nth sine and cosine Fourier coefficients of the form . In general, the Fourier coefficients of any polynomial restricted to are linear combinations of terms of the form . If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a relation between the given family and the Bernoulli polynomials. Using this idea, we give new and simpler proofs of some known identities involving Bernoulli, Euler, and Legendre polynomials. The method can also be applied to certain families of Gegenbauer polynomials. As a result, we obtain new identities for Bernoulli polynomials and Bernoulli numbers.