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

Abstract

The Lindelöf-Wirtinger expansion of the Lerch transcendent function implies, as a limiting case, Hurwitz's formula for the eponymous zeta function. A generalized form of Möbius inversion applies to the Lindelöf-Wirtinger expansion and also implies an inversion formula for the Hurwitz zeta function as a limiting case. The inverted formulas involve the dynamical system of rotations of the circle and yield an arithmetical functional equation.see [8, §1.11, p. 27] or [2, §25.14], for example. Logarithms and complex powers are always assumed to be principal. If |λ| < 1, then for any s ∈ C, the series converges uniformly in z over C\(−∞, 0], thus defining a holomorphic function of z in this region. If |λ| = 1, then the series converges in this same region provided Re s > 1. The value λ = 0 is considered trivial since it yields Φ(0, s, z) = z −s , and thus is usually excluded. There are multiple ways of defining analytic continuations of Φ in each parameter. This function, defined by Mathias Lerch in 1887 in his paper [11], includes as special cases of the parameters the Hurwitz and Riemann zeta functions and the polylogarithms, among others, and has applications ranging from number theory to physics. It is often used to obtain functional identities; see, for instance, [3,7,9,15].One often extends the domain to z ∈ C \ {0, −1, −2, . . . } by including the branch discontinuity of the principal argument. In addition, in this paper we shall only be considering s ∈ C with Re s < 0, in which case the summand (k + z) −s (with k ∈ N) in (1.1) continuously extends to z = −k by defining 0 −s = 0. Thus for Re s < 0 we can define Φ(λ, s, z) for all z ∈ C. For the parameter s, we shall often denote σ = Re s. Also, we shall use the notation C * = C \ {0}.

The Lerch transcendent from the point of view of Fourier analysis

“…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…”

“…The relationships between (1.1) and Fourier series like (1.2) are well known and usually proved via complex analytic methods involving contour integrals. In previous papers (see [11,12]), we have derived some properties of the Lerch function based on Fourier series by direct calculation of the Fourier coefficients. Observe, for instance, the simplicity of the following reasoning.…”

General convolution identities for Apostol-Bernoulli, Euler and Genocchi polynomials

“…It is worth mentioning that the Apostol-Bernoulli polynomials were firstly introduced by Apostol [6] (see also Srivastava [36] for a systematic study) in order to evaluate the value of the Hurwitz-Lerch zeta function. For some nice methods and results on these polynomials and numbers, one is referred to [7,8,26,32,33]. This paper is organized as follows.…”

Remarks on Ritt operators, their $H^\infty$ functional calculus and associated square function estimates

“…where Li s (z) are the poly-logarithmic functions. By Wirtinger's theorem [27] (see also, for example [22]) it has the following asymptotics when |z| < 1, z → 1:…”