Another discrete Fourier transform pairs associated with the Lipschitz-Lerch zeta function

In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: Keywords: Discrete Fourier transform Alternating zeta functions Hurwitz–Lerch zeta function Lipschitz–Lerch zeta function Lerch zeta function Hurwitz zeta function Legendre chi function Bernoulli polynomials a b s t r a c t It is demonstrated that the alternating Lipschitz–Lerch zeta function and the alternating Hurwitz zeta function constitute a discrete Fourier transform pair. This discrete transform pair makes it possible to deduce, as special cases and consequences, many (mainly new) transformation relations involving the values at rational arguments of alternating variants of various zeta functions, such as the Lerch and Hurwitz zeta functions and Legendre chi function. 1. Introduction and definitions In a recent note [1] it was shown that numerous (known or new) seemingly disparate results involving various special functions, such as the Lipschitz–Lerch zeta function, Lerch zeta function, Hurwitz zeta function and Legendre chi function, could be established in a more general context provided by a discrete Fourier transform (DFT). In this sequel to [1], we aim at deriving the analogous transformation formula which would be valid in the case of the alternating counterparts of these functions. The alternating Hurwitz–Lerch zeta function defined by (cf. [2, p. 121 et seq.])