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The Lerch zeta and related functions of non-positive integer order

Author: Djurdje Cvijovic
Journal: Proc. Amer. Math. Soc. 138 (2010), 827-836
MSC (2000): Primary 11M35, 33E20; Secondary 11B83
Published electronically: October 22, 2009
MathSciNet review: 2566548
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Abstract: It is known that the Lerch (or periodic) zeta function of non-positive integer order, $ \ell_{-n}(\xi)$, $ n\in \mathbb{N}_{0}:= \{0, 1, 2, 3, \ldots\}$, is a polynomial in $ \cot(\pi \xi)$ of degree $ n + 1$. In this paper, a very simple explicit closed-form formula for this polynomial valid for any degree is derived. In addition, novel analogous explicit closed-form formulae for the Legendre chi function, the alternating Lerch zeta function and the alternating Legendre chi function are established. The obtained formulae involve the Carlitz-Scoville tangent and secant numbers of higher order, and the derivative polynomials for tangent and secant are used in their derivation. Several special cases and consequences are pointed out, and some examples are also given.

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Additional Information

Djurdje Cvijovic
Affiliation: Atomic Physics Laboratory, Vinča Institute of Nuclear Sciences, P.O. Box 522, 11001$ Belgrade, Republic of Serbia

Keywords: Lerch zeta function, Legendre chi function, alternating Lerch zeta function, alternating Legendre chi function, derivative polynomials, tangent numbers of order $k$, secant numbers of order $k$, higher (generalized) tangent numbers, higher (generalized) secant numbers.
Received by editor(s): April 14, 2009
Received by editor(s) in revised form: July 12, 2009
Published electronically: October 22, 2009
Communicated by: Walter Van Assche
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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