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A note on the asymptotic expansion of the Lerch's transcendent
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Univ Publ Navarra, Dept Estadist Matemat & Informat, Pamplona, Spain;Univ Publ Navarra, INAMAT, Pamplona, Spain.
2019 (English)In: Integral transforms and special functions, ISSN 1065-2469, E-ISSN 1476-8291, Vol. 30, no 10, p. 844-855Article in journal (Refereed) Published
Abstract [en]

In Ferreira and Lopez [Asymptotic expansions of the Hurwitz-Lerch zeta function. J Math Anal Appl. 2004;298(1):210-224], the authors derived an asymptotic expansion of the Lerch's transcendent Phi(z, s, a) for large vertical bar a vertical bar, valid for Ra > 0, Rs > 0 and z is an element of C \ [1, infinity). In this paper, we study the special case z >= 1 not covered in Ferreira and Lopez [Asymptotic expansions of the Hurwitz-Lerch zeta function. J Math Anal Appl. 2004; 298(1): 210-224], deriving a complete asymptotic expansion of the Lerch's transcendent Phi(z, s, a) for z > 1 and Rs > 0 as Ra goes to infinity. We also show that when a is a positive integer, this expansion is convergent for Rz >= 1. As a corollary, we get a full asymptotic expansion for the sum Sigma(m)(n=1) z(n)/n(s) for fixed z > 1 as m -> infinity. Some numerical results show the accuracy of the approximation.

Place, publisher, year, edition, pages
TAYLOR & FRANCIS LTD , 2019. Vol. 30, no 10, p. 844-855
Keywords [en]
Hurwitz-Lerch zeta function, asymptotic expansion, special functions
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:uu:diva-396071DOI: 10.1080/10652469.2019.1627530ISI: 000478210200001OAI: oai:DiVA.org:uu-396071DiVA, id: diva2:1366680
Funder
Knut and Alice Wallenberg FoundationAvailable from: 2019-10-30 Created: 2019-10-30 Last updated: 2019-10-30Bibliographically approved

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