A note on the asymptotic expansion of the Lerch's transcendent. Issue 10 (3rd October 2019) - Cai, Xing Shi; López, José L.
ABSTRACT: In Ferreira and López [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 Φ ( z, s, a ) for large | a |, valid for ℜ a > 0, ℜ s > 0 and z ∈ C ∖ [ 1, ∞ ) . In this paper, we study the special case z ≥ 1 not covered in Ferreira and López [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 Φ ( z, s, a ) for z >1 and ℜ s > 0 as ℜ a goes to infinity. We also show that when a is a positive integer, this expansion is convergent for ℜ z ≥ 1 . As a corollary, we get a full asymptotic expansion for the sum ∑ n = 1 m z n / n s for fixed z > 1 as m → ∞ . Some numerical results show the accuracy of the approximation.