Journal of Computational and Applied Mathematics 173 (2005) 359–363 www.elsevier.com/locate/cam Letter to the Editor Two new asymptotic expansions of the ratio of two gamma functions Julio Abad, Javier Sesma ∗ Departamento de F sica Te orica, Facultad de Ciencias, University of Zaragoza, Zaragoza 50009, Spain Received 25 November 2003; received in revised form 9 February 2004 Abstract Formal expansions, giving as particular cases semiasymptotic expansions, of the ratio of two gamma func- tions are obtained. c 2004 Elsevier B.V. All rights reserved. MSC: 33E50; 41A60 Keywords: Gamma function; Asymptotic expansion The ratio of two gamma functions of (not very dierent) large arguments admits a well-known asymptotic expansion, (z + ) (z + ) ∼ z - ∞ n=0 (-1) n ( - ) n n! B (-+1) n () 1 z n as z →∞; (1) where the symbols B () n (x) stand for the generalized Bernoulli polynomials [5,7]. A modication of that expansion, tending to improve its computational eciency, was suggested by Fields [3], who adopted a related large parameter, namely w=z +(+-1)= 2, to obtain an expansion [5, Section 2.11, Eq. (14)] [7, Eq. (3.32)] that contains only even negative powers of the large parameter. Here, with a quite dierent purpose, we present two expansions, namely (z + ) (z + ) = w - ∞ n=0 ( - ) n n! 2 F 0 (-n;z + ;; w); (2) * Corresponding author. Tel.: +34-976-76-1265; fax: +34-976-76-1264. E-mail address: javier@unizar.es (J. Sesma). 0377-0427/$-see front matter c 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2004.03.017