Asymptotic expansions of Gurland's ratio and sharp bounds for their remainders

In this paper, we establish a new asymptotic expansion of Gurland's ratio of gamma functions, that is, as x -> infinity, Gamma (x + p) Gamma (x + q)/Gamma (x + (p + q)/2)(2 )= exp [Sigma B-n (k=1)2k (s) - B-2k (1/2)/k (2k - 1) (x + r(0))(2k-1) + R-n (x;p, q)] ( )where p, q is an element of R with w = vertical bar p - q vertical bar not equal 0 and s = (1 - w) 12, r(0) = (p + q - 1) /2, B2n+1 (s) are the Bernoulli polynomials. Using a double inequality for hyperbolic functions, we prove that the function x bar right arrow (-1)(n) R-n (x; p, q) is completely monotonic on (-r(0), infinity) if vertical bar p - q vertical bar < 1, which yields a sharp upper bound for vertical bar R-n (x; p, q)vertical bar. This shows that the approximation for Gurland's ratio by the truncation of the above asymptotic expansion has a very high accuracy. We also present sharp lower and upper bounds for Gurland's ratio in terms of the partial sum of hypergeometric series. Moreover, some known results are contained in our results when q -> p. (C) 2020 Elsevier Inc. All rights reserved.