Integral representations and complete monotonicity related to the remainder of Burnside's formula for the gamma function

In the paper, the authors establish integral representations of some functions related to the remainder of Burnside’s formula for the gamma function and find the (logarithmically) complete monotonicity of these and related functions. These results extend and generalize some known conclusions. 1. Motivation and main results A function f is said to be completely monotonic on an interval I if f has derivatives of all orders on I and (−1)nf (x) ≥ 0 for all x ∈ I and n ≥ 0. A function f is said to be absolutely monotonic on an interval I if f has derivatives of all orders on I and f (x) ≥ 0 for all x ∈ I and n ≥ 0. A positive function f(x) is said to be logarithmically completely monotonic on an interval I ⊆ R if it has derivatives of all orders on I and its logarithm ln f(x) satisfies (−1)k[ln f(x)] ≥ 0 for all k ∈ N on I. For more information on these kinds of functions, please refer to the papers and monographs [7, 18, 27, 32, 36, 40, 44] and plenty of references cited therein. It is well known that the classical Euler’s gamma function may be defined by Γ(x) = ∫ ∞ 0 te dt (1.1) for x > 0. The logarithmic derivative of Γ(x), denoted by ψ(x) = Γ ′(x) Γ(x) , is called the psi or digamma function, and ψ(x) for k ∈ N are called the polygamma functions. The noted Binet’s formula [17, p. 11] states that ln Γ(x) = (