The gamma function via interpolation

A new computational framework for evaluation of the gamma function Γ(z) over complex plane is developed. The algorithm based on interpolation by rational functions, and generalizes classical methods Lanczos (SIAM J. Numer. Anal. B 1:86–96, (1964) Spouge Anal., 31(3):931–944, (1994) (which we show are also interpolatory). This utilizes exact poles function. By relaxing this condition allowing to vary, a near-optimal approximation possible, which demonstrated using adaptive Antoulous Anderson (AAA) algorithm, developed in Nakatsukasa et al. (Appl. Math. Comp., 40:1494-1522, (2016)) Trefethen Sci. 42(5):A3157–A3179, (2020)). resulting approximations competitive with Stirling’s formula terms overall efficiency.