Hermite–Padé Approximants to Exponential Functions and an Inequality of Mahler Original Research Article

We improve Mahlerʹs inequalityimagewheregis any sufficiently large positive integer by decreasing the constant 33 to 19.183. This we do by computing precise asymptotics for a set of approximants to the exponential which is slightly different from the classical Hermite–Padé approximants. These approximants are related to the Legendre-type polynomials studied by Hata, which allows us to use his results about the arithmetic of the coefficients.