We construct new rational approximants of Euler’s constant that improve those of
Aptekarev et al. (2007) and Rivoal (2009). The approximants are given in terms of
certain (mixed-type) multiple orthogonal polynomials associated with the exponential
integral. The dual family of multiple orthogonal polynomials leads to new rational
approximants of the Gompertz constant that improve those of Aptekarev et
al. (2007). Our approach is motivated by the fact that we can reformulate
Rivoal’s construction in terms of type-I multiple Laguerre polynomials of the
first kind by making use of the underlying Riemann–Hilbert problem. As a
consequence, we can drastically simplify Rivoal’s approach, which allows us to
study the Diophantine and asymptotic properties of the approximants more
easily.
