Rational Approximations via Hankel Determinants

Abstract

Define the monomials e n (x) := x n and let L be a linear functional. In this paper we describe a method which, under specified conditions, produces approximations for the value L(e 0 ) in terms of Hankel determinants constructed from the values L(e 1 ), L(e 2 ), . . . . Many constants of mathematical interest can be expressed as the values of integrals. Examples include the Euler-Mascheroni constant γ, the Euler-Gompertz constant δ, and the Riemann-zeta constants ζ(k) for k ≥ 2. In many cases we can use the integral representation for the constant to construct a linear functional for which L(e 0 ) equals the given constant and L(e 1 ), L(e 2 ), . . . are rational numbers. In this case, under the specified conditions, we obtain rational approximations for our constant. In particular, we execute this procedure for the previously mentioned constants γ, δ, and ζ(k). We note that our approximations are not strong enough to study the arithmetic properties of these constants.