Rational Approximations via Hankel Determinants

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 $\gamma$, the Euler-Gompertz constant $\delta$, and the Riemann-zeta constants $\zeta(k)$ for $k \ge 2$. 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 $\gamma$, $\delta$, and $\zeta(k)$. We note that our approximations are not strong enough to study the arithmetic properties of these constants.