Partial Results on Hankel Determinants and the Corresponding J-Fractions of a Sequence Related to Bernoulli Numbers

Abstract

When exploring the Hankel determinant of the sequence $μ_{k}=B_{k+1}/(k+1)$, where $B_{k}$ is the $k$-th Bernoulli number, we obtained two interesting results. The first one applies in general to all sequences $(c_{k})_{k\geq0}$ with all even-indexed term $0$, except for $c_{0}$. In this case, the coefficient of the second highest order of the corresponding monic orthogonal polynomials determines the Hankel determinants. Our second result shows, the corresponding J-fractions, obtained from the generating function of $μ_{k}$, is exactly the same as in early work of Cao, on a faster sequence converging to the Euler--Mascheroni constant.