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

When exploring the Hankel determinant of the sequence ${\mu }_k=B_{k+1}\mathrm{/}(k+1)$, where $B_k$ is the $k$-th Bernoulli number, we obtained two interesting results. The first one applies in general to all sequences $(c_k{)}_{k\ge 0}$ 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 ${\mu }_k$, is exactly the same as in early work of Cao, on a faster sequence converging to the Euler–Mascheroni constant.