  Using continued fraction expansions of certain polygamma functions as a main
tool, we find orthogonal polynomials with respect to the odd-index Bernoulli
polynomials $B_{2k+1}(x)$ and the Euler polynomials $E_{2k+\nu}(x)$, for
$\nu=0, 1, 2$. In the process we also determine the corresponding Jacobi
continued fractions (or J-fractions) and Hankel determinants. In all these
cases the Hankel determinants are polynomials in $x$ which factor completely
over the rationals.
