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

Orthogonal polynomials and Hankel Determinants for certain Bernoulli and Euler Polynomials