  We consider the use of random walks as an approach to obtain connection
coefficients for higher-order Bernoulli and Euler polynomials. In particular,
we consider the cases of a $1$-dimensional linear reflected Brownian motion and
of a $3$-dimensional Bessel process. Considering the successive hitting times
of two, three, and four fixed levels by these random walks yields non-trivial
identities that involve higher-order Bernoulli and Euler polynomials.
