  When $p$ is an odd prime, Delbourgo observed that any Kubota-Leopoldt
$p$-adic $L$-function, when multiplied by an auxiliary Euler factor, can be
written as an infinite sum. We shall establish such expressions without
restriction on $p$, and without the Euler factor when the character is
nontrivial, by computing the periods of appropriate measures. As an
application, we will reprove the Ferrero-Greenberg formula for the derivative
$L_p'(0,\chi)$. We will also discuss the convergence of sum expressions in
terms of elementary $p$-adic analysis, as well as their relation to
Stickelberger elements; such discussions in turn give alternative proofs of the
validity of sum expressions.
