Let X be a spectrally negative self-similar Markov process with 0 as an
absorbing state. In this paper, we show that the distribution of the absorption
time is absolutely continuous with an infinitely continuously differentiable
density. We provide a power series and a contour integral representation of
this density. Then, by means of probabilistic arguments, we deduce some
interesting analytical properties satisfied by these functions, which include,
for instance, several types of hypergeometric functions. We also give several
characterizations of the Kesten's constant appearing in the study of the
asymptotic tail distribution of the absorbtion time. We end the paper by
detailing some known and new examples. In particular, we offer an alternative
proof of the recent result obtained by Bernyk, Dalang and Peskir [Ann. Probab.
36 (2008) 1777--1789] regarding the law of the maximum of spectrally positive
L\'{e}vy stable processes.