568

**Density and Generating Functions of a Limiting
Distribution for Quicksort**

**Kok Hooi Tan and Petros Hadjicostas**

### Abstract:

*Let be the random number of comparisons needed to sort a list of length
by Quicksort. Régnier (1989), Rösler (1991), and Hennequin
(1991) showed that converges in distribution to some
random variable . In addition, Hennequin derived a formula for the
cumulants of . Using results of the above authors we derive properties of
the generating functions of . Analytical formulas for these functions are
derived, but the formulas are not in closed form. Using the method of
successive substitution (Eddy and Schervish, 1992), we obtain numerical
estimates for some of the generating functions. Some theoretical aspects of
the last method are investigated. In addition, we prove
that is absolutely continuous with respect to the Lebesgue measure, and
that its distribution is supported on the entire real line.
**
***Keywords:** Gamma function, generating functions,
limiting distribution, Quicksort, Stirling numbers, successive substitution,
zeta function.