  We study Euler transforms of products of two incomplete beta functions,
giving closed forms in a variety of special cases, and series expansions of the
fully general case. The results are applied to derive the moments of the
maximum of two independent beta-distributed random variables and to find inner
products of incomplete beta functions. Restricted to positive integer
parameters, our results also provide solutions to a combinatorial problem of
enumerating strings comprised of three different letters, subject to
constraints.
