An asymptotic evaluation of the specific heat of an ideal Bose gas confined to a thin-film geometry (∞×∞×D) is carried out, under a variety of boundary conditions, without converting summations into integrations. The theoretical results for the Dirichlet boundary conditions (ψS=0) contain all the significant features of the numerical results obtained earlier by Goble and Trainor; in particular, we reproduce the characteristic length D* at which the specific heat of the system is at an absolute maximum. It turns out that D* is directly proportional to the mean interparticle distance l̄, with a constant of proportionality c(~20-30). No such characteristic length appears if boundary conditions other than Dirichlet's are employed. The shift, and the rounding off, of the specific-heat maximum are also studied, and the distinguishing influence of the boundary conditions examined.