  In work of C. A. Tracy and the author asymptotic formulas were derived for
certain operator determinants which gave solutions to the cylindrical Toda
equations. Later the author considered a more general class of operators which
retained some of the properties of those cited and found analogous asymptotics
for the determinants. These results applied only in what we call the regular
case, when the symbol of an associated convolution operator is nonzero and has
zero index. The present work goes further and establishes asymptotics for a
class of singular cases.
