  For a family of compact Riemann surfaces X_t of genus g>1 parametrized by the
Schottky space S_g, we define a natural basis for the holomorphic
n-differentials on X_t which varies holomorphically with t and generalizes the
basis of normalized abelian differentials of the first kind for n=1. We
introduce a holomorphic function F(n) on S_g which generalizes the classical
product \prod(1-q^m)^2 appearing in the Dedekind eta function for n=1 and g=1.
We prove a holomorphic factorization formula expressing the regularized
determinant of the Laplacian as a product of |F(n)|^2, a holomorphic anomaly
depending on the classical Liouville action (a Kahler potential of S_g), and
the determinant of the Gram matrix of the natural basis. The factorization
formula reduces to Kronecker's first limit formula when n=1 and g=1, and to
Zograf's factorization formula for n=1 and g>1.
