Solution of string equations for asymmetric potentials

We consider the large N   expansion of the partition function for the Hermitian one-matrix model. It is well known that the coefficients of this expansion are generating functions F(g)F(g) for a certain kind of graph embedded in a Riemann surface. Other authors have made a simplifying assumption that the potential V   is an even function. We present a method for computing F(g)F(g) in the case that V is not an even function. Our method is based on the string equations, and yields “valence independent” formulas which do not depend explicitly on the potential. We introduce a family of differential operators, the “string polynomials”, which make clear the valence independent nature of the string equations.