A formal asymptotic Hermite series expansion is developed for the conditional state density in a scalar estimation problem which differs from the standard Kalman filtering context only by the addition of a small quadratic term in the dynamicss. This case is one that might result from a higher order description of perturbations in a more general state estimation situation. The parameters of the expansion here are generated recursively from the measurement data by an initial value system of 3 n ordinary differential equations for an expansion of n th order asymptotic accuracy in the coefficient of the above quadratic term. To at least third order, this expansion has the special form of the standard Edgeworth expansion of corresponding order for the conditional state density, and its parameters can be expressed in terms of this density's cumulants, which are generated by a simpler set of recursive equations.