In this work we propose a new filtering approach for linear discrete time non-Gaussian systems that generalizes a previous result concerning quadratic filtering [A. De Santis, A. Germani, and M. Raimondi, IEEE Trans. Automat. Control, 40 (1995) pp. 1274-1278]. A recursive nu th-order polynomial estimate of finite memory Delta is achieved by defining a suitable extended state which allows one to solve the filtering problem via the classical Kalman linear scheme. The resulting estimate will be the mean square optimal one among those estimators that take into account nu-polynomials of the last Delta observations. Numerical simulations show the effectiveness of the proposed method.