We give a purely algebraic proof that the rational functions $P_n(t),\, Q_n(t)$ inductively defined by the recurrence relation (1), (2) respectively, are polynomials. The proof reveals the Hirota bilinear relations satisfied by the $\tau$-functions.
"Special polynomials and the Hirota bilinear relations of the second and the fourth Painlevé equations." Nagoya Math. J. 159 179 - 200, 2000.