We establish new operational formulae of Burchnall type for the complex disk polynomials (generalized Zernike polynomials). We then use them to derive some interesting identities involving these polynomials. In particular, we establish recurrence relations with respect to the argument and the integer indices, as well as Nielsen identities and Runge addition formula. In addition, various new generating functions for these disk polynomials are proved.Notice that the normalization adopted here differs from the one adopted for the disk polynomials considered by Wünsche [13] and denoted P γ m,n (z,z) or by Dunkl [14, 15] and denoted R (γ) m,n (z). In fact, we have (see Remark 4.3): Z γ m,n (z,z) = (γ + 1) m+n P γ m,n (z,z) = (γ + 1) m+n R (γ) m,n (z).