  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.
