  The partial derivatives and Laplacians of the Zernike circle polynomials
occur in various places in the literature on computational optics. In a number
of cases, the expansion of these derivatives and Laplacians in the circle
polynomials are required. For the first-order partial derivatives, analytic
results are scattered in the literature, starting as early as 1942 in Nijboer's
thesis and continuing until present day, with some emphasis on recursive
computation schemes. A brief historic account of these results is given in the
present paper. By choosing the unnormalized version of the circle polynomials,
with exponential rather than trigonometric azimuthal dependence, and by a
proper combination of the two partial derivatives, a concise form of the series
expressions emerges. This form is appropriate for the formulation and solution
of a model wave-front sensing problem of reconstructing a wave-front on the
level of its expansion coefficients from (measurements of the expansion
coefficients of) the partial derivatives. It turns out that the least-squares
estimation problem arising here decouples per azimuthal order $m$, and per $m$
the generalized inverse solution assumes a concise analytic form, thereby
avoiding SVD-decompositions. The preferred version of the circle polynomials,
with proper combination of the partial derivatives, also leads to a concise
analytic result for the Zernike expansion of the Laplacian of the circle
polynomials. From these expansions, the properties of the Laplacian as a
mapping from the space of circle polynomials of maximal degree $N$, as required
in the study of the Neumann problem associated with the Transport-of-Intensity
equation, can be read off within a single glance. Furthermore, the inverse of
the Laplacian on this space is shown to have a concise analytic form.
