Zernike coefficients for concentric, circular scaled pupils: an equivalent expression

We present an alternative formal calculation of the scaled Zernike coefficient expansion by means of the inner product of the Zernike polynomials and the wavefront error corresponding to the scaled pupil. The relationship exhibited by the radial polynomials and Bessel functions leads to a general expression in terms of the Gauss hypergeometric function. Direct properties and index selection rules are established, and easy derivation of the non-normalized coefficients is also straightforward.