  Zernike circular polynomials (ZCP) play a significant role in optics
engineering. The symbolic expressions for ZCP are valuable for theoretic
analysis and engineering designs. However, there are still two problems which
remain open: firstly, there is a lack of sufficient mathematical formulas of
the ZCP for optics designers; secondly the formulas for inter-conversion of
Noll's single index and Born-Wolf's double indices of ZCP are neither uniquely
determinate nor satisfactory. An automatic method for generating symbolic
expressions for ZCP is proposed based on five essential factors: the new
theorems for converting the single/double indices of the ZCP, the robust and
effective numeric algorithms for computing key parameters of ZCP, the symbolic
algorithms for generating mathematical expressions of ZCP, and meta-programming
\& \LaTeX{} programming for generating the table of ZCP. The theorems, method,
algorithms and system architecture proposed are beneficial to both optics
design process, optics software, computer-output typesetting in publishing
industry as well as STEM education.
