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In these presentations (wikipedia and mathworld) the Zernike polynomials are presented in a particular formula and it is stated that they are orthogonal w.r.t. a particular inner product $\langle{}\cdot{},{}\cdot{}\rangle$. I find this development unsatisfying.

Is it possible to derive the Zernike polynomials from something else?

For example, can one apply Gram-Schmidt to a particular basis (say, $\{\rho^n e^{im\theta} \}_{n,m}$) and arrive at the Zernike polynomials?

Alternatively, can one use separation of variables on a particular P.D.E. to obtain them?

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