Zernike Polynomials - Higher Dimensions

Higher Dimensions

The concept translates to higher dimensions if multinomials in Cartesian coordinates are converted to hyperspherical coordinates, multiplied by a product of Jacobi Polynomials of the angular variables. In dimensions, the angular variables are Spherical harmonics, for example. Linear combinations of the powers define an orthogonal basis satisfying

.

(Note that a factor is absorbed in the definition of here, whereas in the normalization is chosen slightly differently. This is largely a matter of taste, depending on whether one wishes to maintain an integer set of coefficients or prefers tighter formulas if the orthogonalization is involved.) The explicit representation is

R_n^{(l)}(\rho) = \sqrt{2n+D}\sum_{s=0}^{(n-l)/2} (-1)^s{(n-l)/2 \choose s}{n-s-1+D/2 \choose (n-l)/2}\rho^{n-2s}
=(-1)^{(n-l)/2}\sqrt{2n+D}\sum_{s=0}^{(n-l)/2} (-1)^s{(n-l)/2 \choose s} {s-1+(n+l+D)/2 \choose (n-l)/2} \rho^{2s+l}
=(-1)^{(n-l)/2}\sqrt{2n+D}{ (D+n+l)/2-1 \choose (n-l)/2}\rho^l {}_2F_1( -(n-l)/2,(n+l+D)/2;l+D/2;\rho^2)

for even, else identical to zero.

Read more about this topic:  Zernike Polynomials

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