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ZernikeR, or Zernike polynomial is eigendunction of the Bessel transform.

Explicit representation

For integer value of parameter \(m\) of the Bessel transform, the Zernike polynomial is expressed as follows:

\(\!\!\!\!\!\!\!\!\!\! (1) \displaystyle ~ ~ ~ R_n(x)= \sum_{k=0}{(n-m)/2} \frac{(-1)^k ~ (n-k)!} {k! ~ \left( \frac{n+m}{2}-k \right) ! ~ \left( \frac{n-m}{2}-k \right) ! } x^{n-2k} \)

at least for integer values of \((n\!-\! m)/2\). [1] The Zernike polynomial is implemented in Mathematica, and can be called as

\( \rm ZernikeR[n,\nu,x]\)

where \(n\) is order of the polynomial, \(\nu\) is parameter of the corresponding Bessel transform and the last argument \(x\) is just argument of the Zernike polynomial. However, not all versions of Mathematica support this option.


For the principal mode (\(mm=0\)), the first Zernike polynomials are

\( R_0(x)=1\)
\( R_2(x)=-1+2 x^2\)
\( R_4(x)=1-6x^2+6x^4\)
\( R_6(x)=-1+12 x^2-30 x^4+20 x^6\)