ZernikeR

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 $(n\!-\! m)/2$. 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.

Examples

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$