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

## Examples

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

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