By decoupling per azimuthal order $m=0,1,\u2026$, and normalization and orthogonality^{4} of $Rnm(\rho ),n=m,m+2,\u2026$, the Zernike coefficients $\alpha nm(\u03f5)$ of $\Phi (\rho \u03f5,\theta )$ and the Zernike coefficients $\alpha nm$ of $\Phi (\rho ,\theta )$ are related byDisplay Formula
10$\alpha nm(\u03f5)=2(n+1)\u2211n\u2032\alpha n\u2032mMnn\u2032m(\u03f5),n=m,m+2,\u2026.$
The summation in Eq. 10 is over $n\u2032=m,m+2,\u2026$, andDisplay Formula11$Mnn\u2032m(\u03f5)=\u222b01Rnm(\rho )Rn\u2032m(\rho \u03f5)\rho d\rho ,n,n\u2032=m,m+2,\u2026.$
We shall show thatDisplay Formula12$Mnn\u2032m(\u03f5)=12(n+1)[Rn\u2032n(\u03f5)\u2212Rn\u2032n+2(\u03f5)];$
in particular, it follows that $Mnn\u2032m(\u03f5)=0$ when $n\u2032<n$ and that $Mnn\u2032m(\u03f5)$ does not depend on $m$, except that in Eq. 10 we only use $n,n\u2032=m,m+2,\u2026$. For this we use^{6}Display Formula13$Rlk(\rho )=(\u22121)(l\u2212k)\u22152\u222b0\u221eJl+1(r)Jk(\rho r)dr,0\u2a7d\rho <1,$
when $k,l$ are integers $\u2a7e0$ with same parity; in the case of $k\u2212l>0$, the right-hand side of Eq. 13 vanishes, which is consistent with the convention that then $Rlk\u22610$. We use Eq. 13 in Eq. 11 to rewrite $Rn\u2032m(\rho \u03f5)$ and interchange integrals to getDisplay Formula14$Mnn\u2032m(\u03f5)=(\u22121)(n\u2032\u2212m)\u22152\u222b0\u221eJn\u2032+1(r)[\u222b01Rnm(\rho )Jm(\rho \u03f5r)\rho d\rho ]dr.$
To the inner integral we apply the resultDisplay Formula15$\u222b01Rnm(\rho )Jm(\rho v)\rho d\rho =(\u22121)(n\u2212m)\u22152[Jn+1(v)v]$
from the Nijboer-Zernike theory,^{4} and we getDisplay Formula16$Mnn\u2032m(\u03f5)=(\u22121)(n\u2032+n\u22122m)\u22152\u222b0\u221eJn\u2032+1(r)Jn+1(\u03f5r)\u03f5rdr.$
Next we use the identity^{7}Display Formula17$Jn+1(\u03f5r)\u03f5r=Jn(\u03f5r)+Jn+2(\u03f5r)2(n+1),$
and use Eq. 13 to rewrite the resulting two integrals in terms of Zernike polynomials. This gives Eq. 12.