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The even and odd Zernike polynomials are defined as follows: $Z^{m}_n(\rho,\varphi) = R^m_n(\rho)\,\cos(m\,\varphi) \!$ and: $Z^{-m}_n(\rho,\varphi) = R^m_n(\rho)\,\sin(m\,\varphi), \!$ with: $R^m_n(\rho) = \! \sum_{k=0}^{(n-m)/2} \!\!\! \frac{(-1)^k\,(n-k)!}{k!\,((n+m)/2-k)!\,((n-m)/2-k)!} \;\rho^{n-2\,k}$ My question: is there a way to express the Zernike polynomials in terms of Legendre polynomials? Thanks in advance

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    Why do you want these in terms of Legendres? The radial polynomials are defined over $[0,1]$ while the Legendres are defined over $[-1,1]$.2013-04-10

1 Answers 1

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First we rewrite your definition for the radial Zernike polynomial in a more convenient form:

$\mathcal R_n^m(\rho)=\sum_{k=0}^{(n-m)/2}(-1)^k \binom{n-k}{k} \binom{n-2k}{(n-m)/2-k} \rho^{n-2k}$

Now, you are asking about how to expand the radial Zernike polynomial as a Legendre series. We first note that $\mathcal R_n^m(\rho)$ can be expressed solely in terms of odd-order Legendre polynomials for odd $n,m$, and in terms of even-order Legendre polynomials for even $n,m$. (Recall also that the radial Zernike polynomials are identically zero if $n,m$ are not of the same parity.)

Now, for the Legendre expansion

$\mathcal R_n^m(\rho)=\sum_{k=0}^n c_k P_k(\rho)$

where the coefficients are given by

$c_k=\left(k+\frac12\right)\int_{-1}^1 \mathcal R_n^m(t)P_k(t)\,\mathrm dt$

we can derive an expression for $c_k$ by inserting the series definition of $\mathcal R_n^m(\rho)$ into the integral expression of $c_k$, and then using the identity

$\int_{-1}^1 u^{n-2j}P_k(u)\,\mathrm du=\frac2{k+1}\frac{\tbinom{(n-2j-1)/2}{(k-1)/2}}{\tbinom{(n-2j+k+1)/2}{(k+1)/2}}$

to yield the expression

$c_k=\frac{2k+1}{k+1}\sum_{j=0}^{(n-m)/2}(-1)^j \frac{\tbinom{n-j}{j}\tbinom{n-2j}{(n-m)/2-j}\tbinom{(n-2j-1)/2}{(k-1)/2}}{\tbinom{(n-2j+k+1)/2}{(k+1)/2}}$

$c_k$ can be expressed in terms of a ${}_4 F_3$ hypergeometric function, but I'll omit that expression for now.

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    That's why I applaud your effort: I played around with this stuff many moons ago and did not wish to do so again.2013-04-10