Zernike and Legendre polynomials 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
 A: 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.
