In this paper,
Honarvar and Paramesran derive an interesting method to compute the radial Zernike polynomials in a very nice recursive way.
The recursion formula is surprisingly straightforward, without division or multiplication by large integers:
$$ R^m_n(\rho) = \rho
\left(R^{|m-1|}_{n-1}(\rho)+R^{m+1}_{n-1}(\rho)\right) - R^{m}_{n-2}(\rho)$$
I'd recommend to have a look at figure 1 in the Honarvar and Paramesran paper, which clearly illustrates the dependencies between the different Zernike polynomials.
This is implemented in the following Octave script:
clear % Tested with Octave instead of Matlab
N = 120;
n_r = 1000;
R = cell(N+1,N+1);
rho = [0:n_r]/n_r;
rho_x_2 = 2*[0:n_r]/n_r;
R{0+1,0+1} = ones(1,n_r+1); % R^0_0 Unfortunately zero based cell indexing is not possible
R{1+1,1+1} = R{0+1,0+1}.*rho; % R^1_1 ==> R{...+1,...+1} etc.
for n = 2:N,
if bitget(n,1) == 0, % n is even
R{0+1,n+1} = -R{0+1,n-2+1}+rho_x_2.*R{1+1,n-1+1}; % R^0_n
m_lo = 2;
m_hi = n-2;
else
m_lo = 1;
m_hi = n-1;
end
for m = m_lo:2:m_hi,
R{m+1,n+1} = rho.*(R{m-1+1,n-1+1}+R{m+1+1,n-1+1})-R{m+1,n-2+1}; % R^m_n
end
R{n+1,n+1} = rho.*R{n-1+1,n-1+1}; % R^n_n
end;
Z = @(m,n,rho) (-1)^((n-m)/2) * rho.^m .* jacobiPD((n-m)/2,m,0,1-2*rho.^2);
m = 22;
n = 112;
figure
plot(rho,Z(m,n,rho))
hold on
plot(rho,R{m+1,n+1},'r');
xlabel("rho")
ylabel("R^{22}_{112}(rho)")
legend("via Jacobi","recursive");
%print -djpg plt.jpg
m = 0;
n = 46;
max_diff_m_0_n_46 = norm(Z(m,n,rho)-R{m+1,n+1},inf)
For example, the figure produced by this code shows that with $m = 22$, and $n = 112$, catastrophic cancellation occurs near $\rho = 0.7$, if Zernike radial polynomials are computed via Jacobi polynomials.
Therefore, one also has to worry about the accuracy of the lower-degree Zernike
polynomials.

The recursive method seems to be much more suitable for computing these higher-order Zernike polynomials in a stable way.
Nevertheless, for $m = 0$ and $n = 46$, the maximum difference between the Jacobi and the recursive method is (only?) 1.4e-10
, which might be accurate enough for your application.