I went with the solution of the MaCarlo suggested. Since this was a requirement for a computational issue the small errors for N, M larger than 10 or so were no problem. As for as i sought, i did not find an exact result.

Thank you, it seems that i have some accuracy issues with my code. Thank you for the clarification.

Thanks for the quick reply. I was incomplete in my question. I am aware of the MP distribution, but i am looking for a general pdf or marginal distribution and not the limiting case since the applications requires all sizes of matrices.