Poincar\'e polynomial of the moduli spaces of parabolic bundles

In this paper we use Weil conjectures (Deligne's theorem) to calculate the Betti numbers of the moduli spaces of semi-stable parabolic bundles on a curve. The quasi parabolic analogue of the Siegel formula, together with the method of Harder-Narasimhan filtration gives us a recursive formula for the Poincar\'e polynomials of the moduli. We solve the recursive formula by the method of Zagier, to give the Poincar\'e polynomial in a closed form. We also give explicit tables of Betti numbers in small rank, and genera.