Loop homological invariants associated to real projective spaces

Let A be a based subspace of Y. Under the assumptions that Y is path-connected and that the reduced diagonal map of A   induces the zero map in all mod 2 reduced homology groups, we compute a formula for the mod 2 reduced Poincaré series of the loop space Ω((A∧RP∞)∪A∧RP1(Y∧RP1))Ω((A∧RP∞)∪A∧RP1(Y∧RP1)). Here RP∞RP∞ and RP1RP1 denote the infinite real projective space and the real projective line respectively.