The physics of ultracold atomic gases is making progress at a rapid pace, which has led to a realization of Bosonfermion mixtures of atomic gases ͓1-4͔. Boson-fermion ͑BF͒ mixtures may exhibit the richest variety of phenomena of all. They may show very different behavior from pure fermion or pure Bose gases ͓5,6͔. Especially interesting is a possible instability of the mixture when there is an attraction between bosons and fermions ͓5,7-9͔, as a recent experiment in fact suggests a collapse of the mixture ͓10͔.In the present work we propose and study quite a different scenario for an attractively interacting boson-fermion mixture. To simplify the problem in a first survey we shall consider the situation where there is no interaction between atoms of the same kind. As we will discuss at the end of the paper, this is not a severe approximation to cases where the interaction between like atoms is repulsive. More precisely we want to address the question of what happens to a mixture of free fermions and bosons when a ͑tunable͒ attraction is switched on between fermions and bosons. We imagine that correlated BF pairs will be created. These BF pairs are composite fermions and as such these BF pairs should form a Fermi gas of composites. Besides in ultracold atomic gases such a situation can exist in other branches of physics. For example, in nuclear systems ͑e.g., neutron stars͒ of highdensity K − mesons and nucleons may form a gas of ⌳'s and the ⌳'s may then form a Fermi gas of their own ͓11͔. Or in a quark-gluon plasma additional quarks may bind to preformed diquarks or color Cooper pairs ͑the "bosons"͒ ͓12͔ to form a gas of nucleons in the so-called hadronization transition. Further examples may be added to this list.For a numerical example, we take a mixture of 40 K ͑fer-mion͒ and 41 K ͑boson͒ atoms throughout the paper. They are known as candidates for a realization of this kind of quantum systems. While their scattering lengths are not well fixed at present, and different values have been reported experimentally ͓13͔, it is not crucial at the moment because our study will be mostly academic, elaborating on the basic phenomenon. Applications to realistic systems will be left for the future.Let us consider a single BF pair propagating in the background of a homogeneous gas of free one-component fermions and spinless bosons. We will formulate our approach for a situation at finite temperature T, though later on in our application we will concentrate on the T = 0 case. We have in mind an analogous study Cooper performed a long time ago ͓14͔ for the propagation of two fermions ͑spin up or down͒ in the background of a homogeneous gas of two-component free fermions. In other words we consider a situation where in the original Cooper problem one fermion type ͑let us say spin down͒ is replaced by spinless bosons. The BF propagator at finite temperature T and finite center-of-mass momentum P of the pair that is added to the system with momenta P /2+p ͑fermion͒ and P /2−p ͑boson͒ is