A new system of biorthogonal polynomials is developed for the angular expansion of the directional flux in the linear Boltzmann transport equation. It is shown in systems infinite in one space dimension that the angular integral in the Boltzmann equation can be reduced to a weighted integral over the unit circle. The corresponding system of orthogonal functions is found to be a system of two sets of polynomials in two variables. Recursion relations and an addition theorem are derived for these polynomials. The angular dependence of the particle flux is expanded in each set of these polynomials. Systems of partial differential equations are derived for the expansion coefficients, that is, for angular moments of the particle flux. One of these systems is shown to be a specific linear combination of the equations obtained when the directional flux is expanded in spherical harmonics functions specialized for the geometry considered. It is shown that this same system, in (x, y) geometry, reduces simply to the spherical harmonics equations in one-dimensional plane geometry.