The solutions of various time-independent one-speed half-space transport problems, such as the generalized Milne and albedo problems, are derived with the help of the Chandrasekhar S-function and the Case eigenfunctions. First the S-function is expressed by Chandrasekhar's H-function and Busbridge's q-polynomials. These polynomials can be separately determined by solving systems of linear equations. Using the S-function, the emerging distribution is obtained by simple physical arguments. Then, the distribution inside the medium is found by applying full-range orthogonality relations between the Case eigenfunctions.