Zeros of linear combinations of Laguerre polynomials from different sequences

We study interlacing properties of the zeros of two types of linear combinations of Laguerre polynomials with different parameters, namely R n = L n α + a L n α ′ and S n = L n α + b L n − 1 α ′ . Proofs and numerical counterexamples are given in situations where the zeros of R n , and S n , respectively, interlace (or do not in general) with the zeros of L k α , L k α ′ , k = n or n − 1 . The results we prove hold for continuous, as well as integral, shifts of the parameter α .