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 Rn=Lnα+aLnα′ and Sn=Lnα+bLn−1α′. Proofs and numerical counterexamples are given in situations where the zeros of RnRn, and SnSn, respectively, interlace (or do not in general) with the zeros of Lkα, Lkα′, k=nk=n or n−1n−1. The results we prove hold for continuous, as well as integral, shifts of the parameter αα.