We study properties of the monic polynomials {Q n } n∈N orthogonal with respect to the Sobolev inner product(p, q)S=∫∞0(p, p′)1μμλqq′xαe−xdx,where λ−μ 2 >0 and α>−1. This inner product can be expressed as(p, q)S=∫∞0p(x)q(x)((μ+1)x−αμ)xα−1e−xdx+λ∫∞0p′q′xαe−xdx,when α>0. In this way, the measure which appears in the first integral is not positive on [0, ∞) for μ∈R\[−1, 0]. The aim of this paper is the study of analytic properties of the polynomials Q n . First we give an explicit representation for Q n using an algebraic relation between Sobolev and Laguerre polynomials together with a recursive relation for k n =(Q n , Q n ) S . Then we consider analytic aspects. We first establish the strong asymptotics of Q n on C\[0, ∞) when μ∈R and we also obtain an asymptotic expression on the oscillatory region, that is, on (0, ∞). Then we study the Plancherel–Rotach asymptotics for the Sobolev polynomials Q n (nx) on C\[0, 4] when μ∈(−1, 0]. As a consequence of these results we obtain the accumulation sets of zeros and of the scaled zeros of Q n . We also give a Mehler–Heine type formula for the Sobolev polynomials which is valid on compact subsets of C when μ∈(−1, 0], and hence in this situation we obtain a more precise result about the asymptotic behaviour of the small zeros of Q n . This result is illustrated with three numerical examples.