This paper deals with polynomials \(L_ n(x)\) orthonormal with respect to the weight function \(| x|^{2\alpha}(b+x)^{\beta}e^{-x}\) on \((a,+\infty)\), \(a\leq 0\), \(\alpha >0\), \(\beta >0\) and \(b+a>0\). The author uses techniques already known to \textit{J. A. Shohat} [Duke Math. J. 5, 401-417 (1939; Zbl 0021.30802)] to show that the coefficients \(q_ n\) and \(j_ n\) in the three term recurrence relation \[ xL_ n(x)=q_{n+1}L_{n+1}(x)+j_ nL_ n(x)+q_ nL_{n-1}(x) \] are such that \(q_ n=n+O(1)\) and \(j_ n=2n+O(1)\). Similar techniques for weights on \((-\infty,+\infty)\) were used by \textit{G. Freud} [Proc. R. Irish Acad. 76, 1-6 (1976; Zbl 0327.33008)] and further developed by \textit{A. P. Magnus} [J. Approximation Theory 46, 65-99 (1986; Zbl 0619.42015)]. A nonhomogeneous second order differential equation is given for these orthogonal polynomials, but the nonhomogeneous term is not specified.