Let \(u_\alpha= e^{- x} x^\alpha\) for \(0< x< \infty\) and let \((l^\alpha_n)\), \(n= 0, 1, 2,\dots\), be a sequence of polynomials, orthogonal on \((0, \infty)\) with respect to the weight \(u_\alpha\). There is considered the space \(\check Z^\alpha\) of functions \(f\in C^\infty(0, \infty)\) such that for every \(p\in \mathbb{R}\) and \(m\in \mathbb{N}\) there exists \(c= c(m, p)\) such that \(|(1+ x^2)^p f^{(m)} \sqrt{u_\alpha}|\leq c\), with convergence \(f_n\to 0\) defined by convergence of \(f^{(m)}_n \sqrt{u_\alpha}\) to zero for every \(m\in \mathbb{N}\) locally uniformly on \((0, \infty)\), where for every \(p\in \mathbb{R}\) and \(m\in \mathbb{N}\) there exists \(c= c(m, p)\) independent of \(n\) for which \(|(1+ x^2)^p f^{(m)}_n \sqrt{u_\alpha}|\leq c\). There is also considered the space \(\check Z^{- \alpha}\) dual to \(\check Z^\alpha\). There are obtained necessary and sufficient conditions in order that a distribution \(\Gamma\) belongs to \(\check Z^{- \alpha}\). It is also shown that a formal series \(\sum^\infty_{k= 0} c_kl^\alpha_k\) is convergent in \(\check Z^{- \alpha}\) if and only if there exists an \(r\in \mathbb{R}\) such that \(c_k= O(k^r)\).