We introduce a method of estimating asymptotic behaviour of polynomials $Q_n^{(\alpha)}(x):=\sum_{k\le n}c_k a_{nk} x^k$, $n\to\infty$, related to a given polynomial $Q_n(x):=\sum_{k\le n} a_{nk}x^k$, where $(c_k)$, $k\in N$ is any regularly varying sequence of index $\alpha$ in the sense of Karamata. Then we apply our results to classical orthogonal polynomials as relevant examples.