On Hadamard type polynomial convolutions with regularly varying sequences

Abstract

For a sequence of polynomials$P_n(x):=\sum_{m\le n}p_mx^m$, $n\ge 1$, we give a necessary and sufficient condition for the asymptotic equivalence$$P_n^{(\alpha)}(x):=\sum_{m\le n}c_mp_mx^m\sim c_nP_n(x) \quad (n\toınfty),$$to hold for each $x\ge A$ and an arbitrary regularly varying sequence $\{c_n\}$of index $\alphaın R$.