On Hadamard type polynomial convolutions with regularly varying sequences

Abstract

For a sequence of polynomials$P_n(x):=\sum _{m\le n}{p}_m{x}^m$, $n\ge 1$, we give a necessary and sufficient condition for the asymptotic equivalence$$P_n^{(\alpha )}(x):=\sum _{m\le n}{c}_m{p}_m{x}^m\sim c_n{P}_n(x)(n\to ınfty),$$to hold for each $x\ge A$ and an arbitrary regularly varying sequence $\{c_n\}$of index $\alpha ınR$.