On hadamard type polynomial convolutions with regularly varying sequences

Abstract

For a sequence of polynomials Pn(x) := Σ m≤npmxm, n ≥ 1, we give a necessary and sufficient condition for the asymptotic equivalence Pn(α) := Σm≤n cmpmxm ∼ cnP n(x) (n → ∞), to hold for each x ≥ A and an arbitrary regularly varying sequence {cn} of index α ∈ R.