ON HADAMARD TYPE POLYNOMIAL CONVOLUTIONS WITH REGULARLY VARYING SEQUENCES

For a sequence of polynomials P-n(x) := Sigma(m <= n) p(m)x(m), n >= 1, we give a necessary and sufficient condition for the asymptotic equivalence P-n((alpha)) (x) := Sigma(m <= n) c(m)p(m)x(m) similar to c(n)P(n)(x) (n -> 8), to hold for each x >= A and an arbitrary regularly varying sequence {c(n)} of index alpha is an element of R.