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 $$ P_n^{(\alpha)}(x):=\sum_{m\le n}c_mp_mx^m\sim c_nP_n(x) \quad (n\to\infty), $$ to hold for each $x\ge A$ and an arbitrary regularly varying sequence $\{c_n\}$ of index $\alpha\in R$.

Keywords: Regular variation, polynomials, asymptotic behavior.

MSC: 26A12

Pages:  13$-$17     

Volume  58 ,  Issue  1$-$2 ,  2006