On Hadamard type polynomial convolutions with regularly varying sequences. (English) Zbl 1119.26002

Let \(P_{n}(x)=\sum_{m\leq n}p_mx^m\), \(n\geq1\), \(P_n^\alpha(x)=\sum_{m\leq n}c_mp_mx^m\), \(x\geq A>0\), where \(c_n\) is an arbitrary regularly varying sequence of index \(\alpha\in R\). The author proves that the asymptotic relation \(P_n^\alpha(x)\sim n^\alpha l_nP_n(x)\), \(n\to\infty\), holds if and only if \(\frac{AP'_n(A)}{P_n(A)}\sim n\), \(n\to\infty\), where \(l_n\) is an arbitrary slowly varying sequence. Some sufficient conditions for this asymptotic equivalence to hold and not to hold are presented too.


26A12 Rate of growth of functions, orders of infinity, slowly varying functions
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