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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.

MSC:

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