Taking the logarithm of binomial type sequences: linear approach

TL;DR

This paper develops a linear method to asymptotically expand the logarithms of binomial type sequences, avoiding complex calculations and providing a new approach to understanding their asymptotic behavior.

Contribution

It introduces a novel linear approach for deriving asymptotic expansions of logarithms of binomial type sequences, simplifying previous methods.

Findings

Derived formal asymptotic expansions for logarithms of binomial type sequences.

Established a linear method that bypasses complex expansion calculations.

Enhanced understanding of the asymptotic properties of these sequences.

Abstract

In this paper we obtain the formal asymptotic expansion of the logarithms $\ln p_s(\alpha)$ of $p_s(\alpha)$, which are canonical continuations of polynomials of binomial type $p_n(\alpha)$. Our approach is based on linear methods which do not require the calculation of expansions $(p_s(\alpha)\alpha^{-s}-1)^k$, as opposed to the direct logarithmization.