Value sharing and Stirling numbers (Q6618174)

Let \(f\) be an entire function and \(L(f)\) be a linear differential polynomial in \(f\) with constant coefficients. Suppose that \(\alpha = \alpha (z)\) is a small function with respect to \(f\). If \(f\), \(f'\) share \(\alpha\) with counting multiplicities and \(f\), \(L(f)\) share \(\alpha\) with counting multiplicities except in a finite set, then the reviewer [Comput. Methods Funct. Theory 23, No. 3, 393--416 (2023; Zbl 1521.30040)] obtained a set of possible characterisation of \(f\) and \(L(f)\). However, one possibility does not provide any specific form of \(f\). In the paper, the authors analyse this possibility in detail with the aid of Stirling numbers and establish that the possibility may appear in some special cases only.