TY - JOUR

T1 - Value Sharing and Stirling Numbers

AU - Hinkkanen, Aimo

AU - Laine, Ilpo

N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.

PY - 2024

Y1 - 2024

N2 - Let f be an entire function and L(f) a linear differential polynomial in f with constant coefficients. Suppose that f, f′, and L(f) share a meromorphic function α(z) that is a small function with respect to f. A characterization of the possibilities that may arise was recently obtained by Lahiri. However, one case leaves open many possibilities. We show that this case has more structure than might have been expected, and that a more detailed study of this case involves, among other things, Stirling numbers of the first and second kinds. We prove that the function α must satisfy a linear homogeneous differential equation with specific coefficients involving only three free parameters, and then f can be obtained from each solution. Examples suggest that only rarely do single-valued solutions α(z) exist, and even then they are not always small functions for f.

AB - Let f be an entire function and L(f) a linear differential polynomial in f with constant coefficients. Suppose that f, f′, and L(f) share a meromorphic function α(z) that is a small function with respect to f. A characterization of the possibilities that may arise was recently obtained by Lahiri. However, one case leaves open many possibilities. We show that this case has more structure than might have been expected, and that a more detailed study of this case involves, among other things, Stirling numbers of the first and second kinds. We prove that the function α must satisfy a linear homogeneous differential equation with specific coefficients involving only three free parameters, and then f can be obtained from each solution. Examples suggest that only rarely do single-valued solutions α(z) exist, and even then they are not always small functions for f.

KW - Linear differential polynomials

KW - Primary 30D35

KW - Secondary 11B73

KW - Stirling numbers

KW - Uniqueness theory

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U2 - 10.1007/s40315-024-00552-5

DO - 10.1007/s40315-024-00552-5

M3 - Article

AN - SCOPUS:85196644761

SN - 1617-9447

JO - Computational Methods and Function Theory

JF - Computational Methods and Function Theory

ER -