Research Problems in Function Theory — Problem 2.29

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Fix $n\ge1$ and entire coefficient functions $a_1,\ldots,a_n$, and consider all entire solutions $y$ of \[y^{(n)}+a_1(z)y^{(n-1)}+\cdots+a_n(z)y=0.\] A function is $p$-valent in a disc if each finite value has at most $p$ preimages there, counted with multiplicity; levels for which the function minus that level is identically zero are excluded from the counting assertion.

Statement recorded in the supplied source.

Find a characterisation, in terms of the value distribution of the solutions, of when all coefficient functions $a_1,\ldots,a_n$ are polynomials, analogous to the constant-coefficient bounded-value-distribution characterisation of Problem 2.28. [S2] Update 2.29 supplies Tijdeman's characterisation: the coefficients are polynomials exactly when there are fixed $p\in\mathbb N$ and $q\ge0$ such that every nonconstant solution is $p$-valent in each disc centred at $z_0$ of radius $1/(1+|z_0|^q)$. The same constants serve all solutions and all centres. [S2]

Short English statement

Characterise polynomial coefficients of a linear differential equation by uniform local valence properties of its solutions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302029 (AMR-022-2029). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 2, Problem 2.29 and Update 2.29; Problem 2.28, equation (2.5). https://arxiv.org/abs/1809.07200.