Research Problems in Function Theory — Problem 2.74

Definitions and mathematical statement

A real entire function has real Taylor coefficients, equivalently $f(\overline z)=\overline{f(z)}$. An entire function has genus at most $q$ when its Hadamard representation uses canonical factors of genus at most $q$ and an exponential of a polynomial of degree at most $q$; finite products are allowed. For $p\ge0$, let $V_{2p}$ be the real entire functions expressible as \[ f(z)=e^{-az^{2p+2}}g(z),\qquad a\ge0, \] where $g$ is a real entire function of genus at most $2p+1$ whose zeros are all real. Put $U_0=V_0$ and $U_{2p}=V_{2p}\setminus V_{2p-2}$ for $p\ge1$. Fix $p\ge1$ and a nonpolynomial $f\in U_{2p}$ with $f(0)=1$, and write $f(z)=\sum_{n\ge0}a_nz^n$. Is it true that no $2p+2$ consecutive coefficients vanish? Explicitly, must every $m\ge0$ satisfy \[ (a_m,a_{m+1},\ldots,a_{m+2p+1})\ne(0,\ldots,0)? \] The nonpolynomial condition is necessary because a polynomial has an eventual string of zero coefficients. The question generalizes the two-consecutive-coefficient restriction for the corresponding nonpolynomial class $U_0$.

Short English statement

Does the real-zero class $U_{2p}$ prevent a nonpolynomial entire function from having $2p+2$ consecutive zero Taylor coefficients?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302074 (AMR-022-2074), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 2.74. Original source indexed by AMR-022-2074. https://arxiv.org/abs/1809.07200.