Consecutive zero Taylor coefficients in Laguerre–Pólya subclasses (Hayman Problem 2.74)
Statement
Let $f$ be a real entire function of finite order with only real zeros. For $p\ge 0$, let $V_{2p}$ be the class of functions $g(z)\,e^{-a z^{2p+2}}$ with $a\ge 0$ and $g$ real entire of genus $\le 2p+1$ with only real zeros; set $U_0=V_0$ and $U_{2p}=V_{2p}\setminus V_{2p-2}$ for $p\ge1$. It is known that a non-polynomial $f\in U_0$ cannot have two consecutive Taylor coefficients equal to zero. Determine whether the analogue holds for $U_{2p}$: can an $f\in U_{2p}$ have $2p+2$ consecutive zero Taylor coefficients, and more generally, what is the maximal run of consecutive zero coefficients attainable in $U_{2p}$?
Acceptance. ADVANCES: settle a single concrete case — exhibit $f\in U_2$ with $4$ consecutive zero Taylor coefficients (an explicit product/coefficient certificate), or prove no $f\in U_2$ has $4$ consecutive zero coefficients (a finite-jet inequality). FULLY RESOLVES: determine, for every $p$, the maximal number of consecutive zero Taylor coefficients attainable in $U_{2p}$. Provide the explicit function or the proof; witnesses are finite and checkable.
Background
Hayman–Lingham, Research Problems in Function Theory (arXiv:1809.07200), Problem 2.74, posed by J. Williamson. The base case $p=0$ is classical: a non-polynomial function in the Laguerre–Pólya class $U_0$ cannot have two consecutive zero Taylor coefficients, proved via a Turán-type inequality among consecutive coefficients. Nothing is published for $p\ge 1$; the book records 'no progress has been reported.' The first concrete open case is $p=1$: can $f\in U_2$ have $4$ consecutive zero Taylor coefficients? Adjacent active work studies second quotients of Taylor coefficients of Laguerre–Pólya functions (Vishnyakova and others) but not this consecutive-zeros question. An attacker must bring: for fixed small $p$, real-algebraic / SDP machinery — Laguerre–Pólya membership (via genus-bounded products $\prod(1-z/x_i)\,e^{-az^{2p+2}}$) and the '$k$ consecutive zero coefficients' condition are semi-algebraic constraints on finite jets; search for a witness with a long zero run, or prove impossibility via a Wronskian / Turán-type inequality generalizing the $p=0$ proof. Decomposes by $p$ and by run-length.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Hayman–Lingham, Research Problems in Function Theory — Problem 2.74 | arxiv |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.