Research Problems in Function Theory — Problem 7.54

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.

For $t\in\mathbb C$ put $\phi_t(z)=e^{tz}-1$, and define its iterates by $\phi_t^{\circ1}=\phi_t$ and $\phi_t^{\circ(n+1)}=\phi_t\circ\phi_t^{\circ n}$. The entire function of $t$ given by $F_n(t)=\phi_t^{\circ n}(-1)$ has Taylor expansion $F_n(t)=\sum_{k=0}^\infty a_{n,k}t^k$ at $0$. For example, $F_1(t)=e^{-t}-1$ and $F_2(t)=\exp(t(e^{-t}-1))-1$.

Statement recorded in the supplied source.

Is $|a_{n,k}|\le1$ for every $n\ge1$ and every $k\ge0$? [S2]

Short English statement

Are all Taylor coefficients, in the parameter, of every iterate evaluated at minus one bounded in modulus by one?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307054 (AMR-022-7054). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 7.54, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.