Research Problems in Function Theory — Problem 2.32

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.

Let $f(z)=\sum_{n=0}^\infty a_nz^n$ be transcendental entire with $a_n\ge0$, and for $x>0$ set $p_n(x)=a_nx^n/f(x)$, so that $\sum_{n\ge0}p_n(x)=1$. The integral constraints below imply $a_n>0$ for every $n$; integrals are improper integrals on the positive real axis.

Statement recorded in the supplied source.

Do the identities \[\int_0^\infty \frac{a_nx^n}{f(x)}\,dx=1\quad(n=0,1,2,\ldots),\] or equivalently \[\int_0^\infty \frac{f(tx)}{f(x)}\,dx=\frac1{1-t}\quad(0<t<1),\] force $f(z)=ce^z$ for some $c>0$? [S2] Update 2.32 explicitly gives the scale-corrected uniqueness question $f(z)=ce^z$, rather than the original literal exclusion of only $e^z$, and records Miles and Williamson's affirmative theorem. [S2]

Short English statement

Do the prescribed integral identities uniquely determine the exponential entire function up to a positive constant factor?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302032 (AMR-022-2032). 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 2.32 and Update 2.32. https://arxiv.org/abs/1809.07200.