Distinguishing Case I from Case II in Kuznetsov's construction of measures with identical moments: intrinsic criteria on the generating entire function

For an indeterminate Hamburger moment problem, all solutions are parametrized by the Nevanlinna formula built on a quartet (A, B, C, D) of entire functions. Kuznetsov (2017) gave an alternative construction: starting from an entire function f in his class M (canonical product with zeros in the upper half-plane and rapid decay in the closed lower half-plane), a contour-integral argument produces a family of measures with identical moments, parametrized in Nevanlinna form by a quadruple (a, b, c, d) of entire functions with ad - bc = 1. The paper shows by example that the classical Nevanlinna functions B, D may or may not be linear combinations of the constructed b, d (Case I versus Case II), and that both cases can generate the very same moment problem. The problem, posed in Section 3, is to find what properties of f in M allow one to distinguish the two cases, and, separately, to determine the connection between the quartet (A, B, C, D) and the constructed quadruple (a, b, c, d); the section closes with the verbatim question: "It is an interesting problem to find what properties of a function f ∈ M would allow one to distinguish between these two cases."

Background

The Hamburger moment problem asks for positive Borel measures on the real line whose power moments reproduce a prescribed sequence; it is called indeterminate when more than one such measure exists (in fact, then uncountably many exist). For an indeterminate problem, the Nevanlinna parametrization describes all solutions in terms of Pick functions - analytic functions on the open upper half-plane $\mathbb{H}^+$ that map it into the closed upper half-plane (the class $\mathcal{P}$, taken together with the value $\infty$; the parametrization is a bijection onto the full solution set). Writing $P_n$ for the orthonormal polynomials of the problem (obtained by orthonormalizing $1, x, x^2, \dots$; they are determined by the moment sequence alone), one defines four entire functions $A, B, C, D$ (the Nevanlinna quartet, or Nevanlinna matrix) by series in the $P_n$; they satisfy $A(z)D(z)-B(z)C(z)\equiv 1$, and every solution $\nu(\cdot;\phi)$ of the moment problem is obtained from exactly one $\phi\in\mathcal{P}\cup\{\infty\}$ via

$$\int_{\mathbb{R}} \frac{\nu(dx;\phi)}{x-z} = -\frac{A(z)\phi(z)-C(z)}{B(z)\phi(z)-D(z)}, \qquad z \in \mathbb{H}^+.$$

The quartet functions are real entire functions (entire functions taking real values on the real axis) of minimal exponential type - that is, growing more slowly than $e^{\varepsilon|z|}$ for every $\varepsilon>0$ - and sharing a common order and type (the standard growth characteristics of an entire function), as established by Berg and Pedersen; classical background is in the monographs of Akhiezer and Levin. Only a small number of indeterminate problems have had their quartet computed explicitly.

Kuznetsov (2017) approached these problems from the side of entire functions. He introduced a class $\mathcal{M}$ of entire functions $f$ of the form $f(z)=C\prod_{n\geq 1}(1-z/z_n)$ with all zeros $z_n$ in $\mathbb{H}^+$ (the canonical product converging), such that for every $n\geq 0$ one has $z^n/f(z)\to 0$ as $z\to\infty$, uniformly on the closed lower half-plane $\mathbb{H}^-\cup\mathbb{R}$. Such an $f$ admits a unique decomposition $f=b-i\,d$ with $b, d$ real entire, and $-d/b$ is a Pick function; the class sits inside the Hermite-Biehler class of entire functions studied in Levin's monograph (entire functions, writable as $b-i\,d$ with $b,d$ real entire, all of whose zeros lie in the open upper half-plane; by the Hermite-Biehler theorem this is equivalent to $b$ and $d$ having real, simple, interlacing zeros). For $f\in\mathcal{M}$ and $\phi$ in a dense subclass $\mathcal{P}_b$ of the Pick functions (those with uniform boundary limits on compact subsets of the real axis and bounded image closure), Kuznetsov defines densities

$$\mu(x;\phi,f)=\frac{\Im(\phi(x))/\pi}{|d(x)-\phi(x)b(x)|^2}, \qquad x\in\mathbb{R},$$

and proves, by a contour-deformation and residue computation, that all these measures have identical moments: they are solutions of a single indeterminate Hamburger moment problem. His second main theorem shows that the function $g$ entering the construction extends to an entire function $g=a-i\,c$ with $a, c$ real entire, that

$$a(z)d(z)-b(z)c(z)\equiv 1,$$

and that the same family is parametrized in exact Nevanlinna form by the quadruple $(a,b,c,d)$.

This coexistence of two quadruples parametrizing the same solution set raises a structural question, which Section 3 of the paper shows is genuinely non-trivial. For the indeterminate moment problem of Ismail and Valent attached to elliptic-function weights - taking $f(z)=\tfrac{2}{\sqrt{\pi}}\cos(\sqrt{z}(K-iK')/2)$ with $K=K(k)$ the complete elliptic integral of the first kind, $K(k)=\tfrac{\pi}{2}\,{}_2F_1(\tfrac{1}{2},\tfrac{1}{2};1;k^2)$, $k' = \sqrt{1-k^2}$, $K'=K(k')$ - the constructed $f$ falls into Case I: the Nevanlinna functions $B, D$ are linear combinations of $b, d$. But the modified function $\tilde f = f + iz\,b$ also belongs to $\mathcal{M}$ and yields Case II, even though it generates the same moment problem. Hence the desired classification is a property of the function $f$ itself, not of the moment problem it induces.

Problem Statement

Let $\mathcal{M}$ be Kuznetsov's class of entire functions: $f(z)=C\prod_{n\geq 1}(1-z/z_n)$ with all zeros $z_n$ in the open upper half-plane $\mathbb{H}^+$ (the canonical product converging), such that for every $n\geq 0$, $z^n/f(z)\to 0$ as $z\to\infty$ uniformly on the closed lower half-plane $\mathbb{H}^-\cup\mathbb{R}$. Write the unique decomposition $f=b-i\,d$ with $b, d$ real entire, so that $-d/b$ is a Pick function. The construction of Kuznetsov (2017) associates to $f$ the family of measures $\mu(x;\phi,f)\,dx$ with $\mu(x;\phi,f)=\frac{\Im(\phi(x))/\pi}{|d(x)-\phi(x)b(x)|^2}$ for $\phi$ in a dense subclass $\mathcal{P}_b$ of the Pick functions; all these measures have identical moments, so they solve one indeterminate Hamburger moment problem, and the construction yields entire functions $a, b, c, d$ with $a(z)d(z)-b(z)c(z)\equiv 1$ parametrizing this family in Nevanlinna form. The same moment problem carries its classical Nevanlinna quartet $(A,B,C,D)$, with $A(z)D(z)-B(z)C(z)\equiv 1$, built from its orthonormal polynomials.

Say that $f$ is of Case I if the Nevanlinna functions $B$ and $D$ of the induced moment problem are linear combinations of the constructed functions $b$ and $d$, and of Case II otherwise; both cases occur.

(1) Find properties of $f\in\mathcal{M}$ - intrinsic to $f$ or to the pair $(b,d)$, for instance in terms of the zero sequence, the growth and indicator behavior (the Phragmen-Lindelof indicator, which measures the directional growth of an entire function), or the Pick function $-d/b$ - that allow one to distinguish Case I from Case II; give a necessary and sufficient condition.

(2) Separately, and independently of (1), determine the connection between the Nevanlinna quartet $(A,B,C,D)$ and the constructed quadruple $(a,b,c,d)$: characterize when the two parametrizations of the same solution set coincide up to their natural normalization (the normalization freedom of the Nevanlinna formula, i.e. a simultaneous sign change of all four functions, under which the formula is invariant). Coincidence forces Case I but is strictly stronger than it - Case I does not force coincidence - so (1) and (2) are independent demands and both are required.

Any answer must be consistent with the two worked examples of the source paper. For $k\in(0,1)$, let $K=K(k)$ be the complete elliptic integral of the first kind, $K(k)=\tfrac{\pi}{2}\,{}_2F_1(\tfrac{1}{2},\tfrac{1}{2};1;k^2)$, and let $k'=\sqrt{1-k^2}$, $K'=K(k')$; then $f(z)=\tfrac{2}{\sqrt{\pi}}\cos(\sqrt{z}(K-iK')/2)$ lies in $\mathcal{M}$ and is of Case I, with $B(x)=\tfrac{\sqrt{\pi}}{2}b(x)-\tfrac{1}{\sqrt{\pi}}\ln(k/k')\,d(x)$ and $D(x)=\tfrac{2}{\sqrt{\pi}}d(x)$, while $\tilde f(z)=f(z)+iz\,b(z)$ also lies in $\mathcal{M}$ and is of Case II (there $\tilde b=b$ and $\tilde d=d-zb$), even though $\tilde f$ generates the same indeterminate moment problem as $f$. In particular, a correct criterion cannot be a property of the induced moment problem alone: it must genuinely depend on $f$.

References

1. Alexey Kuznetsov, "Constructing measures with identical moments", Proceedings of the American Mathematical Society 145(10) (2017), 4431-4441, doi:10.1090/proc/13585, https://doi.org/10.1090/proc/13585 (author manuscript: arXiv:1607.08003, https://arxiv.org/abs/1607.08003)
2. M. E. H. Ismail and G. Valent, "On a family of orthogonal polynomials related to elliptic functions", Illinois Journal of Mathematics 42(2) (1998), 294-312, doi:10.1215/ijm/1256045046, https://doi.org/10.1215/ijm/1256045046
3. Christian Berg, "Indeterminate moment problems and the theory of entire functions", Journal of Computational and Applied Mathematics 65 (1995), 27-55, doi:10.1016/0377-0427(95)00099-2, https://doi.org/10.1016/0377-0427(95)00099-2
4. Christian Berg and Henrik L. Pedersen, "On the order and type of the entire functions associated with an indeterminate Hamburger moment problem", Arkiv för Matematik 32 (1994), 1-11, doi:10.1007/BF02559520, https://doi.org/10.1007/BF02559520
5. B. Y. Levin, "Lectures on Entire Functions", Translations of Mathematical Monographs 150, American Mathematical Society, 1996, doi:10.1090/mmono/150, https://doi.org/10.1090/mmono/150
6. N. I. Akhiezer, "The Classical Moment Problem and Some Related Questions in Analysis", Oliver & Boyd, Edinburgh and London, 1965 (SIAM reissue: doi:10.1137/1.9781611976397, https://doi.org/10.1137/1.9781611976397)
7. Alexey Kuznetsov, "A Direct Evaluation of an Integral of Ismail and Valent", in Frontiers in Orthogonal Polynomials and q-Series, 2018, 417-424, doi:10.1142/9789813228887_0021, https://doi.org/10.1142/9789813228887_0021
8. Martin Nicholson, "On a specific family of orthogonal polynomials of Bernstein-Szegő type", arXiv:2505.05545 (2025), https://arxiv.org/abs/2505.05545
9. Yicao Wang, "Complex analysis of symmetric operators. II: entire operators with deficiency index 1", arXiv:2510.20249 (2025), https://arxiv.org/abs/2510.20249