Structure and Zero Asymptotics of Differential Operators Associated with $Ξ_n$ and $Λ_n$

Abstract: We study the second-order differential operators (\mathcal D_Ξ) and (\mathcal D_Λ) associated with the rescaled polynomial families ((\widetildeΞn)) and ((\widetildeΛ_n)), and more generally the polynomial sequences generated by iterating these operators from an arbitrary linear initial datum (cx-d). We establish structural properties of (\mathcal DΞ) and (\mathcal D_Λ), including factorizations into first-order operators, weighted divergence forms, formal self-adjointness, and hypergeometric descriptions of the corresponding formal eigenvalue equations. We also show that both operators preserve hyperbolicity, preserve zeros in ((0,b)) for (b\ge 1), and preserve proper position. For the iterated polynomial sequences, we derive explicit closed formulae in terms of the auxiliary families ((\widetildeΞ_n)) and ((\widetildeΛ_n)), prove strict interlacing of consecutive zeros under explicit conditions on (d/c), and obtain asymptotic formulae for the normalized logarithmic derivatives. As a consequence, the associated zero counting measures converge weakly to the same limiting probability measure as in the auxiliary case.

• The paper establishes the algebraic factorization and detailed operator structure for Ξ and Λ differential operators derived from specific polynomial families.
• It demonstrates that these operators preserve hyperbolicity and exhibit strict interlacing properties, ensuring stability and orthogonality under weighted measures.
• It reveals that the normalized zero distributions converge to a universal measure with an explicit density, linking hypergeometric functions to analytic number theory.

Structural Properties and Zero Asymptotics of Operators Linked to $\Xi_n$ and $\Lambda_n$

Overview and Context

The paper "Structure and Zero Asymptotics of Differential Operators Associated with $\Xi_n$ and $\Lambda_n$" (2604.13117) provides a comprehensive operator-theoretic treatment of two parametric families of second-order differential operators, $\mathcal D_{\Xi}$ and $\mathcal D_{\Lambda}$. These operators are intrinsically tied to previously introduced polynomial families $(\Xi_n)$ and $(\Lambda_n)$, which are constructed from explicit integral representations associated with the Dirichlet beta function and the Riemann zeta function, respectively.

The manuscript develops both a detailed algebraic characterization of these operators and an extensive asymptotic analysis of the zero distributions of the polynomials generated by iterated application of these operators, even when initiated from arbitrary linear data rather than constants.

Operator-Theoretic Structure

Algebraic Properties

The core objects, $\mathcal D_{\Xi}$ and $\mathcal D_{\Lambda}$, are defined as specific second-order differential operators with polynomial coefficients, acting naturally on $\Lambda_n$0. Both operators strictly increment the degree of any nonzero input polynomial by one. They admit explicit expressions in terms of monomial actions and preserve significant algebraic structure, particularly the subspace $\Lambda_n$1.

The paper establishes that these operators possess nontrivial factorizations into compositions of first-order differential operators: $\Lambda_n$2 and $\Lambda_n$3, where $\Lambda_n$4 and $\Lambda_n$5. This factorization is leveraged throughout the analysis of their spectral and stability properties.

Formal Self-Adjointness

Employing weighted divergence constructions, both operators are shown to be formally self-adjoint with respect to explicit inner products: $\Lambda_n$6 relative to $\Lambda_n$7 and $\Lambda_n$8 relative to the weight $\Lambda_n$9. As a consequence, eigenfunctions corresponding to distinct eigenvalues are orthogonal with respect to these measures under mild boundary behavior assumptions.

Hypergeometric Connection

The paper shows that the formal eigenvalue problems for both operators are reducible to Gauss hypergeometric differential equations. Explicitly, for appropriate spectral parameters, the general eigenfunction can be expressed in terms of $\Xi_n$0 with shifted parameters dependent on the eigenvalue. This identifies the structure of the solution space and links the recursion-driven polynomials to classical special functions—a significant analytic insight.

Real-Rootedness and Interlacing

Preservation Properties

A major algebraic result is that both $\Xi_n$1 and $\Xi_n$2, as well as the constituent first-order operators, preserve hyperbolicity (i.e., real-rootedness) of polynomials. The precise interval preservation result is that these operators map polynomials with zeros in $\Xi_n$3 to polynomials whose zeros remain in $\Xi_n$4 for any $\Xi_n$5; this fails for $\Xi_n$6, and the paper proves sharpness via explicit counterexamples.

Interlacing

The iterated sequences exhibit strict interlacing properties under explicit affine conditions on the ratio $\Xi_n$7 for the linear initial datum $\Xi_n$8. A detailed algebraic argument using sign sequences and explicit evaluations yields sharp constraints:

• For $\Xi_n$9-type polynomials, strict interlacing for all $\Lambda_n$0 holds if and only if $\Lambda_n$1.
• For $\Lambda_n$2-type polynomials, strict interlacing is present if and only if $\Lambda_n$3.

This characterization is extended inductively: positivity adjustments ensure that proper position (in the sense of Borcea–Brändén) is preserved throughout the hierarchy. Thereby, the polynomials in the sequence maintain maximally ordered zero configurations.

Asymptotic Analysis of Zero Distributions

Closed Form and Recursion

For arbitrary linear initial data, the iterated polynomial sequences are shown to admit explicit representations in terms of the auxiliary families $\Lambda_n$4 and $\Lambda_n$5, with proportionality constants and shifts given in terms of $\Lambda_n$6. The logarithmic derivatives of the normalized polynomials are decomposed into contributions from the auxiliary family and a term that vanishes in the $\Lambda_n$7 limit.

Weak Convergence of Empirical Zero Measures

The crucial analytic component is the demonstration that, for $\Lambda_n$8, the normalized zero counting measures of both families' general-iterated polynomials converge weakly to a universal probability measure $\Lambda_n$9 supported in $\mathcal D_{\Xi}$0. This limiting measure has an explicit density:

$\mathcal D_{\Xi}$1

with cumulative distribution function given in closed form. The quantile functions for the zeros are also provided explicitly, showing that the (properly scaled) zeros are distributed according to a nontrivial function involving hyperbolic tangent and arctangent, reflecting the connection to the hypergeometric background and the underlying integral structures.

Universality

A key technical claim is that these asymptotics are independent of the specific sequence of normalization coefficients or the affine parameter $\mathcal D_{\Xi}$2 (as long as $\mathcal D_{\Xi}$3 and conditions for strict interlacing hold). The limiting measure coincides with that previously established for the even-polynomial auxiliary sequences arising from the original Malmsten-type and polylogarithmic settings.

Implications and Theoretical Significance

The study demonstrates how operator-theoretic and spectral-analytic methods illuminate the deep structure of recursively generated polynomial families, with ramifications in analytic number theory (via connections to special values of $\mathcal D_{\Xi}$4 and $\mathcal D_{\Xi}$5 functions), quasi-orthogonality, and potential theory (through weak convergence of zero measures). Further, the sharp interlacing and hyperbolicity preservation results create the possibility for generalized stability analyses of operators directly arising from special functions or zeta integrals.

Given the transcendental nature of the asymptotic zero density, one may anticipate further applications in investigating universality classes of limiting measures for non-standard polynomials and their connections to random matrix theory and large deviations. On the operator-analytic side, the explicit weighted self-adjointness and hypergeometric ties suggest links to spectral problems in Sturm–Liouville theory.

Conclusion

This paper (2604.13117) rigorously characterizes the structure and asymptotics of differential operators associated with the $\mathcal D_{\Xi}$6 and $\mathcal D_{\Xi}$7 polynomial families, revealing intricate algebraic properties, explicit interlacing regimes, and universal zero distributions. The synthesis of factorization, spectral analysis, and asymptotics provides a robust framework with the potential for further development in both analytic and algebraic directions.