Integer Moments of Mixed Derivatives

• Integer moments of mixed derivatives are precise quantifications of cross-derivative behaviors in functions, stochastic processes, and L-functions.
• They employ combinatorial, representation-theoretic, analytic, and probabilistic methods to derive explicit formulas and asymptotic laws in contexts like random matrices and zeta functions.
• The framework bridges analysis, integrable systems, and inverse problems by enabling measure reconstruction, determinant evaluations, and the study of high-dimensional derivative interactions.

Integer moments of mixed derivatives arise across analysis, probability, random matrix theory, and number theory as precise quantifications of higher-order or cross-derivative behaviors in functions, stochastic processes, and L-functions. The computation, algebraic structure, asymptotic laws, and functional-analytic implications of such moments have been deeply explored using combinatorial, representation-theoretic, analytic, and probabilistic methods.

1. Algebraic and Analytic Definitions

The integer moments of mixed derivatives typically refer to expressions involving expectations or averages of products of derivatives evaluated either at points or over distributions:

• For random matrix characteristic polynomials: Mixed moments are functionals of the form

$$M_{n_1,\ldots,n_k} = \mathbb{E}_{U\sim U(N)}\left[\prod_{j=1}^k Z_U^{(n_j)}(\theta_j)\right]$$

where $Z_U(\theta)$ is the characteristic polynomial, and $n_j$ are non-negative integers representing derivative orders.

• For the Riemann zeta function and L-functions: Integer moments encompass sums like

$$\sum_{0 < \gamma \leq T} \prod_{j=1}^k \zeta^{(n_j)}\left(\frac{1}{2} + i\gamma\right)$$

where the sum is over non-trivial zeros.

• In analysis (e.g., functions of multiple variables): The $L^2$-integrability or existence almost everywhere of $f_{xy}$, $f_{yx}$, and higher mixed partials connects to moment conditions on the underlying function or process.

In each context, these moments characterize the joint behavior of the function and its derivatives—either as mixed algebraic objects (as in partition symmetric function moments), or as explicit averages/expectations.

2. Combinatorial and Representation-Theoretic Frameworks

a. Schur Functions, Partitions, and Hook Contents

For random matrix moments, a core insight is the use of Schur functions and the representation theory of unitary groups. Products and derivatives of characteristic polynomials can be rewritten via Schur functions, leading to combinatorial formulas:

• The evaluation of moments utilizes orthogonality relations, Cauchy identities, and the Generalized Binomial Theorem for Schur functions. This produces the formula:

$$\frac{\mathcal{M}_N(2k, r)}{\mathcal{M}_N(2k,0)} i^r= \sum_{\mu \vdash r}\frac{r!}{h_\mu^2}\frac{(N\uparrow\mu)(-k\uparrow\mu)}{(-2k\uparrow\mu)}$$

with generalized Pochhammer symbols $k \uparrow \mu = \prod_{(i,j)\in\mu} (k + j - i)$ and hook-lengths $h_\mu$ (Dehaye, 2010).

b. Jack Polynomials and Hypergeometric Series

In several results, the moments can be reformulated as determinants (e.g., of matrices whose entries are Bessel or hypergeometric functions), or as hypergeometric series involving generalized binomial coefficients and partition data. This approach connects integer moments of mixed derivatives to special functions and highlights deep combinatorial structures (Altug et al., 2012, Keating et al., 2023, Bailey et al., 2019).

c. Moments of Partitions in Higher Derivative Formulas

For higher-order derivatives of composite functions, moments of integer partitions (power-sum, elementary symmetric) appear in the Faà di Bruno-type formulas, especially in generalizations involving products and mixed compositions:

$$(f\circ p\cdot g\circ p^{(s)})^{(n)}(t) = n! \sum_{0\leq r < n} \sum_{A\in \mathcal{P}(n+r s)} e_r(A^{>s})\, f^{(n-r)}(p(t)) g^{(r)}(p^{(s)}(t))\, \prod_{i=1}^\infty \frac{1}{m_i(A)! (i!)^{m_i(A)}}$$

where $A$ is a partition and $e_r$ is an elementary symmetric function associated with $A$ (Zemel, 2019).

3. Asymptotic Laws and Number Theory Links

a. Random Matrix Theory

The asymptotics for the integer moments of mixed derivatives of characteristic polynomials (in particular ensembles like CUE) are explicitly computable as $N \to \infty$, with leading-order expressions involving combinatorial factors tied to the representation-theoretic framework: - For CUE:

$$\mathbb{E}_{U(N)}\left[|Z_U^{(n_1)}(1)|^{2k-2M} |Z_U^{(n_2)}(1)|^{2M}\right] \sim N^{k^2 + 2(k-M)n_1 + 2M n_2} \cdot (\text{explicit determinant/combinatorial factor})$$

(Keating et al., 2023, Bailey et al., 2019).

• Significant structural feature: The combinatorial constants in these leading terms, after normalization, are often ratios of factorials or determinants of special function matrices.

b. Zeta Function Moments at Zeros

The moments of mixed derivatives of $\zeta(s)$ at nontrivial zeros are governed by explicit asymptotic conjectures:

$$\sum_{0 < \gamma \leq T} \prod_{j=1}^{k} \zeta^{(n_j)}\left(\frac{1}{2}+i\gamma\right) \sim (-1)^{n_1+\cdots+n_k+k}\frac{n_1!\cdots n_k!}{\left(\sum_{j=1}^{k} n_j+1\right)!}\frac{T}{2\pi}\left(\log\frac{T}{2\pi}\right)^{\sum_{j=1}^{k} n_j+1}$$

(Hughes et al., 9 Sep 2025, Hughes et al., 12 Jun 2024).

This shape mirrors the structure found in the random matrix analogues, and both random matrix theory and the Ratios Conjecture provide derivations (or strong motivation) for the appearance of the factorial combinatorics and log-powers.

c. Extensions to Automorphic and L-functions

Related methods yield exact or asymptotic formulas for mixed moments of automorphic L-functions and their symmetric powers:

• Mixed moments (e.g., $L(f,1/2)L(\operatorname{sym}^2 f,1/2)$) decompose into diagonal, non-diagonal, and dual moments, with the dual term often represented as an integral of a kernel involving hypergeometric functions, estimated by advanced analytic methods (Balkanova et al., 2018).

4. Probabilistic and Functional Analytic Contexts

a. Continuous-State Branching Processes

For multi-type branching processes in random (Lévy) environments, integer mixed moments satisfy closed polynomial recursions:

• The $n$th moment of process $X(t)$ is a polynomial in its initial data of degree at most $n$. For two-type processes,

$$\mathbb{E}[X_1(t)]^n = \mathbb{E}[X_1(0)]^n \cdot e^{\eta_1(t)} + \sum_j \int_0^t \mathbb{E}[X_1(s)^{j+1} X_2(s)^{n-j-1}] e^{\eta_1(t)-\eta_1(s)} ds + \ldots$$

capturing lower degree cross-moments and the influence of the random environment (Chen et al., 2022).

b. Fractional Polynomial Processes

The extension to fractional time processes (e.g., polynomial Markov processes time-changed by the inverse of a subordinator) replaces the semigroup exponential in moment evolution with the matrix Mittag-Leffler function:

$$\mathbb{E}_x[u(X_{L_t})] = H(x)^T E_\alpha(t^\alpha A) u$$

where $E_\alpha$ is the matrix Mittag-Leffler function and $A$ the generator on the polynomial space. Cross-moments and equilibrium correlations have closed form representations. Long-range dependence arises, with autocorrelations decaying as $\sim (t+s)^{-\alpha}$ (Assefa et al., 23 Jan 2025).

5. Analytical and Summability Properties

Moment differentiation and summability concepts extend the classical derivative operator to Carleman classes:

• The moment derivative operator $\partial_{m_e, z}$ acts on formal and analytic functions as

$$\partial_{m_e, z} \left(\sum_{p=0}^\infty \frac{u_p}{m_e(p)} z^p\right) = \sum_{p=0}^\infty \frac{u_{p+1}}{m_e(p)} z^p$$

and preserves summability in ultraholomorphic classes. Related integral representations permit explicit upper bounds for mixed-moment derivatives under growth constraints; these concepts feed into generalized moment PDEs (Lastra et al., 2020).

In harmonic and Fourier analysis, square integrability of pure derivatives (e.g., $f_{xx}$, $f_{yy}$ in two variables) ensures the existence and $L^2$-control of mixed derivatives (integer moments of their Fourier coefficients), with almost everywhere equality $f_{xy} = f_{yx}$, and implications for joint continuity (Mykhaylyuk, 2015).

6. Algorithmic and Inverse Problem Aspects

The algebraic framework for derivatives of moments enables:

• Reconstruction of measures (including Gaussian mixtures) from truncated sequences of moments and their derivatives, often via eigenvalue problems or Hankel matrices.
• Sharp lower and upper bounds for the minimal number of components needed to represent these functionals, linked to the Carathéodory number, even in high-dimensional or multivariate polynomial settings (Dio, 2019).

This methodology generalizes classical moment problems to atomic, Gaussian, and polyhedral measures, synthesizing distribution theory, symmetric function theory, and matrix analysis.

7. Connections to Integrable Systems and Special Functions

A persistent connection emerges between the moments of (mixed) derivatives and integrable systems:

• Determinant structures governing moments are often $\tau$-functions for Painlevé equations (notably Painlevé III′), satisfied by Bessel or hypergeometric functions.
• Differential recurrence relations for the determinant expressions exemplify Toda lattice equations, supporting computational advances and deeper structural understanding (Altug et al., 2012, Bailey et al., 2019).

These connections reinforce the interplay between random matrix theory, the spectral theory of integrable operators, and number theory.

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In summary, the theory of integer moments of mixed derivatives forms an interdisciplinary bridge connecting combinatorics, representation theory, asymptotic analysis, probability, analytic number theory, and integrable systems. The central techniques leverage partition theory, special function determinants, fractional calculus, and summability theory, providing rich, explicit formulas, deep symmetry, and pathways for both theoretical and computational exploration.