Derivative of the Gram Series

• The derivative of the Gram series is an analytical tool that differentiates series linked to Gram matrices, unveiling spectral sensitivities and underlying combinatorial structures.
• It connects methodologies from generating functions, matrix calculus, and integral transforms to analyze parameter perturbations in quantum groups, probability, and number theory.
• This approach provides actionable insights into the behavior of determinants and reproducing kernels, facilitating new approximations and proofs in diverse mathematical contexts.

The derivative of the Gram series is an analytical and algebraic concept arising across combinatorics, quantum groups, probability, number theory, harmonic analysis, and mathematical physics. It refers to the differentiation—either formal or analytic—of a series or product naturally associated with Gram matrices, which themselves encapsulate geometric, probabilistic, or group-theoretic data. The paper of the derivative of the Gram series reveals how infinitesimal changes in underlying parameters (such as group rank, distributional moment, deformation parameter, or index) affect the structure, invariants, or spectral properties of Gram matrices and their associated objects, notably determinants, cumulants, generating functions, reproducing kernels, or arithmetic sums.

1. Combinatorial Decomposition and the Gram Series Derivative in Quantum Groups

In the context of “easy quantum groups,” including $S_n$, $O_n$, $B_n$, their free, and half-liberated analogs, Gram matrices $G_{kn}$ are constructed from the combinatorics of set partitions or related diagrams. The determinant of these matrices decomposes as a product over partitions:

$\det(G_{kn}) = \prod_{\pi\in P(k)} \phi(\pi)$

with $\phi(\pi) = n!/(n-|\pi|)!$ in the classical case. The Gram trace, $$T_k(t) = \operatorname{Tr}(G_k^{t}) = \sum_{r=1}^k S_{kr} t^r$$, encodes the blocks via the generalized Stirling numbers $S_{kr}$.

The derivative of the Gram series (in $n$ or $t$) probes the behavior of each $\phi(\pi)$, providing

$$\frac{d}{dn}\log\det(G_{kn}) = \sum_{\pi\in P(k)} \frac{\phi'(\pi)}{\phi(\pi)}$$

This operation measures the sensitivity of the determinant to changes in the group parameter, and, equivalently, how the spectrum, encoded via the Gram series expansion, responds to infinitesimal perturbations. In asymptotics, the subleading terms such as $z_k$ in

$$\det(G_{kn}) = n^{s_k}(1 + \frac{z_k}{n} + O(n^{-2}))$$

(where $s_k = \sum_\pi |\pi|$), are controlled by the derivative structure, linking combinatorial decompositions to analytic variables (Banica et al., 2010).

2. Generating Functions, Recursions, and Probabilistic Gram Series

For random Gram matrices $G_n = A^T A$ formed by independently sampled vectors, the Gram series encompasses the exponential generating functions (EGF) for the expected determinant $a_n$ and the expected permanent $p_n$: $$\sum_{n=0}^{\infty} a_n \frac{x^n}{n!} = \exp\bigg(\frac{t_1x}{1} - \frac{t_2 x^2}{2} + \frac{t_3 x^3}{3} - \cdots \bigg)$$

$$\sum_{n=0}^{\infty} p_n \frac{x^n}{n!} = \exp\bigg(\frac{t_1x}{1} + \frac{t_2 x^2}{2} + \frac{t_3 x^3}{3} + \cdots \bigg)$$

where $t_j = \operatorname{trace}(M^j)$ for the second moment matrix $M$.

Differentiating these generating functions with respect to $x$ yields recursions: $$a_{n+1} = \sum_{j=0}^n \binom{n}{j}(-1)^j j! a_{n-j} t_{j+1}$$ Thus, the “derivative” of the Gram series in this context is the mechanism generating recursions for spectral/statistical invariants, tightly connecting the combinatorial and probabilistic perspectives. The sensitivity of Gram series coefficients (expected characteristic and permanental polynomials) to changes in the underlying measure is thereby encoded in these differential recursions (Martin et al., 2013).

3. Analytic Number Theory: Derivative of the Gram Series and Sums over Primes

In number theory, the Gram series appears as an analytic surrogate for prime counting functions, for instance

$$H(x) = 1 + \sum_{n=1}^{\infty} \frac{(\log x)^n}{n! \, n \, \zeta(n+1)}.$$

Its derivative,

$H'(x),$

is a smoothed “density of primes” and is used as an approximation for $\pi'(x)$. Ramanujan’s formula explicitly connects the formal derivative to Möbius-weighted sums: $$\pi'(x) \approx \frac{1}{x \log x} \sum_{n=1}^{\infty} \frac{\mu(n)}{n} x^{1/n}.$$ Applying Riemann–Stieltjes integration, one relates integrals involving the prime-counting function to those involving $H'(x)$. The paper introduces new formulas expressing $H'(x)$ in terms of the Riesz function: $$\sum_{n=1}^{\infty} \frac{\mu(n)}{n^2} e^{-x/n^2} = \frac{1}{2\pi i} \int_{(c)} \frac{\Gamma(s)}{\zeta(2-2s)} x^{-s} ds,$$ and explicitly links Möbius series with Mellin–Fourier transforms involving the Gram series’ derivative. This construction not only reexpresses prime sums in terms of rapidly convergent Möbius series but also facilitates the analysis of their asymptotics, singularity structure, and their deep connections to the Riemann Hypothesis via integral transforms (Patkowski, 19 Aug 2025).

4. Integral and Functional Representations of the Gram Series Derivative

Gram series arising from analytic number theory and harmonic analysis often admit integral representations. For instance, in the Nyman–Beurling approach to the Riemann Hypothesis, the Gram series

$S(x) = \sum_{n=1}^\infty R(nx)$

(with $R_1(x)$ involving the logarithmic, fractional part, and harmonic sum terms) admits both direct series and integral forms. Under mild conditions, termwise differentiation yields

$S'(x) = \sum_{n=1}^\infty n R'(n x),$

and, via change of variables and Mellin-type integrals,

$$S_1(r) = -\int_r^\infty \varphi_1(t) \frac{dt}{t^2}, \quad \varphi_1(t) = \sum_{k=1}^\infty \frac{\{kt\} - 1/2}{k}$$

The derivative’s analytic structure, including singularities (for instance, arising from jumps or fractional part discontinuities), is tractable because they cancel when summing over $n$. These analytic continuations and reciprocity relations play a vital role in explicit formulae for quadratic forms associated with zeta functions and Dirichlet polynomials (Ehm, 10 May 2024).

5. Algebraic and Geometric Aspects of the Gram Series Derivative

In situations where Gram matrices encode geometric or physical invariants—such as Gram matrices of isotropic vectors in conformal field theories or perturbative quantum field theory—the derivative of the Gram series is an algebraic object. Given that the determinant and minors of such matrices can be written in terms of invariant polynomials (e.g., $P_{ij}$, $H_{ij}$, $V_{i,jk}$), the derivative with respect to deformation parameters is computed via matrix calculus and the chain rule on these algebraic generators: $$\frac{d}{dt}\det(X(t)) = \operatorname{tr}\left(\operatorname{adj}(X(t))\, \frac{dX(t)}{dt}\right)$$ Such derivatives live in the coordinate rings of determinantal varieties, constrained by algebraic relations among the invariants, and are central to the computation and analysis of scattering amplitudes and conformal correlators. The structural properties of the Gram matrix ensure that these derivatives remain physically meaningful and conformally invariant (Maazouz et al., 13 Nov 2024).

6. Kernel Methods, Exact Inversion, and Derivative Propagation

In the context of reproducing kernel Hilbert spaces and interpolation, the Gram matrix’s inversion (especially after transformation to an orthogonal polynomial basis) and the consequent construction of kernels

$$K(x, y) = \sum_{i, j} b_{ij} \varphi_i(x) \varphi_j(y)$$

allow derivatives of the kernel to be computed termwise: $$\partial_x K(x, y) = \sum_{i, j} b_{ij} \varphi_i'(x) \varphi_j(y)$$ Numerical evidence demonstrates that error variances for derivatives of such kernels are significantly smaller than those found in Taylor expansions, both for trigonometric and exponential models. The exact inversion of the Gram matrix in this setting precisely controls the propagation of error to derivatives, reinforcing the utility of the series’ derivative in approximation and analysis (Spitzer, 20 Feb 2024).

7. Vector-Valued and Multivariate Generalizations

In multivariate settings, the Gram–Charlier series and its generalizations are expanded via repeated application of Kronecker product–based differentiation operators—the “K-derivative”. The formalism produces higher-order derivatives encoded as vectorized arrays instead of multisymmetric tensors: $$f(x) = \sum_{m=0}^{\infty}\frac{1}{m!}c(m,d)^{\prime}x^{\otimes m}$$ where $c(m,d)$ vectorizes higher derivatives via Kronecker calculus. In this representation, differentiation remains central, both for constructing the expansion and for updating the series as cumulants/moments shift in high-dimensional probability spaces. These methodologies allow direct extension from univariate to multivariate, streamlining statistical modeling and inference for vector-valued densities (C, 2015).

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In summary, the derivative of the Gram series unifies and generalizes differentiation phenomena across a spectrum of mathematical disciplines—combinatorics, probability, analysis, geometry, and physics. Whether probing spectral sensitivity, capturing analytic structure, enabling efficient computation, or reflecting algebraic invariance, the derivative operates as a central tool in extracting refined quantitative and qualitative information from Gram-associated entities. This concept continues to inspire novel connections, most recently linking Gram series derivatives to the Riesz function, Möbius-weighted prime summation, and the analytic theory of numbers (Patkowski, 19 Aug 2025).