Hadamard Generating Function Calculus

• Hadamard Generating Function Calculus is a framework that employs algebraic, analytic, and geometric tools to compute coefficientwise (Hadamard) products and extract diagonals of generating functions.
• It establishes structural classifications via diagonal and Hadamard grades and links the analytic properties of D-finite series to the monodromy of differential equations.
• The calculus provides explicit algorithms, including integral representations and partial-fraction methods, to analyze and transform rational, algebraic, and D-finite generating functions.

The Hadamard-Generating-Function Calculus encompasses a framework of algebraic, analytic, and geometric tools for describing and computing coefficientwise (Hadamard) products and diagonal extractions of generating functions, with deep connections to the monodromy of differential equations and to the algebraic and D-finite hierarchies of series. This calculus systematizes central computational rules, structural classification (via diagonal and Hadamard grades), closure properties, and concrete algorithms for the analysis of rational, algebraic, and D-finite generating functions and their coefficientwise transformations.

1. Definitions: Hadamard Product, Diagonals, and Grades

Given two power series $f(x)=\sum_{n\ge0}a_n x^n$ and $g(x)=\sum_{n\ge0}b_n x^n$ over a field $K$, their Hadamard product is defined by

$(f \odot g)(x) = \sum_{n \ge 0} a_n b_n x^n.$

This operation is bilinear and preserves convergence radii. The Hadamard product is also known as the coefficientwise or Schur product (Gessel et al., 2023, Prodinger et al., 2019).

For a multivariate rational function $R(x_0,\dots,x_n)$, the $n$-th diagonal is

$$\operatorname{Diag}_n R(x) = \sum_{i \ge 0} a_{i,i,\dots,i} x^i,$$

where $a_{i_0,\dots,i_n}$ are the coefficients of the power series expansion of $R$, convergent in a polydisk.

A series $f \in K[[x]]$ is D-finite if it satisfies a nontrivial linear differential equation over $K(x)$.

Two key structural grades are:

• Diagonal grade $\operatorname{dg}(f)$: least $n$ such that $f(x)$ is the diagonal of a rational function in $n$ variables.
• Hadamard grade $\operatorname{hg}(f)$: least $k$ such that $f(x) = h_1(x) \odot \cdots \odot h_k(x)$ with each $h_i$ algebraic over $K(x)$; $0$ if rational.

The algebraic series in $K[[x]]$ comprise exactly $\mathcal{D}_1$, with $\mathcal{H}_1 = \mathcal{D}_1$ by Furstenberg's theorem (Harder et al., 14 Apr 2025).

2. Theoretical Foundations and Monodromy Bounds

For D-finite series $f$ annihilated by a minimal differential operator $L$ with regular singularity at $x=0$, the nilpotence index $\mathrm{Nil}(f)$ is defined as one plus the maximal multiplicity of logarithms in the local solutions at $x=0$. It measures the maximal size of Jordan blocks in the local monodromy.

The principal geometric theorem states that if $f(x) = \operatorname{Diag}_n R(x_0,\ldots,x_n)$ for $R$ algebraic (in particular, rational) over $K(x_0,\ldots,x_n)$, then

$\mathrm{Nil}(f) \le n+1,$

and if $R$ is rational, $\mathrm{Nil}(f) \le n$ (Harder et al., 14 Apr 2025). Thus, for D-finite $f$,

$$\operatorname{dg}(f) \ge \mathrm{Nil}(f),\quad \operatorname{hg}(f) \ge \mathrm{Nil}(f).$$

This connects the analytic structure (via monodromy) with the algebraic-combinatorial structure (grades) of generating functions.

A prototypical example is given by the hypergeometric function

$${}_n F_{n-1}(\alpha; 1,\ldots, 1\mid x) = (1-x)^{-\alpha_1} \odot \cdots \odot (1-x)^{-\alpha_n},$$

which has grade $n$ (both diagonal and Hadamard), and whose associated differential monodromy has a single Jordan block of size $n$ in the non-resonant case (Harder et al., 14 Apr 2025).

3. Explicit Calculus: Rational and D-finite Series

Rational Function Case

If $f=P/Q$, $g=R/S$ are rational power series, their Hadamard product remains rational. One can always write $f$ and $g$ in partial-fraction form as sums of monomials and terms of the form $(1-\alpha z)^{-k-1}$. The Hadamard product table consists of explicit evaluations:

• $z^n \odot z^m = \delta_{n,m} z^n$,
• $$z^n \odot (1-\alpha z)^{-m-1} = \alpha^n \binom{n+m}{m} z^n$$,
• $$(1-\alpha z)^{-k-1} \odot (1-\beta z)^{-\ell-1} = \sum_{n=0}^\infty \binom{n+k}{k} \binom{n+\ell}{\ell} (\alpha \beta)^n z^n$$ (0810.3641).

For any finite sum of such terms, $f \odot g$ can be represented as a rational function, obtained by explicit summation and differential operator calculus (0810.3641).

D-finite Sequences and Recurrences

For $a_n$ satisfying an order-$r$ linear recurrence with generating function $A(z)=P(z)/Q(z)$ and $m$-th power Hadamard product,

$(A\odot \cdots \odot A)(z)\text{ (%%%%48%%%% times)},$

the result is rational, of degree $rm$ in both numerator and denominator (Prodinger et al., 2019). The generating function for partial sums, $\sum_{k=0}^n a_k^m$, is given by dividing by $(1-z)$. Partial fraction decomposition then yields a closed form (Binet-type formula) for these sums, with the roots of the denominator and the corresponding residues directly determining the solution (Prodinger et al., 2019).

4. Computational Techniques and Algorithmic Framework

Distinct computational strategies exist for explicit Hadamard product evaluation (Gessel et al., 2023):

• Resultant Method: Compute the denominator as the resultant $\operatorname{Res}_y(U(y), y^n V(x/y))$, yielding $\prod_{i,j}(1-a_ib_jx)$ for $A(x) = P(x)/U(x)$, $B(x) = Q(x)/V(x)$.
• Symmetric Function Method: By expressing generating function denominators via elementary symmetric functions and computing mixed power sums, one uses Newton's identities to determine the resulting coefficients.
• Partial-Fraction Method: Extract the coefficient of $t^0$ in $A(t) B(x/t)$, reformulated via partial fraction decomposition in $t$.

All these methods provide rational generating functions for Hadamard products of rational series (Gessel et al., 2023).

Integral Representations

Integral calculus methods realize the Hadamard product via:

• Fourier integral: For analytic $F(z),G(z)$ in $|z|<1$,

$$(F\odot G)(z) = \frac{1}{2\pi} \int_{-\pi}^{\pi} F(z e^{i\theta}) G(e^{-i\theta}) d\theta.$$

• Contour integral: If analytic in a suitable annulus,

$$(F\odot G)(z) = \frac{1}{2\pi i} \int_{|w|=\rho} F(w) G(z/w) \frac{dw}{w}.$$

Integral kernels thus encode OGF-to-EGF conversion, weighted Hadamard products, various special function transforms, and diagonal extractions of multivariate series (Schmidt, 2018).

5. Hierarchies, Closure Properties, and Structural Theorems

Grade Hierarchies and Strict Inclusions

The family $\mathcal{D}_n$ of univariate series with diagonal grade $\le n$ and $\mathcal{H}_k$ with Hadamard grade $\le k$ are nested: $$\mathcal{D}_1 \subsetneq \mathcal{D}_2 \subsetneq \cdots, \quad \mathcal{H}_1 \subsetneq \mathcal{H}_2 \subsetneq \cdots,$$ with strict inclusions established by $$_nF_{n-1}\left(\frac{1}{2},\dots,\frac{1}{2};1,\dots,1\mid x\right)$$ (Harder et al., 14 Apr 2025). The closure theorem asserts that if $f\in \mathcal{D}_k$, $g\in \mathcal{D}_\ell$, then $f\odot g \in \mathcal{D}_{k+\ell}$ (Gessel et al., 2023, Harder et al., 14 Apr 2025).

Notably, every rational power series is closed under the Hadamard product, and the subalgebra of rational series is preserved (0810.3641).

Zero Divisor and Subsequence Rules

Zero divisor characterization relies on residue classes, and the extraction of subsequences (via $e_{r,m}(f)$) preserves D-finiteness, diagonal, and Hadamard grades. These operations enable decompositions into “clean” irreducible residues and provide an eigen-decomposition tool for grade analysis (Harder et al., 14 Apr 2025).

6. Applications, Examples, and Extensions

Classical Examples

• Tetranacci Sequence: The generating function for Tetranacci numbers $T_n$,

$G(z) = \frac{z}{1 - z - z^2 - z^3 - z^4},$

yields, via Hadamard product, the generating function for $T_n^2$, which is also rational. Partial sum generating functions follow by division by $1-z$, and closed-form expressions are obtained through partial-fraction techniques (Prodinger et al., 2019).

• Apéry Sequence: The generating function for $A_n = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2$ is

$$G(x) = \operatorname{Diag}_3\left(\frac{1}{(1-x_0-x_1)(1-x_2-x_3)+x_0x_1x_2x_3}\right),$$

with both diagonal and Hadamard grade 3 (Harder et al., 14 Apr 2025).

Broader Applications

Hadamard-generating-function calculus underpins residue and period computations in algebraic geometry, analysis of D-finite closures, automated generation of identities for combinatorial sequences, and transformations for special function expansions and OGFs/EGFs (Gessel et al., 2023, Schmidt, 2018).

7. Open Problems and Future Directions

• Determination of the precise relationship between nilpotence index and diagonal/Hadamard grades (especially upper bounds).
• Structure theory for $\mathcal{D}$ and $\mathcal{H}$ as filtered rings, refined zero divisor analysis, and classification of unit groups.
• Extension of calculus techniques to multivariate, noncommutative, and $p$-adic contexts, with possible links to Christol–Dwork theory.
• Broader explorations in quantum field combinatorics, periods in arithmetic geometry, and mirror symmetry computations.

The Hadamard-generating-function calculus thus forms a rich, highly structured system, central to modern enumerative combinatorics, analytic algebra, and computational number theory, equipping researchers with robust algebraic, analytic, and geometric methodology for the classification and manipulation of generating functions and their algebraic-combinatorial invariants (Harder et al., 14 Apr 2025, 0810.3641, Gessel et al., 2023, Prodinger et al., 2019, Schmidt, 2018).