An integrality phenomenon

1. Problem Statement

The paper addresses the phenomenon of integrality in sequences generated by linear recurrence relations with polynomial coefficients.

• Context: While many sequences arising in number theory (like the Apéry numbers for $\zeta(3)$) are integers despite being defined by recursions that involve division by $n^k$, this is generally rare. Most such recursions only guarantee that $n!^k u_n$ is an integer.
• Specific Motivation: The authors investigate a specific conjecture related to the Hörmander–Bernhardsson extremal function. This function arises from a differential equation with irregular singularities. The coefficients of the power series solution to this equation satisfy a recurrence relation (denoted as Eq. 8 in the paper).
• The Conjecture: Numerical evidence suggested that the sequence $u_n$ generated by this specific recurrence is not just rational with factorial denominators, but consists of polynomials with integer coefficients in the parameters of the equation. This is a stronger integrality property than typically observed in such contexts.

2. Methodology

The authors employ a combination of generating function analysis, differential equations, and algebraic combinatorics to prove the integrality.

A. Analysis of the Specific Case (Hörmander–Bernhardsson)

1. Derivation of Recurrence: They start with a differential equation with irregular singularities. Through substitution and symmetry arguments (specifically looking at the symmetric square of the differential equation), they derive a recurrence relation for the coefficients $u_n$:
   $$n u_n - 2(2n-1)(n(n-1)-b)u_{n-1} + 4c(n-1)u_{n-2} = 0$$
   where $b$ and $c$ are parameters derived from the original equation's constants.
2. Transformation Strategy: Instead of proving integrality directly via induction (which is difficult due to the division by $n$), they construct an explicit identity relating the generating function of $u_n$ to the generating function of an auxiliary sequence $w_n$.
3. Explicit Formula: They derive a closed-form expression for $u_n$ as a finite sum involving binomial coefficients and the auxiliary sequence $w_n$:
   $$u_n = (-1)^n \sum_{k=0}^{\lfloor n/2 \rfloor} (-1)^k c^k (n-2k)! w_{n-2k} \binom{n-k}{k} \binom{2n-2k}{n-k}$$
   Since it is known that $n! w_n$ has integer coefficients, the combinatorial factors in the sum ensure that $u_n$ itself has integer coefficients.

B. Generalization (Theoretical Framework)

To generalize this result, the authors introduce a broader class of recursions:
$$n u_n = \sum_{i=1}^n p_i(n-i/2) u_{n-i}$$
where $p_i$ are odd polynomials.

• Algebraic Structure: They define a sequence of rational functions $\langle Q \rangle_m$ recursively.
• Generating Functions: They construct a multivariate generating function $F$ and analyze it using a specific derivation operator $\delta = \sum x_i z_i \frac{\partial}{\partial z_i}$.
• Differential Identity: By manipulating the differential equation satisfied by $F$, they prove a quadratic identity of the form:
  $$F^2 = 1 + \sum z_i \tilde{P}_i(F)$$
  where $\tilde{P}_i$ are polynomials with coefficients in $\mathbb{Z}[1/2]$.
• Inductive Argument: Using this identity, they prove by induction on the degree of the multi-index $m$ that the coefficients of the generating function lie in $\mathbb{Z}[1/2]$.

3. Key Contributions and Results

Theorem 1: Integrality of the Specific Sequence

The authors prove that the sequence $u_n$ generated by the recurrence related to the Hörmander–Bernhardsson function consists of polynomials in parameters $b$ and $c$ with integer coefficients ($u_n \in \mathbb{Z}[b, c]$).

• Significance: This confirms the conjecture that this specific "irregular" example possesses a strong integrality property, contrasting with the auxiliary sequence $w_n$ (which only satisfies $n! w_n \in \mathbb{Z}[b, c]$).

Theorem 2 & 3: General Integrality Statement

The paper establishes a general theorem for sequences defined by:
$$n u_n = \sum_{i=1}^n p_i(n-i/2) u_{n-i}$$
where $p_i$ are odd polynomials with integer coefficients.

• Result: The sequence $u_n$ lies in $\mathbb{Z}[1/2]$ (integers with powers of 2 in the denominator).
• Refinement: The power of 2 in the denominator grows at most linearly with the index $n$.
• Mechanism: The proof relies on the fact that the generating function satisfies a specific algebraic differential equation that forces the coefficients to be integral (up to powers of 2).

4. Significance and Implications

1. Resolution of a Conjecture: The paper provides a rigorous proof for a conjecture regarding the integrality of coefficients in the Hörmander–Bernhardsson extremal function, a problem previously only supported by numerical evidence.
2. Unification of Phenomena: It demonstrates that integrality is not limited to differential equations with regular singularities (like Apéry's case) but can also occur in systems with irregular singularities, provided the recurrence structure satisfies specific symmetry conditions (odd polynomials).
3. New Proof Techniques: The method of using the symmetric square of a differential equation to transform a difficult recurrence into a solvable identity (Eq. 10) offers a new tool for analyzing integrality in combinatorial sequences.
4. General Theory: The general theorems (2 and 3) provide a robust framework for identifying integrality in a wide class of recursively defined sequences, expanding the known landscape of "Apéry-like" integrality phenomena.

Conclusion

The paper successfully bridges the gap between numerical observation and rigorous proof for a specific integrality conjecture arising from mathematical physics (extremal functions). By generalizing the underlying algebraic structure, the authors show that this phenomenon is a consequence of specific symmetries in the generating functions of these sequences, specifically relating to the behavior of odd polynomials under differential operators.