PaperlayerView exact revision on arXiv

Source-linked AI summary

An Infinite Family of Apéry-like Recurrences for Euler's Constant

Rotem Kalisch, Tali Monderer, Elyasheev Leibtag, Shachar Weinbaum, Michael Shalyt, Ido Kaminer

arXiv:2610.13919v1math.NTcs.SC

TL;DR

Euler’s constant remains without a proof of irrationality, and few Apéry-like recurrences approximating it are known. The paper uses a Meijer G-function identity and a Conservative Matrix Field to generate holonomic sequences; convergence criteria are proven for a specified trajectory family, but the approximations do not yield an irrationality proof.

  • Problem

    Euler’s constant has no proven irrationality result, and only a few known Apéry-like recurrences approximate it.

    Source: §2 · p. 1

  • Method

    The paper combines a Meijer G-function identity with a Conservative Matrix Field construction to generate holonomic sequences converging to γ.

    Source: §2 · p. 1, §4 · p. 8

  • Results

    The dual convergence criteria are proven for trajectories in T*_3, while convergence for the broader T_r family is conjectured.

    Source: §9 · p. 24

  • Takeaways & Limitations

    The construction provides a systematic framework for generating families of Apéry-like sequences converging to γ.

    Source: §2 · p. 1, §9 · p. 24

    Empirical evaluations indicate that the sequences do not directly contribute to an irrationality proof for γ.

    Source: §8.3 · p. 22

Abstract

from arXiv · show

Euler's constant $γ$, also known as the Euler-Mascheroni constant, remains one of the most elusive constants in mathematics. Despite extensive study, its irrationality is still unproven, and Apéry-like approximations of it are remarkably scarce. We present a novel identity that expresses $γ$ in terms of a Meijer G-function and its first few derivatives. This identity is then generalized and used to systematically generate infinitely many Apéry-like approximations of $γ$ using a Conservative Matrix Field construction based on the Meijer G-function.

2 Introduction

Euler’s constant remains without an irrationality proof, and only a few Apéry-like recurrences for γ are known. This work uses a Meijer G-function identity and Conservative Matrix Field construction to generate holonomic sequences converging to γ.

2 Introduction

  • The rank-r Conservative Matrix Field has dimension 2r − 1 and assigns a trajectory matrix to each source and trajectory choice.The trajectory matrices represent linear difference systems.Source: §2 · p. 2
  • A novel identity connecting the Meijer G-function to γ supplies initial value vectors for the construction.The paper presents this identity as the algebraic component of its approach.Source: §2 · pp. 1–2
  • The broader family T_r is proposed for arbitrary rank r ≥ 3, but its convergence is conjectured and supported by numerical evidence.It contains T*_3 when r = 3.Source: §2 · p. 2
  • For trajectories in T*_3 and generic vectors, the dual system isolates the initial state up to scaling and gives the stated limit to γ.The convergence criteria are proven for T*_3; convergence for the broader T_r family is conjectured.Source: §2 · p. 2
  • Empirical convergence of the approximations is insufficient to establish an irrationality proof for γ.The authors hope that broader sequence families and matrix analysis may contribute to such a proof.Source: §2 · p. 2
  • The four-stage pipeline turns analytic contiguous relations into discrete systems, then uses the cyclic vector method to produce order-r recurrences.It constructs a CMF, derives initial values, selects a trajectory, and converts the system into a holonomic sequence.Source: §2 · p. 2

3 Background and Notation

The background introduces the special functions and recurrence systems used to build Apéry-like sequences. It also establishes how Conservative Matrix Fields encode parameter shifts and how their trajectories yield difference systems.

3.1 Special Functions

  • The Meijer G-function is D-finite and is defined by a contour integral whose contour must be chosen for convergence.The contour may separate gamma-function poles or loop around one family of poles; different valid choices give the same function.Source: §3.1 · pp. 3–4, Figure 2 · p. 4

3.2 Difference Equations and Holonomic Sequences

  • Holonomic sequences satisfy polynomial-coefficient recurrences, which can be represented as first-order matrix systems and uncoupled into Apéry-like recurrences by the cyclic vector method.The cyclic vector method constructs a companion-form system with rational functions of n in its rightmost column.Source: §3.2 · p. 5
  • A unique recessive solution of a difference system becomes dominant in the dual system, whose iteration isolates that solution’s initial state up to scaling.The isolation holds for any vector generic to the dual system.Source: §3.2 · p. 6

3.3 The Conservative Matrix Field

  • A Conservative Matrix Field is a discrete analogue of a conservative vector field, with basis matrices satisfying a path-independence conservation property.The basis matrices play the role of a discrete gradient, and their shifts obey the stated compatibility relation.Source: §3.3 · pp. 6–7

3.4 D-finite Conservative Matrix Fields

  • For a D-finite function, contiguous relations express parameter shifts as rational combinations of its derivatives, yielding matrices that form a CMF basis.The state vector uses derivatives through one less than the differential equation’s order; the discrete parameter and continuous variable are encoded separately.Source: §3.4 · pp. 7–8

4 Approximation Construction Framework

The framework combines a Meijer G-function identity for γ with a Conservative Matrix Field (CMF) whose trajectories can isolate the initial state. For rank 3, this yields infinitely many γ approximations that can be converted into Apéry-like recurrences under a regular, invertible gauge.

4 Approximation Construction Framework

  • The CMF uses Meijer G-function parameters (m, n, p, q) = (r, r − 1, r − 1, r), with rank r and lattice dimension 2r − 1.Its initial value vectors p_r and q_r are defined explicitly, and trajectories are parameterized by upper and lower entries.Source: §4 · p. 8
  • Convergence is proved for a subfamily of rank-3 trajectories, while the claim for all r ≥ 3 and all trajectories in T_r is conjectured.The theorem’s stated scope is rank 3 and T*3; the broader result is Conjecture 1.Source: §4 · pp. 8–9
  • For rank 3, every trajectory in T*3 and generic dual vector yields a CMF limit evaluating to γ, and qualifying systems convert to Apéry-like recurrences.The conversion preserves the evaluation when the resulting gauge is regular and invertible for every n in N0.Source: §4 · p. 9
  • The construction combines an algebraic identity linking the CMF state vector at the origin to γ with a dynamical result that isolates that state vector.The dynamical result applies to infinitely many trajectories; the algebraic identity is stated for r ≥ 3.Source: §4 · p. 9

5 The Meijer G-function and Euler’s Constant Identity

This section develops a Meijer G-function identity for Euler’s constant and uses it to obtain initial value vectors for the CMF. It supports both a general operator formulation and a rank-three expression for γ.

5 The Meijer G-function and Euler’s Constant Identity

  • The section proves a novel identity connecting the Meijer G-function and Euler’s constant, the algebraic component used in the paper’s construction.The CMF state vector at the origin encodes derivatives of the basis function, allowing polynomial differential operators to be represented by coefficient vectors.Source: §5 · p. 10, Then the numerator and denominator… · p. 10

Then the numerator and denominator operators

  • The numerator and denominator operators are N_p = (1 + θ)^(p−2) + (−1)^p θ^(p−1) and D_p = (1 + θ)^(p−1) + (−1)^p θ^p.Their coefficient vectors give the initial value vectors p_r and q_r.Source: Then the numerator and denominator… · p. 10

5.1 Integral Representation and WZ Certificates

  • WZ certificates modify the Meijer G-function integrand through differential operators, with the residue expansion remaining absolutely convergent at z = 1.The resulting WZ pair telescopes when the contour is closed to the right, leaving the residue at s = −1 after the boundary term vanishes.Source: §5.1 · pp. 10–11, §5.2 · p. 12

5.2 Proof of Proposition 1

  • For p = 2, the proposition gives the rank-three value form (1 + θ)G_2(1) = γ.The operator 1 + θ shifts one upper Meijer G-function parameter from 0 to −1.Source: §5.2 · p. 12

5.3 Examples

  • The examples yield p_3 = (1, 1, 0)^T and q_3 = (1, 1, 1)^T, and p_6 = (1, 3, 3, 1, −1, 0)^T and q_6 = (1, 4, 6, 4, 1, −1)^T.For the p = 5 example, the displayed operator D_5 evaluates to 1 at G_5(1).Source: §5.3 · p. 13

6 The Meijer G-function CMF

This section gives a symbolic construction of a D-finite CMF for the Meijer G-function. It derives basis matrices from the function’s differential equation and parameter-shift identities.

6 The Meijer G-function CMF

  • The CMF construction uses Meijer G-function parameters satisfying 0 ≤ m ≤ q and 0 ≤ n ≤ p, and orders them as the a-parameters followed by the b-parameters.The rank is defined as max(p, q), and the lattice dimension as p + q.Source: §6 · p. 13
  • The construction defines a state vector and uses the Meijer G-function differential equation and parameter-shift identities to derive basis matrices.Negative-direction basis matrices are inverted after construction to complete the CMF.Source: §6 · pp. 13–14, We extract our coefficients · p. 15

We extract our coefficients

7 Dual Limit Convergence

The section proves that selected Meijer G-function state vectors are strongly recessive solutions of trajectory systems, enabling dual iteration to isolate initial states. The isolation is preserved by qualifying rational gauges and by conversion to Apéry-like recurrence form.

7 Dual Limit Convergence

  • The dual-convergence proof applies Miller’s algorithm to the asymptotic behavior of the Meijer G-function in the paper’s parameterization.For a generic vector, the dual system isolates the initial state up to scaling; cyclic-vector conversion preserves the isolation when the gauge is regular and invertible on N0.Source: §7 · p. 15, §7.3 · p. 19

7.1 Converging Trajectories

  • For trajectories t = (a_1, a_2; 1, 1, 1), the Meijer G-function state vector S_3(nt) is a strongly recessive solution of the trajectory system.The result follows by identifying it with I_3(n), which decays faster than any power of n relative to the other solutions.Source: §7.1 · pp. 15, 17
  • The three auxiliary branches satisfy the same trajectory system, and the canonical branch is the Meijer G-function state vector; the three solutions are linearly independent.The middle branch requires a sign adjustment, while the branches with m = 1 and m = 3 satisfy the same system directly.Source: §7.1 · pp. 15–17

7.2 Rational Gauge Invariance

  • A rational gauge transformation preserves a strongly recessive asymptotic when the gauge is regular and invertible on the relevant nonnegative integers.The transformed asymptotic is the original one multiplied by the gauge matrix.Source: §7.2 · p. 18

7.4 Example

  • In the rank-three example, the computed asymptotic splitting agrees with the predicted splitting, and cyclic-vector conversion preserves exponential scales up to polynomial factors.The displayed trajectory uses upper parameters (0, 0) and lower parameters (1, 1, 1).Source: §7.4 · p. 19

8 Examples

The examples show how the rank-3 Conservative Matrix Field generates holonomic sequences for γ, and contrast guaranteed convergence with trajectories that fail the criteria. Convergence rates vary across trajectories, and one converging trajectory is gauge-related to Rivoal’s recurrence.

8.1 Full Algorithm Showcase

  • The rank-3 CMF has lattice dimension 5 and yields order-3 holonomic sequences; the example fixes the source at the origin and selects an initial-value matrix for convergence to γ.The showcase uses t = (0, 0; 1, 1, 1)T, a trajectory in a family whose convergence is mathematically guaranteed.Source: §8 · pp. 19–20, §8.1 · p. 20
  • The full showcase derives a trajectory matrix, transforms initial conditions with the CVM gauge matrix, and applies a gauge transform to obtain a companion matrix representing a holonomic sequence.The selected trajectory belongs to the verified converging family.Source: §8.1 · p. 20

8.2 Trajectories Violating the Convergence Criteria

  • The violating-trajectory examples test the convergence criteria through companion-form asymptotics; a rational gauge preserves the separation hierarchy when the unique recessive solution is exponentially separated or more.The analysis uses the Birkhoff-Trjitzinsky engine, validated against RISC’s Asymptotics package.Source: §8.2 · pp. 20–21

8.2.1 Example 1

  • The trajectory t1 = (0, 0; 0, 0, 1)T has a unique recessive solution that is not the state vector, so its dual system is expected to converge to a different constant.Figure 3 shows this trajectory converging to a different constant.Source: §8.2.1 · p. 21, Figure 3 · p. 22

8.2.2 Example 2

  • For t2 = (0, 0; 0, 1, 1)T, two conjugate recessive solutions in the primal system lead to an expected oscillation in the dual system.Figure 3 reports that this trajectory oscillates and diverges.Source: §8.2.2 · p. 21, Figure 3 · p. 22

8.2.3 Example 3

  • For t3 = (0, 0; 1, 2, 2)T, the two conjugate recessive primal solutions likewise predict oscillation in the dual system.Figure 3 reports that this trajectory oscillates and diverges.Source: §8.2.3 · p. 22, Figure 3 · p. 22

8.3 Convergence Rate Comparison

  • Convergence per step increases with larger |a_i| and decreases with larger |b_j|; Rivoal’s and Aptekarev et al.’s recurrences have rates similar to existing trajectories.Figure 4 compares logarithmic error decay by trajectory depth and by lattice-step complexity; its references are omitted from the normalized plot because they are not CMF lattice trajectories.Source: §8.3 · p. 22, Figure 4 · p. 23

8.4 A connection to Rivoal’s recurrence

  • The trajectory t = (0, 0; 1, 1, 1)T is connected to Rivoal’s recurrence [16] by a gauge transform up to a scalar factor, consistent with their similar convergence rates.The authors give the transform relating the companion forms.Source: §8.3 · p. 22, §8.4 · p. 23

9 Conclusion

The conclusion presents a framework for generating candidate γ-approximating sequences, combining an exact Meijer G-function identity with dual-system convergence. It distinguishes a proven converging subfamily from a larger conjectured family and outlines extensions and open structural questions.

9 Conclusion

  • For r ≥ 3, the framework conjectures initial vectors and infinitely many trajectories whose dual CMF limits converge to γ; convergence is proven for T*3, while convergence of the larger Tr family is conjectured.The initial vectors are derived from a Meijer G-function identity, and the dual system isolates the initial state up to scaling when the convergence criteria hold.Source: §9 · p. 24

9.1 The Discovery Process

  • Numerical exploration of 2F2 within CMFs revealed γ, and PSLQ searches over Meijer G-function evaluations helped the authors reverse-engineer and prove their identity.The Meijer G-function perspective was motivated by its relation to the generalized hypergeometric function pFq.Source: §9.1 · p. 24

9.2 Future Work

  • Future work includes extending the identity to γ + log(z), testing other D-finite parameterizations, embedding recurrences in CMFs, and strengthening CMF theory, whose properties remain partly supported by numerical evidence.The authors also ask whether known Apéry-like approximations can be represented as discrete CMF trajectories.Source: §9.2 · pp. 24–25

Declaration of AI use

The first author reports using GPT-5.6 sol while developing a proof variant for Lemma 8.

Declaration of AI use

  • While developing Lemma 8, the first author used GPT-5.6 sol to study unfamiliar mathematical techniques and develop a proof variant later written by the authors.The authors locate the final proof in Section A.Source: Declaration of AI use · p. 25

A Proof of Lemma 8

The proof of Lemma 8 constructs saddle-point contours for the integral representations, applies uniform saddle asymptotics, and establishes the resulting solutions’ linear independence.

A Proof of Lemma 8

  • The parameter-uniform Laplace method is used because the phase and amplitude still depend on the asymptotic parameter after normalization.The cited method provides a uniform expansion for a fixed contour when phase and amplitude vary with an auxiliary parameter in a compact set.Source: A Proof of Lemma 8 · p. 27
  • The proof identifies exact saddle points, builds limiting steepest-descent contours, pulls them back to Mellin–Barnes contours, and applies uniform Laplace asymptotics to obtain the solutions’ growth.The contour construction is arranged to pass through the exact saddles.Source: A Proof of Lemma 8 · pp. 28, 30, 32
  • The three solution components share the same complex exponential amplitude, while polynomial factors determine the state vector’s asymptotic shape.The proof uses these asymptotics, together with exponential separation, to establish linear independence.Source: A Proof of Lemma 8 · p. 32
  • The proof concludes that the three functions are linearly independent.Source: A Proof of Lemma 8 · p. 32
Loading 2610.13919v1…