p-adic analyticity of the Apéry and Franel numbers along the towers mp^r
Year: 2026
Type: preprint
Abstract: Let A(n) = Σ_k C(n,k)² C(n+k,k)² be the Apéry numbers attached to ζ(3) (1, 5, 73, 1445, 33001, …). Gessel proved that A(mp) ≡ A(m) (mod p³) for every prime p ≥ 5, and Coster extended this to the whole p-adic tower: A(mp^r) ≡ A(mp^(r−1)) (mod p^(3r)). Recent work has determined one or two further p-adic digits of these differences. Each of these results controls a fixed number of additional digits of A(mp^r) at each step of the tower. This paper shows that, for the Apéry and the Franel numbers, all the digits are controlled at once, by a single p-adic analytic function. Main results. Theorem A (Apéry numbers). Let p ≥ 5 be a prime and 0 < σ < κ_p := (1/3 − 1/(p−1))/2. There is a constant K = K(p,σ) such that, for every integer m ≥ 1, there is a power series Φ_m(z) = Σ_a c_a(m) z^a, |c_a(m)| ≤ K·p^(−σa), with A(mp^r) = Φ_m(p^r) for every r ≥ 0. The same holds for all the moments A_b(n) = Σ_k k^b C(n,k)² C(n+k,k)². So r ↦ A(mp^r) extends to a p-adic analytic function of p^r, with a bound that does not depend on m: all the p-adic digits of A(mp^r), for all r, are encoded in the single sequence of coefficients c_a(m). These coefficients are not known to be p-integral, so the theorem does not by itself reprove the classical supercongruences; the two are compatible, and the paper explains exactly what would be needed to deduce one from the other. Theorem B (Frobenius structure). For the vector of moments V(x) = (A_0(x), A_1(x), A_2(x)) there exist matrices Θ(y), analytic on the disc |y| ≤ p^σ, and H(y), analytic on |y| < p^σ, such that:(i) V(py) = Θ(y)V(y) for every integer y ≥ 0;(ii) Θ(0) = [[1, 0, p³I₂], [0, p, 0], [0, 0, p³]], where I₂ = ∫_{Z_p^×} u^(−2) du is a Volkenborn integral;(iii) det Θ(y) = p⁴ for every y ∈ Z_p;(iv) H(0) = I, det H(y) = 1 and H(py)Θ(y) = Θ(0)H(y).Consequently V(mp^r) = H(mp^r)^(−1) Θ(0)^r H(m) V(m) for every m ≥ 1 and r ≥ 0. Theorem C (Franel numbers). The same holds for the Franel numbers f(n) = Σ_k C(n,k)³ and all their moments, now for every 0 < σ < 1/3, with Θ_F(0) = diag(1, p, p²), det Θ_F(y) = p³·8^((p−1)y) and det H_F(y) = exp(−y·L₈), where L₈ = log(8^(p−1))/(p−1). Theorem D (Domb numbers: where the method stops). For the Domb numbers D(n) = Σ_k C(n,k)² C(2k,k) C(2n−2k,n−k), the module of moments still has dimension 3, but at every positive integer n the polynomials of degree ≤ 2n span only a two-dimensional subspace of it. Consequently no family of three functionals depending analytically on the parameter can be linearly independent at any point of a p-adic disc, and, for every parameter y ≠ 0, some Frobenius weights have a simple pole at the half-integer k′ = y − 1/2. The approach of Theorems B and C therefore cannot work for the Domb numbers, whatever polynomial basis is chosen. Whether the Domb numbers are analytic along the towers remains open. Why this matters. As far as we know, this is the first result that controls all the p-adic digits of A(mp^r) along the towers at once, rather than a fixed number of them. It gives a p-adic analytic interpolation of r ↦ A(mp^r) in the variable p^r, uniformly in m. The method applies Dwork's ideas in the summation index k, with the upper index n as the parameter, instead of the variable t of the generating function Σ A(n)t^n. The splitting k = pk′ + t with Morita's gamma function is classical (it goes back to Gessel), and very recently H. Zheng (arXiv:2609.23355) used a transfer operator in the summation variable, with analytic weights, to prove a two-step supercongruence for another Apéry-like sequence. What is new here is the use of such a structure to control all the digits at once: the analytic Frobenius matrix on the module of moments, the Volkenborn calculus of its resonant coefficients, the analytic normal form and the determinant identity. The decisive identity det Θ = p⁴ comes from a Casoratian that is identically equal to 1. It is built from three explicit functionals, one of which is a regularized Volkenborn integral with weights H_j^(3) = Σ_{i≤j} i^(−3) and H_j^(2) = Σ_{i≤j} i^(−2). The first is the harmonic number that builds Apéry's second solution in his proof of the irrationality of ζ(3), where the key identity is also a Casoratian; here it is a Casoratian in the transverse direction, that of the summation variable. The constant entry of the Frobenius matrix is p³I₂ with I₂ = ∫_{Z_p^×} u^(−2) du = 2ζ_p(3), a p-adic counterpart of Apéry's ζ(3); modulo p it is (2/3)B_{p−3}, the Bernoulli number in J.-C. Liu's congruence for the next digit of the Gessel–Coster differences. The paper also identifies the exact boundary of the method: the Domb numbers, for a structural reason that is proved, not only observed. Method. (1) Frobenius structure in the summation variable. Writing k = pk′ + t with 0 ≤ t < p, Morita's p-adic gamma function factors the summand as T(py, pk′+t) = T(y,k′)·ω_t(y,k′), where ω_t is analytic in the two variables (y, k′). The recurrence of the summand in k reduces every moment to the first three, with coefficients that are polynomials in y. Together these give V(py) = Θ(y)V(y) with Θ analytic.(2) Volkenborn calculus. At y = 0 the reduction to three moments is given by the Volkenborn integral, and its Taylor expansion in y is computed by indefinite sums. This gives Θ(0) and the eight Taylor coefficients of Θ that control the resonances. All eight vanish, for reasons as simple as Σ_{t<p} 1/t − Σ_{t<p} 1/t = 0 and ∫_{Z_p^×} u^(−1) du = 0.(3) Analytic normal form. Since the resonant coefficients vanish, an analytic change of basis H(y) conjugates Θ(y) to the constant matrix Θ(0) along Frobenius.(4) The determinant. The invertibility of H, and hence the closed formula from the first step r = 0, depends on det Θ. It is proved to be p⁴ with three explicit linear functionals built from the rational solution w₂(y,k) of the recurrence satisfied by the summand in k. Frobenius transports them with factors 1, p, p³ through a rational identity between w₂(y,·) and w₂(py,·), and a telescoping certificate in y with determinant 1 shows that their Casoratian equals 1. Strassmann's theorem then extends det Θ = p⁴ from the integers to Z_p.For the Franel numbers the same scheme works with alternating residue functionals, a certificate of determinant 8 and the Casoratian 8^y. As a by-product, the moments of the Apéry numbers satisfy an explicit first-order recurrence V(y+1) = R(y)V(y) with det R(y) = 1, and at the integers Θ(y) = P(py)Θ(0)P(y)^(−1) with P(y) = R(y−1)⋯R(0). Verification. The proofs do not depend on any computation. As a safeguard, every object and identity used in them was recomputed independently with the Python scripts in this record (exact rational arithmetic, truncated p-adic arithmetic and SymPy; about 40 seconds in total; run_all.sh runs them all and the logs are included). For p = 5 and 7 they compute the Frobenius matrices from their definitions and check V(py) = Θ(y)V(y), Θ(0), the eight resonant zeros, det Θ(y) = p⁴ and det Θ_F(y) = p³·8^((p−1)y) coefficient by coefficient, the normal form and the closed formula for V(mp^r) with m, r ≤ 3. In exact arithmetic they check the certificates, C(y) = 1 and C_F(y) = 8^y, the rational transport identities and the transport of the functionals, including the Volkenborn correction term p³I₂. They also check the propositions on the Domb numbers. Limits and open questions. The Domb numbers: Theorem D rules out the present construction, but not analyticity itself, which remains open. The other sporadic Apéry-like sequences have not been treated. The primes p = 2 and 3 are excluded, and the radius p^σ with σ < κ_p is not optimal: numerically the Taylor coefficients of Θ have valuation at least 0.8n. Integrality of the Taylor coefficients of H (which holds numerically) would let one deduce the classical supercongruences from the closed formula. Authorship. The author of this work is Roberto Blanco Gómez. The exploratory computations that led to it were carried out with SyntheticMind, the mathematical reasoning engine that the author created, designed and programmed himself. The AI system Claude (Anthropic; models Claude Fable 5.1 and Claude Opus 5.5) collaborated as an assistant, under the author's direction, in finding and checking the arguments, in independent review passes over the proofs, in the literature search, in the verification scripts and in the writing. The author has checked the arguments and computations and takes full responsibility for the content. Files: Blanco_Gomez_2026_Apery_Franel_analyticity_EN.pdf: the paper. Blanco_Gomez_2026_Apery_Franel_analyticity_ES.pdf: the Spanish version, with the same content. verification.zip: the verification scripts (Python 3 with SymPy), README.txt, run_all.sh and the logs. sources.zip: LaTeX sources of both versions.
Author: Blanco Gómez, Roberto
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225188653