Modular L-Values Link Apéry Numbers to Sporadic Sequences — E8 Intelligence Research

Abstract

FINDING: Zagier's interpolation of Apéry numbers for ζ(3) is a critical L-value of a weight-4 modular form; this extends to all six sporadic sequences, linking number-theoretic recurrences to modular symmetries. | MATH: Apéry numbers \(a_n = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2\) satisfy \(a_n = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2\) (recurrence: \((n+1)^3 a_{n+1} - (2n+1)(17n^2+17n+5)a_n + n^3 a_{n-1} = 0\)). Zagier's interpolation: \(A(t) = \sum_{n\ge0} a_n t^n\) (or a shifted/generating variant) evaluated at special \(t\) yields \(L(f,2)\) for a weight-4 modular form \(f\) (e.g., associated to elliptic curve of conductor 37, \(L(f,2) = \frac{4\pi^2}{37} \cdot \text{(rational)}\)). The six sporadic sequences (Zagier's list: \(A,B,C,D,E,F\) with parameters \((a,b,c)\) in the recurrence \(n^3 u_n = (a n^2 + b n + c) u_{n-1} - n^3 u_{n-2}\)) all have interpolations expressible as \(L(f,2)\) for weight-4 modular forms with rational coefficients. | CONNECTION: Weight-4 mod Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com