2F1 Hadamard Product Database

Database of Hadamard Products of Modified Hypergeometric Functions

This database catalogs selected Hadamard products of modified hypergeometric functions $f_{k,r,s}$, defined by

$$f_{k,r,s}(z) = (1-z)^k {_2F_1}\left( {r,s \atop 1}; z \right).$$

For datum $(k_1,r_1,s_1) \star (k_2,r_2,s_2)$, the database records cases whose truncated Hadamard-product sums at $z=1$ exhibit supercongruence behavior and often match Fourier coefficients of modular forms modulo $p^2$.

$$\sum_{n=0}^{p-1} f_{n,1} f_{n,2} \equiv a_p(k_1,r_1,s_1,k_2,r_2,s_2) \pmod{p^2}.$$

When $k_1=k_2=0$, the Hadamard product reduces to a generalized hypergeometric function ${_4F_3}$, including examples related to one-parameter hypergeometric Calabi-Yau threefolds.

Database structure

Datum: $(k_1,r_1,s_1,k_2,r_2,s_2)$.
Series expansion: coefficients through the displayed initial terms.
Equivalence classes: data with identical initial expansions are grouped.
Residues: supercongruence coefficients for primes $5 \leq p \leq 199$.
Labels: OldLabel records the observed $|a_p|$ pattern.
Picard-Fuchs data: operator degree, discriminant, and Riemann-symbol data.