Database of Hadamard Products of Modified Hypergeometric Functions

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

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

We conducted a preliminary investigation into Hadamard products $(k_1,r_1,s_1) \star (k_2,r_2,s_2)$ exhibiting notable supercongruence properties at $z=1$. Denoting the series expansions $f_{k_i,r_i,s_i} = \sum_{n=0}^\infty f_{n,i} z^n$ ($i=1,2$), we observe that for certain parameters, the truncated sum

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

corresponds modulo $p^2$ to the $p$-th Fourier coefficient of a modular form.

When $k_1 = k_2 = 0$, the Hadamard product reduces to a generalized hypergeometric function:

$${_2F_1}\left( {r_1,s_1 \atop 1}; z \right) \star {_2F_1}\left( {r_2,s_2 \atop 1}; z \right) = {_4F_3}\left( {r_1,s_1,r_2,s_2 \atop 1,1,1}; z \right).$$

Specific parameter choices, such as $(r_1,s_1,r_2,s_2) = (\frac{1}{4},\frac{1}{2},\frac{1}{2},\frac{3}{4})$, correspond to one-parameter families of hypergeometric Calabi-Yau threefolds (e.g., $X_{4,2}(1^6)$, the complete intersection of a quartic and a quadric in $\mathbb{P}^5(1,1,1,1,1,1)$). The conifold fiber at $z=1$ admits smooth rigid Calabi-Yau resolutions, inducing supercongruences via modularity. For the $(\frac{1}{4},\frac{1}{2},\frac{1}{2},\frac{3}{4})$ case, the associated modular form is $\mathsf{16.4.a.a} = \mathsf{8.4.a.a} \otimes \chi_{-4}$ (LMFDB label), yielding

$$a_p\left(0,\tfrac{1}{4},\tfrac{1}{2},0,\tfrac{3}{4},\tfrac{1}{2}\right) \equiv a_p(\mathsf{16.4.a.a}) \pmod{p^2}.$$

Database Structure

1. Datum: A tuple $(k_1,r_1,s_1,k_2,r_2,s_2)$ parametrizing a Hadamard product.
2. Series Expansion: For each datum, we compute the series expansion $F(z) = 1 + f_1 z + f_2 z^2 + f_3 z^3 + f_4 z^4 + f_5 z^5 + O(z^6)$.
3. Equivalence Classes: Data yielding identical series expansions up to $z^5$ are grouped, as they share supercongruence properties.
4. Residue Computations: For primes $5 \leq p \leq 199$, we compute the supercongruence coefficient $a_p$.
5. Labels:
   • Old Label: Describes the $|a_p|$ sequence (minimal absolute residue in $[0, (p^2+1)/2]$).
   • Uses LMFDB labels or notations like $\mathsf{p*(1 \text{ or } 5)}$ (indicating $|a_p| \equiv p$ or $5p \pmod{p^2}$).
6. Picard-Fuchs Operator: For each $F(z)$, we compute the minimal-order linear differential operator $D$ satisfying $D(F(z)) = 0$. This operator is expressed as a polynomial in $z$ and $t = z \frac{d}{dz}$.
7. Degree Notation:
   • $t$-degree: Highest power of $t$ in $D$.
   • $z$-degree: Highest power of $z$ in $D$.
8. Discriminant: The leading coefficient (as a polynomial in $z$) of the highest $t$-degree term in $D$.
9. Riemann Symbol: Computed at critical points $z \in \mathbb{Q}$ or $z$ quadratic algebraic.