Finite Hypergeometric Sums

• Finite hypergeometric sums are finite field analogues of classical hypergeometric series constructed from Gauss, Jacobi, and exponential sums.
• They are applied in arithmetic geometry to count points on varieties and to connect with modular forms, p-adic series, and hypergeometric motives.
• Their rich transformation formulas and symmetric properties mirror classical identities while enabling efficient combinatorial and algorithmic computations.

Finite hypergeometric sums are finite field analogues of classical generalized hypergeometric series, describing a wide range of arithmetic phenomena via explicit character sums constructed from Gauss sums, Jacobi sums, and exponential sums over finite fields. Their formulation, transformation properties, and applications—particularly in arithmetic geometry and the theory of modular forms—reflect deep parallels with the classical theory over ℂ, while also revealing new phenomena unique to positive characteristic arithmetic and finite group representations.

1. Fundamental Definitions and Character Sum Formulation

Finite hypergeometric sums generalize the classical hypergeometric function

$${}_{r}F_{s}(a_1,\ldots,a_r; b_1,\ldots,b_s; z) = \sum_{n=0}^\infty \frac{(a_1)_n\cdots(a_r)_n}{(b_1)_n\cdots(b_s)_n}\frac{z^n}{n!}$$

by using character sums instead of Pochhammer symbols and gamma functions. For a finite field $\mathbb{F}_q$ and multiplicative characters $A_1, \ldots, A_m,$ $B_1, \ldots, B_n$ of $\mathbb{F}_q^{\times}$, one typical normalized form, due to McCarthy, is

$${}_{n+1}F_n\left( \begin{matrix} A_0, A_1, \ldots, A_n\ B_1, \ldots, B_n \end{matrix}; x\right)^* = \frac{1}{q-1} \sum_{\chi} \frac{\prod_{i=0}^n g(A_i\chi)}{\prod_{j=1}^n g(B_j\chi)} g(\chi^{-1})\chi\big(({-1})^{n+1} x\big)$$

where $g(\chi)$ denotes the Gauss sum of $\chi$, and the sum is over all multiplicative characters $\chi$ of $\mathbb{F}_q^{\times}$, extended to zero by $\chi(0) = 0$ (McCarthy, 2012). Greene's classical construction uses Jacobi sums rather than Gauss sums, and further variants (such as Katz's or the 'A-hypergeometric' functions) can be defined via Fourier expansions of twisted exponential sums (Adolphson, 2012).

Further generalizations connect these functions to the geometry of toric varieties and exponential sums on algebraic tori, linking the sums to the notion of a gamma triple $(\gamma, \delta, N)$ encoding parameters via the Gale dual configuration associated to a Laurent polynomial with $d+2$ monomials (Abdelraouf, 22 Sep 2025).

2. Transformation and Summation Formulas

Finite hypergeometric sums satisfy a rich collection of transformation and summation formulas analogous to those found in the classical theory (Gauss, Kummer, Dixon, Whipple, Pfaff-Saalschütz, and others). Examples include:

• Finite Field Analogues of Classical Transformations: A Whipple-type formula reduces certain $_4F_3^*$ functions to products involving $_3F_2^*$ (McCarthy, 2012). Kummer, Dixon, Pfaff–Saalschütz, and other transformation identities hold with finite-field-specific normalization factors and delta terms that account for degenerate cases (Fuselier et al., 2015).
• Summation Formulas: Expressions for $\sum_x \varphi(x) \, {}_{n+1}F_n(x)$ yield closed-form results, sometimes involving lower-level hypergeometric functions (Pal et al., 2021).
• p-adic Analogues: Finite hypergeometric sums are related to $p$-adic hypergeometric series. Transformations have been shown for $p$-adic analogues mirroring classical relations, such as Kummer and Clausen transformations, with explicit connections via the Gross–Koblitz formula (Barman et al., 2018).

Transformation formulas often rest on properties of Gauss and Jacobi sums, notably the Davenport–Hasse multiplication formula, which replaces the classical multiplication and reflection formulas of the gamma function (Otsubo, 2021).

3. Relationships with Classical and p-adic Hypergeometric Series

Finite hypergeometric sums mirror their analytic counterparts in several ways:

• Structural Analogy: Both theories involve parameters in numerator/denominator and satisfy analogous transformation laws. The finite field theory replaces Pochhammer symbols and gamma functions with Gauss and Jacobi sums.
• Symmetries and Fields of Definition: Finite sums possess symmetries among parameters corresponding to automorphisms in the associated cyclotomic fields. This symmetry is crucial for understanding their field of definition and invariance properties (Beukers, 2018).
• Extension to p-adic Series: Through the Gross–Koblitz and Hasse–Davenport formulas, finite sums relate to $p$-adic hypergeometric series, underlining deep connections with supercongruences and modularity phenomena (Barman et al., 2018, Fuselier et al., 2015).

In specific normalization regimes, finite field hypergeometric functions can be specialized to recover $p$-adic and classical (truncated) series, and vice versa.

4. Arithmetic and Geometric Applications

Finite hypergeometric sums have broad implications in arithmetic geometry, modular forms, and zeta/L-functions:

• Point-Counting on Varieties: Rational point counts for families such as the Dwork and K3 surfaces, diagonal and toric hypersurfaces, and hyperelliptic curves can be expressed via finite hypergeometric sums. These point counts, in turn, determine the zeta-functions of the varieties and encode deep cohomological invariants (Abdelraouf, 22 Sep 2025, Fuselier et al., 2015, Kewat et al., 2021, Otsubo, 2021).
• Modular Forms and Hecke Eigenvalues: The values of finite sums can, in many cases, be interpreted as (normalized) traces of Frobenius acting on the cohomology of curves, hence aligning them with Fourier coefficients of modular forms via the Eichler–Selberg trace formula. This connection is fully realized in families such as the Legendre or Clausen elliptic curves (Dawsey et al., 2021, Pal et al., 2021, Ono et al., 2021).
• Hypergeometric Motives: The framework of hypergeometric motives links these sums to $\ell$-adic sheaves whose Frobenius traces realize the hypergeometric sums, unifying the motivic, cohomological, and arithmetic pictures (Hoffman et al., 2020, Fu, 2012).
• Supercongruences and Arithmetic Superidentities: Special values and congruences for finite hypergeometric sums underlie a wide class of supercongruence phenomena for truncated hypergeometric series, often implicated in Apéry-style irrationality proofs and explicitly computed in the form of mod $p^k$ congruences (Ono et al., 2021, Dawsey et al., 2021, Pal et al., 2021).

5. Value Distribution and Statistical Properties

The distribution of finite hypergeometric sums (across field parameters or over character tuples) shows statistical behaviors matching random matrix ensembles:

• Semicircular Distribution: For the family of normalized $_2F_1$ sums, the limiting even moments as $q \to \infty$ are Catalan numbers, corresponding to the moments of the semicircular law for $SU(2)$ (Ono et al., 2021).
• Batman Distribution: For $_3F_2$-type sums, the moments correspond to the trace statistics of the real orthogonal group $O_3$, which has a distinctive "Batman" shape in its limiting distribution (Ono et al., 2021).
• Moments and Moment Formulas: Explicit formulas for first and second moment sums have been computed; second moments relate directly to arithmetic quantities such as the squared traces of Frobenius for associated families of curves (Pal et al., 2021).

Such distributional results reflect the equidistribution of Frobenius eigenvalues in the relevant monodromy groups, as predicted by the Langlands program and random matrix conjectures.

6. Algorithmic and Combinatorial Aspects

Finite hypergeometric sums admit diverse computational frameworks:

• Generating Function and Polynomial Methods: Explicit generating functions, often hypergeometric in form, encode finite sums over (powers of) binomial coefficients, Bernoulli numbers, Stirling numbers, Franel, Catalan, and other combinatorial families (Simsek, 2019, Simsek, 2021).
• Telescopic Algorithms for Nested Summation: Efficient telescopic algorithms have been developed for expanding nested sums appearing in hypergeometric series, converting such sums into combinations of generalized polylogarithms, Z-sums, and other special functions (McLeod et al., 2020).
• Differentiation Techniques: Using parameter differentiation of classical reduction formulas (e.g., Chu–Vandermonde), closed-form identities involving digamma and related functions have been obtained for finite hypergeometric sums (González-Santander, 2022).

Tables of hypergeometric moments, transformation rules, and explicit combinatorial polynomials further augment the theoretical framework.

7. Extensions and Future Directions

Recent developments point to several future research avenues:

• Gamma Triples and Cyclotomic Field Extensions: The gamma triple formalism enables the definition of finite hypergeometric sums for general parameters defined over cyclotomic fields, supporting point counts for toric hypersurfaces and the realization of more general motives as sums (Abdelraouf, 22 Sep 2025).
• Relations among Sums across Field Extensions: By relating values of hypergeometric sums over extensions $\mathbb{F}_{q^r}$ and applying Newton identities, relations among sums over various finite fields are established, extending the analogy with the Davenport–Hasse relation for Gauss sums (Nakagawa, 2021).
• Trace Formulas and Automorphic Sheaves: Uniform geometric techniques connect hypergeometric character sums to traces of Hecke operators for both modular and Shimura curves, furthering the connections to the theory of automorphic forms and the modularity of Galois representations (Hoffman et al., 6 Aug 2024).
• Connections to p-adic and $L$-functions: The full range of implications for $p$-adic analysis, $L$-functions, and potential links to mirror symmetry and modularity conjectures remain active research areas.

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Table: Key Constructs in Finite Hypergeometric Sums

Notion	Definition/Example	Reference
Finite Hypergeometric Sum	$$\frac{1}{q-1} \sum_\chi \frac{\prod_i g(A_i\chi)}{\prod_j g(B_j\chi)} g(\chi^{-1})\chi((-1)^{n+1}x)$$	(McCarthy, 2012)
Gauss/Jacobi Sums in Hypergeom. Sums	$g(\chi)$ and $J(A, B)$ (products appear in numerator/denominator)	(Fuselier et al., 2015)
Gamma Triple	$(\gamma, \delta, N)$ encoding parameters, cyclotomic field	(Abdelraouf, 22 Sep 2025)
Toric Hypersurface Newton Polytope	Defines ambient variety for geometric realization	(Abdelraouf, 22 Sep 2025)
Moment Formulas	$$\lim_{p\to\infty} \mathbb{E}[({}_2F_1(X)_q)^{2n}] = \frac{(2n)!}{n!(n+1)!}$$	(Ono et al., 2021)
Semicircular/Batman Distributions	Limiting law for $_2F_1$ and $_3F_2$ function values	(Ono et al., 2021)

The theory of finite hypergeometric sums thus forms a central organizing principle illuminating the connection between special functions, arithmetic geometry, modular forms, Galois representations, exponential sum distributions, and $L$-functions, underpinned by explicit explicit character sum formulas and their rich transformation theory.