Distribution of values of Gaussian hypergeometric functions

Ono, Ken; Saad, Hasan; Saikia, Nipen

doi:10.48550/arxiv.2108.09560

Cited by 4 publications

(13 citation statements)

References 61 publications

(69 reference statements)

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“…In this section, we show how to use the inductive representation introduced in §3 to evaluate first and second moments of Gaussian hypergeometric functions over F q . In [13], explicit representations of moments of the hypergeometric functions 2 F 1 and 3 F 2 were given. Namely, the sums of the form…”

Section: Evaluation Of Moments Of Gaussian Hypergeometric Functionsmentioning

confidence: 99%

“…Remark 1. Proposition 4.2 was proved for n = 1 and q ≡ 3 (mod 4) in [13,Proposition 2.11 (3)]. When q ≡ 1 (mod 4), comparing Proposition 4.2 to [13, Proposition 2.11 (4)] implies the existence of a rational number D(q) ∈ [−6, 6] such that gcd(s,q)=1 s≡q+1 (mod 8)…”

Section: Traces Of Elliptic Curves and Momentsmentioning

confidence: 99%

“…Let ǫ and φ be the trivial and quadratic character, respectively, over a finite field k. In [13], the asymptotic behaviour of moments of certain Gaussian hypergeometric function was studied.…”

Section: Introductionmentioning

confidence: 99%

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Moments of Gaussian hypergeometric functions over finite fields

Pal¹,

Roy²,

Sadek³

2021

Preprint

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We prove explicit formulas for certain first and second moment sums of families of Gaussian hypergeometric functions n+1Fn, n ≥ 1, over finite fields with q elements where q is an odd prime. This enables us to find an estimate for the value 6F5(1). In addition, we evaluate certain second moments of traces of the family of Clausen elliptic curves in terms of the value 3F2(−1). These formulas also allow us to express the product of certain 2F1 and n+1Fn functions in terms of finite field Appell series which generalizes current formulas for products of 2F1 functions. We finally give closed form expressions for sums of Gaussian hypergeometric functions defined using different multiplicative characters.

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Section: Evaluation Of Moments Of Gaussian Hypergeometric Functionsmentioning

confidence: 99%

Section: Traces Of Elliptic Curves and Momentsmentioning

confidence: 99%

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Moments of Gaussian hypergeometric functions over finite fields

Pal¹,

Roy²,

Sadek³

2021

Preprint

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“…One of the important fact about these functions is that they satisfy many analogous hypergeometric type identities. Recently Ono, Saad and the second author [21] initiated a study of value distributions of certain families of these functions over random large finite fields F q . More precisely, they investigated the distributions of the normalized values of the following two Gaussian hypergeometric functions over F q that are defined by…”

Section: Introductionmentioning

confidence: 99%

“…is the normalized Jacobi sum. Ono et al [21] showed that for the 2 F 1 (λ) q function the limiting distribution is semicircular whereas for the 3 F 2 (λ) q function the distribution is Batman. In this paper our goal is to study similar questions for p-adic hypergeometric functions over random large finite fields.…”

Section: Introductionmentioning

confidence: 99%

Sato-Tate Distribution of $p$-adic hypergeometric functions

Pujahari¹,

Saikia²

2022

Preprint

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Recently Ono, Saad and the second author [21] initiated a study of value distribution of certain families of Gaussian hypergeometric functions over large finite fields. They investigated two families of Gaussian hypergeometric functions and showed that they satisfy semicircular and Batman distributions. Motivated by their results we aim to study distributions of certain families of hypergeometric functions in the p-adic setting over large finite fields. In particular, we consider two and six parameters families of hypergeometric functions in the p-adic setting and obtain that their limiting distributions are semicircular over large finite fields. In the process of doing this we also express the traces of pth Hecke operators acting on the spaces of cusp forms of even weight k ≥ 4 and levels 4 and 8 in terms of p-adic hypergeometric function which is of independent interest. These results can be viewed as p-adic analogous of some trace formulas of [1,2,6].

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Sato-Tate distribution of p-adic hypergeometric functions

Pujahari

Saikia

2022

Res. number theory

View full text Add to dashboard Cite

Recently Ono, Saad and the second author [21] initiated a study of value distribution of certain families of Gaussian hypergeometric functions over large finite fields. They investigated two families of Gaussian hypergeometric functions and showed that they satisfy semicircular and Batman distributions. Motivated by their results we aim to study distributions of certain families of hypergeometric functions in the p-adic setting over large finite fields. In particular, we consider two and six parameters families of hypergeometric functions in the p-adic setting and obtain that their limiting distributions are semicircular over large finite fields. In the process of doing this we also express the traces of pth Hecke operators acting on the spaces of cusp forms of even weight $$k\ge 4$$ k ≥ 4 and levels 4 and 8 in terms of p-adic hypergeometric function which is of independent interest. These results can be viewed as p-adic analogous of some trace formulas of [1, 2, 6].

show abstract

Distribution of values of Gaussian hypergeometric functions

Cited by 4 publications

References 61 publications

Moments of Gaussian hypergeometric functions over finite fields

Moments of Gaussian hypergeometric functions over finite fields

Sato-Tate Distribution of $p$-adic hypergeometric functions

Sato-Tate distribution of p-adic hypergeometric functions

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