Congruences for generalized Apéry numbers and Gaussian hypergeometric series

Kalita, Gautam; Chetry, Arjun Singh

doi:10.1007/s40993-016-0069-z

Cited by 5 publications

(2 citation statements)

References 19 publications

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“…These properties include transformation laws, explicit evaluations, and contiguous relations. These functions have played central roles in the study of combinatorial supercongruences [1,3,36,43,46,47,51,54,55,56,57,58], Dwork hypersurfaces [9,45], Galois representations [40,41], L-functions of elliptic curves [6,10,11,25,39,44,52,60,63], hyperelliptic curves [7,8], K3 surfaces [4,19,52], Calabi-Yau threefolds [2,3,64], the Eichler-Selberg trace formula [24,25,26,27,38,48,58,59], among other topics.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Distribution of values of Gaussian hypergeometric functions

Ono¹,

Saad²,

Saikia³

2021

Preprint

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In the 1980's, Greene defined hypergeometric functions over finite fields using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss and Kummer. These functions have played important roles in the study of Apéry-style supercongruences, the Eichler-Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. We study the value distribution (over large finite fields) of natural families of these functions. For the 2 F 1 functions, the limiting distribution is semicircular, whereas the distribution for the 3 F 2 functions is the more exotic Batman distribution.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Distribution of values of Gaussian hypergeometric functions

Ono¹,

Saad²,

Saikia³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In [3], Ahlgren and Ono proved a conjecture of Beukers [9] which stated that, if p is an odd prime, then for primes p 5 and integers m, r 1. Other supercongruences of this type have been studied in numerous works (for example, see [4], [8], [15], [16], [20], [25], [31]- [34], [43]). In order to view (1•4)-(1•7) from a general perspective, we discuss the setup from [13].…”

Section: Introductionmentioning

confidence: 99%