A generalized supercongruence of Kimoto and Wakayama

Liu, Ji-Cai

doi:10.1016/j.jmaa.2018.05.031

Cited by 13 publications

(7 citation statements)

References 10 publications

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“…Also, recently, certain nice congruence relations among these Apéry-like numbers that are quite resembled to the Rodriguez-Villegas type congruence [30] and conjectured in [20] are proved in [29]. Further interesting congruence that involves Bernoulli numbers has been obtained in [28] (see also [43]). The congruence in [28] can be considered as a one step deep congruence of the one proved in [29] corrected by the remainder term.…”

Section: Introductionmentioning

confidence: 96%

“…We remark that, however, it is quite difficult to expect an exact generalization of the congruence relation (i.e. of the same shape which is relevant to the hypergeometric series) shown by employing p-adic analysis in [29] (and [28]) for ζ (2).…”

Section: Introductionmentioning

confidence: 97%

“…Further interesting congruence that involves Bernoulli numbers has been obtained in [28] (see also [43]). The congruence in [28] can be considered as a one step deep congruence of the one proved in [29] corrected by the remainder term.…”

Section: Introductionmentioning

confidence: 99%

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Apéry-like numbers for non-commutative harmonic oscillators and automorphic integrals

Kimoto,

Wakayama

2019

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The purpose of the present paper is to study the number theoretic properties of the special values of the spectral zeta functions of the non-commutative harmonic oscillator (NcHO), especially in relation to modular forms and elliptic curves from the viewpoint of Fuchsian differential equations, and deepen the understanding of the spectrum of the NcHO. We study first the general expression of special values of the spectral zeta function ζ Q (s) of the NcHO at s = n (n = 2, 3, . . . ) and then the generating and meta-generating functions for Apéry-like numbers defined through the analysis of special values ζ Q (n). Actually, we show that the generating function w 2n of such Apéry-like numbers appearing (as the "first anomaly") in ζ Q (2n) for n = 2 gives an example of automorphic integral with rational period functions in the sense of Knopp, but still a better explanation remains to be clarified explicitly for n > 2. This is a generalization of our earlier result on showing that w 2 is interpreted as a Γ(2)-modular form of weight 1. Moreover, certain congruence relations over primes for "normalized" Apéry-like numbers are also proven. In order to describe w 2n in a similar manner as w 2 , we introduce a differential Eisenstein series by using analytic continuation of a classical generalized Eisenstein series due to Berndt. The differential Eisenstein series is actually a typical example of the automorphic integral of negative weight. We then have an explicit expression of w 4 in terms of the differential Eisenstein series. We discuss also shortly the Hecke operators acting on such automorphic integrals and relating Eichler's cohomology group.

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Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 97%

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Apéry-like numbers for non-commutative harmonic oscillators and automorphic integrals

Kimoto,

Wakayama

2019

Preprint

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“…Besides, quite a few conjectures on congruence relations and integer-valuedness involving D n (x, y) or S n (x, y) were posed. During the past few years, many of them have been proved, see [3,4,6,7], while some of them are still remaining open.…”

Section: Introductionmentioning

confidence: 99%

Congruences Related to Dual Sequences and Catalan Numbers

Wang¹,

Zhong²

2020

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“…Kimoto and Wakayama [8] also showed that these Apéry-like numbers satisfy some interesting congruences similar to those for the Apéry numbers. For more congruence properties of these Apéry-like numbers, one can refer to [10,12,13].…”

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confidence: 99%