q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon

被引：32

作者：

Gorodetsky, Ofir ^{[1
]}

机构：

[1] Tel Aviv Univ, Raymond & Beverly Sackler Sch Math Sci, IL-6997801 Tel Aviv, Israel

来源：

INTERNATIONAL JOURNAL OF NUMBER THEORY | 2019年 / 15卷 / 09期

关键词：

Gauss congruence; supercongruence; q-analogue; Apery numbers; cyclic sieving phenomenon; FIBONACCI NUMBERS; Q-ANALOGS; SERIES; FERMAT;

D O I：

10.1142/S1793042119501069

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding q-supercongruence. Similar q-supercongruences are established for binomial coefficients and the Apery numbers, by means of a general criterion involving higher derivatives at roots of unity. Our methods lead us to discover new examples of the cyclic sieving phenomenon, involving the q-Lucas numbers.

引用

页码：1919 / 1968

页数：50

共 60 条

[1] Almkvist G., 2006, AMS/IP Stud. Adv. Math., V38, P481
[2] GENERALIZATIONS OF CLAUSEN'S FORMULA AND ALGEBRAIC TRANSFORMATIONS OF CALABI-YAU DIFFERENTIAL EQUATIONS
Almkvist, Gert
Van Straten, Duco
Zudilin, Wadim
[J]. PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, 2011, 54 : 273 - 295
[3] Supercongruences for the Almkvist-Zudilin numbers
Amdeberhan, Tewodros
Tauraso, Roberto
[J]. ACTA ARITHMETICA, 2016, 173 (03) : 255 - 268
[4] Andrews G. E., 1990, PROGR MATH, V85, P1
[5] Andrews G E, 1990, J AM MATH SOC, V3, P653
[6] Andrews GE, 2004, FIBONACCI QUART, V42, P3
[7] LATTICE GAS GENERALIZATION OF THE HARD HEXAGON MODEL .3. Q-TRINOMIAL COEFFICIENTS
ANDREWS, GE
BAXTER, RJ
[J]. JOURNAL OF STATISTICAL PHYSICS, 1987, 47 (3-4) : 297 - 330
[8] q-analogs of the binomial coefficient congruences of Babbage, Wolstenholme and Glaisher
Andrews, GE
[J]. DISCRETE MATHEMATICS, 1999, 204 (1-3) : 15 - 25
[9] [Anonymous], 1988, THESIS
[10] [Anonymous], 2017, SEMIN LOTHAR COMB

← 1 2 3 4 5 6 →