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

被引：33
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
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页码：1919 / 1968
页数：50
相关论文
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 [J].
Almkvist, Gert ;
Van Straten, Duco ;
Zudilin, Wadim .
PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, 2011, 54 :273-295
[3]   Supercongruences for the Almkvist-Zudilin numbers [J].
Amdeberhan, Tewodros ;
Tauraso, Roberto .
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 [J].
ANDREWS, GE ;
BAXTER, RJ .
JOURNAL OF STATISTICAL PHYSICS, 1987, 47 (3-4) :297-330
[8]   q-analogs of the binomial coefficient congruences of Babbage, Wolstenholme and Glaisher [J].
Andrews, GE .
DISCRETE MATHEMATICS, 1999, 204 (1-3) :15-25
[9]  
[Anonymous], 1988, THESIS
[10]  
[Anonymous], 2017, SEMIN LOTHAR COMB
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