Supercongruences for polynomial analogs of the Apéry numbers

Straub, Armin

doi:10.1090/proc/14301

Supercongruences for polynomial analogs of the Apéry numbers
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Proc. Amer. Math. Soc. 147 (2019), 1023-1036 Request permission

Abstract:

We consider a family of polynomial analogs of the Apéry numbers, which includes $q$-analogs due to Krattenthaler–Rivoal–Zudilin and Zheng, and show that the supercongruences that Gessel and Mimura established for the Apéry numbers generalize to these polynomials. Our proof relies on polynomial analogs of classical binomial congruences of Wolstenholme and Ljunggren. We further indicate that this approach generalizes to other supercongruence results.

References

B. Adamczewski, J. P. Bell, É. Delaygue, and F. Jouhet, Congruences modulo cyclotomic polynomials and algebraic independence for $q$-series, Sém. Lothar. Combin. 78B (2017), Art. 54, 12 (English, with English and French summaries). MR 3678636
Scott Ahlgren, Shalosh B. Ekhad, Ken Ono, and Doron Zeilberger, A binomial coefficient identity associated to a conjecture of Beukers, Electron. J. Combin. 5 (1998), Research Paper 10, 1 unnumbered page. MR 1600106
Scott Ahlgren and Ken Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math. 518 (2000), 187–212. MR 1739404, DOI 10.1515/crll.2000.004
Gert Almkvist, Duco van Straten, and Wadim Zudilin, Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations, Proc. Edinb. Math. Soc. (2) 54 (2011), no. 2, 273–295. MR 2794653, DOI 10.1017/S0013091509000959
Tewodros Amdeberhan and Roberto Tauraso, Supercongruences for the Almkvist-Zudilin numbers, Acta Arith. 173 (2016), no. 3, 255–268. MR 3512855, DOI 10.4064/aa8215-2-2016
George E. Andrews, $q$-analogs of the binomial coefficient congruences of Babbage, Wolstenholme and Glaisher, Discrete Math. 204 (1999), no. 1-3, 15–25. MR 1691860, DOI 10.1016/S0012-365X(98)00364-1
Roger Apéry, Irrationalité de $\zeta 2$ et $\zeta 3$, Astérisque 61 (1979), 11–13 (French). Luminy Conference on Arithmetic. MR 3363457
F. Beukers, Some congruences for the Apéry numbers, J. Number Theory 21 (1985), no. 2, 141–155. MR 808283, DOI 10.1016/0022-314X(85)90047-2
F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25 (1987), no. 2, 201–210. MR 873877, DOI 10.1016/0022-314X(87)90025-4
Heng Huat Chan, Shaun Cooper, and Francesco Sica, Congruences satisfied by Apéry-like numbers, Int. J. Number Theory 6 (2010), no. 1, 89–97. MR 2641716, DOI 10.1142/S1793042110002879
Heng Huat Chan and Wadim Zudilin, New representations for Apéry-like sequences, Mathematika 56 (2010), no. 1, 107–117. MR 2604987, DOI 10.1112/S0025579309000436
S. Chowla, J. Cowles, and M. Cowles, Congruence properties of Apéry numbers, J. Number Theory 12 (1980), no. 2, 188–190. MR 578810, DOI 10.1016/0022-314X(80)90051-7
Wenchang Chu, A binomial coefficient identity associated with Beukers’ conjecture on Apéry numbers, Electron. J. Combin. 11 (2004), no. 1, Note 15, 3. MR 2114196
Matthijs J. Coster, Supercongruences, Ph.D. thesis, Universiteit Leiden, 1988.
Jacques Désarménien, Un analogue des congruences de Kummer pour les $q$-nombres d’Euler, European J. Combin. 3 (1982), no. 1, 19–28 (German, with English summary). MR 656008, DOI 10.1016/S0195-6698(82)80005-X
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Number Theory 117 (2006), no. 1, 191–215. MR 2204742, DOI 10.1016/j.jnt.2005.06.005
Sam Formichella and Armin Straub, Gaussian binomial coefficients with negative arguments, Preprint (2018), arXiv:1802.02684.
Ira Gessel, Some congruences for Apéry numbers, J. Number Theory 14 (1982), no. 3, 362–368. MR 660381, DOI 10.1016/0022-314X(82)90071-3
Ofir Gorodetsky, $q$-congruences, with applications to supercongruences and the cyclic sieving phenomenon, Preprint (2018), arXiv:1805.01254.
Andrew Granville, Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers, Organic mathematics (Burnaby, BC, 1995) CMS Conf. Proc., vol. 20, Amer. Math. Soc., Providence, RI, 1997, pp. 253–276. MR 1483922
Victor J. W. Guo and Wadim Zudilin, A $q$-microscope for supercongruences, Preprint (2018), arXiv:1803.01830.
Frédéric Jouhet and Elie Mosaki, Irrationalité aux entiers impairs positifs d’un $q$-analogue de la fonction zêta de Riemann, Int. J. Number Theory 6 (2010), no. 5, 959–988 (French, with English and French summaries). MR 2679452, DOI 10.1142/S1793042110003332
Victor Kac and Pokman Cheung, Quantum calculus, Universitext, Springer-Verlag, New York, 2002. MR 1865777, DOI 10.1007/978-1-4613-0071-7
C. Krattenthaler, T. Rivoal, and W. Zudilin, Séries hypergéométriques basiques, $q$-analogues des valeurs de la fonction zêta et séries d’Eisenstein, J. Inst. Math. Jussieu 5 (2006), no. 1, 53–79 (French, with English and French summaries). MR 2195945, DOI 10.1017/S1474748005000149
Percy A. MacMahon, Combinatory analysis, Chelsea Publishing Co., New York, 1960. Two volumes (bound as one). MR 0141605
Yoshio Mimura, Congruence properties of Apéry numbers, J. Number Theory 16 (1983), no. 1, 138–146. MR 693400, DOI 10.1016/0022-314X(83)90038-0
Gloria Olive, Generalized powers, Amer. Math. Monthly 72 (1965), 619–627. MR 178267, DOI 10.2307/2313851
Robert Osburn and Brundaban Sahu, Congruences via modular forms, Proc. Amer. Math. Soc. 139 (2011), no. 7, 2375–2381. MR 2784802, DOI 10.1090/S0002-9939-2010-10771-2
Robert Osburn, Brundaban Sahu, and Armin Straub, Supercongruences for sporadic sequences, Proc. Edinb. Math. Soc. (2) 59 (2016), no. 2, 503–518. MR 3509244, DOI 10.1017/S0013091515000255
Hao Pan, A generalization of Wolstenholme’s harmonic series congruence, Rocky Mountain J. Math. 38 (2008), no. 4, 1263–1269. MR 2436722, DOI 10.1216/RMJ-2008-38-4-1263
Alfred van der Poorten, A proof that Euler missed$\ldots$Apéry’s proof of the irrationality of $\zeta (3)$, Math. Intelligencer 1 (1978/79), no. 4, 195–203. An informal report. MR 547748, DOI 10.1007/BF03028234
Ling-Ling Shi and Hao Pan, A $q$-analogue of Wolstenholme’s harmonic series congruence, Amer. Math. Monthly 114 (2007), no. 6, 529–531. MR 2321255, DOI 10.1080/00029890.2007.11920441
Armin Straub, A $q$-analog of Ljunggren’s binomial congruence, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), Discrete Math. Theor. Comput. Sci. Proc., AO, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2011, pp. 897–902 (English, with English and French summaries). MR 2820769
Armin Straub, Multivariate Apéry numbers and supercongruences of rational functions, Algebra & Number Theory 8 (2014), no. 8, 1985–2008.
De-Yin Zheng, An algebraic identity on $q$-Apéry numbers, Discrete Math. 311 (2011), no. 23-24, 2708–2710. MR 2837631, DOI 10.1016/j.disc.2011.08.004