We prove a supercongruence modulo between the th Fourier coefficient of a weight 6 modular form and a truncated -hypergeometric series. Novel ingredients in the proof are the comparison of two rational approximations to to produce non-trivial harmonic sum identities and the reduction of the resulting congruences between harmonic sums via a congruence relating the Apéry numbers to another Apéry-like sequence.
On démontre une supercongruence modulo entre le -ième coefficient de Fourier d’une forme modulaire de poids 6 et une série hypergéométrique tronquée. Les nouveaux ingrédients de la preuve sont la comparaison de deux approximations rationnelles de pour produire des identités non triviales entre sommes harmoniques, et la réduction des congruences qui en résultent entre des sommes via une congruence qui relie les nombres d’Apéry á une autre suite du type de celle d’Apéry.
Accepted:
Published online:
Keywords: supercongruence, Apéry numbers, Apéry-like numbers, hypergeometric function

@article{AIF_2018__68_5_1987_0, author = {Osburn, Robert and Straub, Armin and Zudilin, Wadim}, title = {A modular supercongruence for $_6F_5$: {An} {Ap\'ery-like~story}}, journal = {Annales de l'Institut Fourier}, pages = {1987--2004}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {5}, year = {2018}, doi = {10.5802/aif.3201}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3201/} }
TY - JOUR TI - A modular supercongruence for $_6F_5$: An Apéry-like story JO - Annales de l'Institut Fourier PY - 2018 DA - 2018/// SP - 1987 EP - 2004 VL - 68 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3201/ UR - https://doi.org/10.5802/aif.3201 DO - 10.5802/aif.3201 LA - en ID - AIF_2018__68_5_1987_0 ER -
Osburn, Robert; Straub, Armin; Zudilin, Wadim. A modular supercongruence for $_6F_5$: An Apéry-like story. Annales de l'Institut Fourier, Volume 68 (2018) no. 5, pp. 1987-2004. doi : 10.5802/aif.3201. https://aif.centre-mersenne.org/articles/10.5802/aif.3201/
[1] A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math., Tome 518 (2000), pp. 187-212 | DOI | MR | Zbl
[2] Irrationalité de et (Astérisque) Tome 61, Société Mathématique de France, 1979, pp. 11-13 | Zbl
[3] Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, Tome 32, Stechert-Hafner, 1964, v+108 pages | MR
[4] Another congruence for the Apéry numbers, J. Number Theory, Tome 25 (1987) no. 2, pp. 201-210 | DOI | MR | Zbl
[5] Irrationality proofs using modular forms, Journées arithmétiques de Besançon (Besançon, 1985) (Astérisque) Tome 147-148, Société Mathématique de France, 1987, pp. 271-283 | MR | Zbl
[6] Hypergeometric series and harmonic number identities, Adv. Appl. Math., Tome 34 (2005) no. 1, pp. 123-137 | DOI | MR | Zbl
[7] Sporadic sequences, modular forms and new series for , Ramanujan J., Tome 29 (2012) no. 1-3, pp. 163-183 | DOI | MR | Zbl
[8] Gaussian hypergeometric functions and traces of Hecke operators, Int. Math. Res. Not., Tome 2004 (2004) no. 60, pp. 3233-3262 | DOI | MR | Zbl
[9] Hypergeometric functions over finite fields (2015) (http://arxiv.org/abs/1510.02575)
[10] Hypergeometric type identities in the -adic setting and modular forms, Proc. Am. Math. Soc., Tome 144 (2016) no. 4, pp. 1493-1508 | DOI | MR | Zbl
[11] Hypergeometric functions over finite fields, Trans. Am. Math. Soc., Tome 301 (1987) no. 1, pp. 77-101 | DOI | MR | Zbl
[12] Some conjectures concerning partial sums of generalized hypergeometric series, -adic functional analysis (Nijmegen, 1996) (Lecture Notes in Pure and Appl. Math.) Tome 192, Dekker, 1997, pp. 223-236 | MR | Zbl
[13] Supercongruences and complex multiplication, J. Number Theory, Tome 164 (2016), pp. 166-178 | DOI | MR | Zbl
[14] An extension of the Apéry number supercongruence, Acta Arith., Tome 123 (2006) no. 4, pp. 335-348 | DOI | MR | Zbl
[15] Hypergéométrie et fonction zêta de Riemann, Mem. Am. Math. Soc., Tome 186 (2007) no. 875, x+87 pages | DOI | MR | Zbl
[16] Binomial coefficient-harmonic sum identities associated to supercongruences, Integers, Tome 11 (2011), A37 (Art A37, 8 p.) | DOI | MR | Zbl
[17] Extending Gaussian hypergeometric series to the -adic setting, Int. J. Number Theory, Tome 8 (2012) no. 7, pp. 1581-1612 | DOI | MR | Zbl
[18] On a supercongruence conjecture of Rodriguez-Villegas, Proc. Am. Math. Soc., Tome 140 (2012) no. 7, pp. 2241-2254 | DOI | MR | Zbl
[19] Some remarks on , Mat. Zametki, Tome 59 (1996) no. 6, pp. 865-880 | DOI | MR | Zbl
[20] Gaussian hypergeometric series and supercongruences, Math. Comput., Tome 78 (2009) no. 265, pp. 275-292 | DOI | MR | Zbl
[21] On the (K.2) supercongruence of Van Hamme, J. Math. Anal. Appl., Tome 433 (2016) no. 1, pp. 706-711 | DOI | MR | Zbl
[22] Computer proofs of a new family of harmonic number identities, Adv. Appl. Math., Tome 31 (2003) no. 2, pp. 359-378 | DOI | MR | Zbl
[23] , Peters, 1996, xii+212 pages (With a foreword by Donald E. Knuth, With a separately available computer disk) | MR | Zbl
[24] A proof that Euler missed: Apéry’s proof of the irrationality of , Math. Intell., Tome 1 (1979) no. 4, pp. 195-203 | DOI | MR | Zbl
[25] Propriétés diophantinnes des valeurs de la fonction zêta de Riemann aux entiers impairs (2001) (Ph. D. Thesis)
[26] Hypergeometric families of Calabi–Yau manifolds, Calabi–Yau varieties and mirror symmetry (Toronto, ON, 2001) (Fields Inst. Commun.) Tome 38, American Mathematical Society, 2003, pp. 223-231 | MR | Zbl
[27] Symbolic summation assists combinatorics, Sémin. Lothar. Comb., Tome 56 (2007), B56b http://www.mat.univie.ac.at/slc/wpapers/s56schneider.html (Art. B56b, 36 p.) | Zbl
[28] The On-Line Encyclopedia of Integer Sequences, 2017 (published electronically at http://oeis.org)
[29] On the supercongruence conjectures of van Hamme, Res. Math. Sci., Tome 2 (2015), 18 (Art. 18, 21 p.) | DOI | MR | Zbl
[30] Integral solutions of Apéry-like recurrence equations, Groups and symmetries (CRM Proc. Lecture Notes) Tome 47, American Mathematical Society, 2009, pp. 349-366 | MR | Zbl
[31] Apéry’s theorem. Thirty years after, Int. J. Math. Comput. Sci., Tome 4 (2009) no. 1, pp. 9-19 | MR | Zbl
[32] A generating function of the squares of Legendre polynomials, Bull. Aust. Math. Soc., Tome 89 (2014) no. 1, pp. 125-131 | DOI | MR | Zbl
[33] Hypergeometric heritage of W. N. Bailey. With an appendix: Bailey’s letters to F. Dyson (2016) (http://arxiv.org/abs/1611.08806)
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