$N$-tuple sum analogues for Ramanujan-type congruences
HTML articles powered by AMS MathViewer
- by Mohamed El Bachraoui;
- Proc. Amer. Math. Soc. 151 (2023), 1-16
- DOI: https://doi.org/10.1090/proc/16061
- Published electronically: July 15, 2022
- HTML | PDF | Request permission
Abstract:
In this paper we prove supercongruence relations for truncated $N$-tuple sums of basic hypergeometric series. As an application, we give double, triple, and quadruple sum analogues of some Ramanujan-type supercongruences.References
- Bruce C. Berndt, Ramanujanโs notebooks. Part IV, Springer-Verlag, New York, 1994. MR 1261634, DOI 10.1007/978-1-4612-0879-2
- Mohamed El Bachraoui, On supercongruences for truncated sums of squares of basic hypergeometric series, Ramanujan J. 54 (2021), no.ย 2, 415โ426. MR 4204764, DOI 10.1007/s11139-019-00226-0
- George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR 2128719, DOI 10.1017/CBO9780511526251
- Jesรบs Guillera and Wadim Zudilin, โDivergentโ Ramanujan-type supercongruences, Proc. Amer. Math. Soc. 140 (2012), no.ย 3, 765โ777. MR 2869062, DOI 10.1090/S0002-9939-2011-10950-X
- Victor J. W. Guo, A $q$-analogue of the (L.2) supercongruence of Van Hamme, J. Math. Anal. Appl. 466 (2018), no.ย 1, 749โ761. MR 3818142, DOI 10.1016/j.jmaa.2018.06.023
- Victor J. W. Guo, A $q$-analogue of the (J.2) supercongruence of Van Hamme, J. Math. Anal. Appl. 466 (2018), no.ย 1, 776โ788. MR 3818144, DOI 10.1016/j.jmaa.2018.06.021
- Victor J. W. Guo, $q$-analogues of the (E.2) and (F.2) supercongruences of Van Hamme, Ramanujan J. 49 (2019), no.ย 3, 531โ544. MR 3979689, DOI 10.1007/s11139-018-0021-z
- Victor J. W. Guo, Common $q$-analogues of some different supercongruences, Results Math. 74 (2019), no.ย 4, Paper No. 131, 15. MR 3963751, DOI 10.1007/s00025-019-1056-1
- Victor J. W. Guo and Su-Dan Wang, Some congruences involving fourth powers of central $q$-binomial coefficients, Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), no.ย 3, 1127โ1138. MR 4091055, DOI 10.1017/prm.2018.96
- Victor J. W. Guo and Wadim Zudilin, Ramanujan-type formulae for $1/\pi$: $q$-analogues, Integral Transforms Spec. Funct. 29 (2018), no.ย 7, 505โ513. MR 3806553, DOI 10.1080/10652469.2018.1454448
- Victor J. W. Guo and Wadim Zudilin, A $q$-microscope for supercongruences, Adv. Math. 346 (2019), 329โ358. MR 3910798, DOI 10.1016/j.aim.2019.02.008
- Victor J. W. Guo and Wadim Zudilin, Dwork-type supercongruences through a creative $q$-microscope, J. Combin. Theory Ser. A 178 (2021), Paper No. 105362, 37. MR 4183862, DOI 10.1016/j.jcta.2020.105362
- Long Li, Some $q$-supercongruences for truncated forms of squares of basic hypergeometric series, J. Difference Equ. Appl. 27 (2021), no.ย 1, 16โ25. MR 4224772, DOI 10.1080/10236198.2020.1862808
- Yudong Liu and Xiaoxia Wang, $q$-analogues of the (G.2) supercongruence of Van Hamme, Rocky Mountain J. Math. 51 (2021), no.ย 4, 1329โ1340. MR 4298850, DOI 10.1216/rmj.2021.51.1329
- Robert Osburn and Wadim Zudilin, On the (K.2) supercongruence of Van Hamme, J. Math. Anal. Appl. 433 (2016), no.ย 1, 706โ711. MR 3388817, DOI 10.1016/j.jmaa.2015.08.009
- S. Ramanujan, Modular equations and approximations to $\pi$, Quart. J. Math. 45 (1914), 350โ372.
- Hanfei Song and Chun Wang, Some $q$-supercongruences modulo the fifth power of a cyclotomic polynomial from squares of $q$-hypergeometric series, Results Math. 76 (2021), no.ย 4, Paper No. 222, 17. MR 4328757, DOI 10.1007/s00025-021-01536-w
- Holly Swisher, On the supercongruence conjectures of van Hamme, Res. Math. Sci. 2 (2015), Art. 18, 21. MR 3411813, DOI 10.1186/s40687-015-0037-6
- L. van Hamme, Some conjectures concerning partial sums of generalized hypergeometric series, $p$-adic functional analysis (Nijmegen, 1996) Lecture Notes in Pure and Appl. Math., vol. 192, Dekker, New York, 1997, pp.ย 223โ236. MR 1459212
- Wadim Zudilin, Ramanujan-type supercongruences, J. Number Theory 129 (2009), no.ย 8, 1848โ1857. MR 2522708, DOI 10.1016/j.jnt.2009.01.013
Bibliographic Information
- Mohamed El Bachraoui
- Affiliation: Department of Mathematical Sciences, United Arab Emirates University, Al-Ain, United Arab Emirates
- MR Author ID: 708599
- Email: melbachraoui@uaeu.ac.ae
- Received by editor(s): November 13, 2021
- Received by editor(s) in revised form: March 3, 2022
- Published electronically: July 15, 2022
- Communicated by: Mourad Ismail
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 1-16
- MSC (2020): Primary 11B65, 11F33, 33C20
- DOI: https://doi.org/10.1090/proc/16061
- MathSciNet review: 4504603