Abstract
Congruences of truncated sums of infinite series do not directly extend to congruences of the truncated sums of higher powers of these infinite series. Guo and Zudilin recently established a variety of supercongruences for truncated sums of certain basic hypergeometric series. In this note we extend some of these supercongruences to the truncated sums of the squares of the corresponding series.
Similar content being viewed by others
References
Berndt, B.: Ramanujan’s Notebooks, Part IV. Springer, New York (1994)
Guillera, J., Zudilin, W.: Divergent Ramanujan-type supercongruences. Proc. Am. Math. Soc. 140, 765–777 (2012)
Guo, V.J.W.: A \(q\)-analgue of the (L.2) supercongruence of Van Hamme. J. Math. Anal. Appl. 466, 749–761 (2018)
Guo, V.J.W.: A \(q\)-analgue of the (J.2) supercongruence of Van Hamme. J. Math. Anal. Appl. 466, 776–788 (2018)
Guo, V.J.W.: \(q\)-Analgues of the (E.2) and (F.2) supercongruences of Van Hamme. Ramanujan J. 49, 531–544 (2019)
Guo, V.J.W.: Common \(q\)-analgues of some different supercongruences. Results Math. 74, Art. 131 (2019)
Guo, V.J.W., Wang, S.-D.: Some congruences involving fourth powers of central \(q\)-binomial coefficients. Proc. R. Soc. Edinburgh Sect. A (2019). https://doi.org/10.1017/prm.2018.96
Guo, V.J.W., Zudilin, W.: Ramanujan-type formulae for \(1/\pi \): \(q\)-analogues. Integr. Transform. Spec. Funct. 29, 505–513 (2018)
Guo, V.J.W., Zudilin, W.: A \(q\)-microscope for supercongruences. Adv. Math. 346, 329–358 (2019)
Osburn, R.: On the (K.2) supercongruence of Van Hamme. J. Math. Anal. Appl. 433, 706–711 (2016)
Ramanujan, S.: Modular equations and approximations to \(\pi \). Q. J. Math. 45, 350–372 (1914)
Shi, L.L., Pan, H.: A \(q\)-analogue of Wolstenholme’s harmonic series congruence. Am. Math. Mon. 114, 529–531 (2007)
Straub, A.: Supercongruences for polynomial analogs of the Apery numbers. Proc. Am. Math. Soc. 147, 1023–1036 (2019)
Van Hamme, L.: Some conjectures concerning partial sums of generalized hypergeometric series. In: \(p\)-Adic Functional Analysis, Nijmegen, 1996, Lecture Notes in Pure and Appl. Math., vol. 192, pp. 223–236. Dekker, New York (1997)
Zudilin, W.: Ramanujan-type supercongruences. J. Number Theory 129, 1848–1857 (2009)
Acknowledgements
The author is very grateful to the referees for valuable comments and interesting suggestions which have improved the quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
El Bachraoui, M. On supercongruences for truncated sums of squares of basic hypergeometric series. Ramanujan J 54, 415–426 (2021). https://doi.org/10.1007/s11139-019-00226-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-019-00226-0