Proof of a Dwork-type supercongruence by induction

Yong Zhang; Peisen Yuan; Yong Zhang; Peisen Yuan

doi:10.3934/math.2021671

AIMS Mathematics

2021, Volume 6, Issue 10: 11568-11583. doi: 10.3934/math.2021671

Previous Article Next Article

Research article

Proof of a Dwork-type supercongruence by induction

Yong Zhang ¹,
Peisen Yuan ^{2
,
,}

1.
Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing 211167, China
2.
College of Artificial Intelligence, Nanjing Agricultural University, Nanjing 210095, China

Received: 08 June 2021 Accepted: 05 August 2021 Published: 10 August 2021
MSC : 11A07, 11B68, 11E25, 05A10, 11B65, 11B75

In this paper we prove a Dwork-type supercongruence: for any prime $ p\geq3 $ and integer $ r\geq 1 $,

$ \begin{align*} \sum\limits_{k = 0}^{p^r-1}\frac{3k+1}{16^k}{\binom{2k}{k}}^3\equiv p\sum\limits_{k = 0}^{p^{r-1}-1}\frac{3k+1}{16^k}{\binom{2k}{k}}^3\pmod{p^{3r+1-\delta_{p, 3}}}, \end{align*} $

which extends a result of Guo and Zudilin.
- supercongruences,
- Dwork-type supercongruences,
- generalized harmonic numbers of order $ m $,
- induction,
- Jacobsthal's binomial congruence
Citation: Yong Zhang, Peisen Yuan. Proof of a Dwork-type supercongruence by induction[J]. AIMS Mathematics, 2021, 6(10): 11568-11583. doi: 10.3934/math.2021671

Related Papers:

Abstract

In this paper we prove a Dwork-type supercongruence: for any prime $ p\geq3 $ and integer $ r\geq 1 $,

$ \begin{align*} \sum\limits_{k = 0}^{p^r-1}\frac{3k+1}{16^k}{\binom{2k}{k}}^3\equiv p\sum\limits_{k = 0}^{p^{r-1}-1}\frac{3k+1}{16^k}{\binom{2k}{k}}^3\pmod{p^{3r+1-\delta_{p, 3}}}, \end{align*} $

which extends a result of Guo and Zudilin.

References

[1]	H. Alzer, D. Karayannakis, H. M. Srivastava, Series representations for some mathematical constants, J. Math. Anal. Appl., 320 (2006), 145–162. doi: 10.1016/j.jmaa.2005.06.059
[2]	A. O. L. Atkin, H. P. F. Swinnerton-Dyer, Modular forms on noncongruence subgroups, Combinatorics, Amer. Math. Soc., Providence, R.I., 19 (1971), 1–25.
[3]	F. Beukers, Some congruences for the Apéry numbers, J. Number Theory, 25 (1985), 141–155.
[4]	T. X. Cai, A congruence involving the quotients of Euler and its applications (I), Acta Arith., 103 (2002), 313–320. doi: 10.4064/aa103-4-1
[5]	B. Dwork, $p$-adic cycles, Inst. Hautes Études Sci. Publ. Math., 37 (1969), 27–115.
[6]	I. Gessel, Some congruences for generalized Euler numbers, Canad. J. Math., 35 (1983), 687–709. doi: 10.4153/CJM-1983-039-5
[7]	A. Granville, Arithmetic properties of binomial coefficients I: Binomial coefficients modulo prime powers, CMS Conf. Proc., 20 (1997), 253–275.
[8]	J. Guillera, W. Zudilin, "Divergent" Ramanujan-type supercongruences, Proc. Amer. Math. Soc., 14 (2012), 765–777.
[9]	V. J. W. Guo, $q$-Analogues of Dwork-type supercongruences, J. Math. Anal. Appl., 487 (2020), 124022. doi: 10.1016/j.jmaa.2020.124022
[10]	V. J. W. Guo, $q$-Analogues of two "divergent" Ramanujan-type supercongruences, Ramanujan J., 52 (2020), 605–624. doi: 10.1007/s11139-019-00161-0
[11]	V. J. W. Guo, W. Zudilin, Dwork-type supercongruences through a creative $q$-microscope, J. Combin. Theory Ser. A, 178 (2020), 105362.
[12]	R. Gy, Extended congruences for harmonic numbers, 2019, arXiv: 1902.05258.
[13]	X. Z. Lin, $p$-adic $L$-functions and classical congruences, Acta Arith., 194 (2020), 29–49. doi: 10.4064/aa181207-2-5
[14]	J. C. Liu, Congruences for truncated hypergeometric series ${}_2F_1$, Bull. Austral. Math. Soc., 96 (2017), 14–23. doi: 10.1017/S0004972717000181
[15]	G. S. Mao, T. Zhang, Proof of Sun's conjectures on supercongruences and the divisibility of certain binomial sums, Ramanujan J., 50 (2019), 1–11. doi: 10.1007/s11139-019-00138-z
[16]	H. X. Ni, A $q$-Dwork-type generalization of Rodriguez-Villegas' supercongruences, 2020, arXiv: 2008.02541.
[17]	R. Osburn, B. Sahu, A. Straub, Supercongruences for sporadic sequences, Proc. Edinburgh Math. Soc. (2), 59 (2016), 503–518. doi: 10.1017/S0013091515000255
[18]	A. Straub, Multivariate Apéry numbers and supercongruences of rational functions, Algebra Number Theory, 8 (2014), 1985–2008. doi: 10.2140/ant.2014.8.1985
[19]	Z. W. Sun, Arithmetic theory of harmonic numbers, Proc. Amer. Math. Soc., 140 (2012), 415–428.
[20]	Z. W. Sun, Open conjectures on congruences, Nanjing Univ. J. Math. Biquarterly, 36 (2019), 1–99.
[21]	Z. W. Sun, Super congruences and Euler numbers, Sci. China Math., 54 (2011), 2509–2535. doi: 10.1007/s11425-011-4302-x
[22]	X. Wang, M. Yue, A $q$-analogue of a Dwork-type supercongruence, Bull. Austral. Math. Soc., 103 (2020), 1–8.
[23]	Y. Zhang, H. Pan, On the Atkin and Swinnerton-Dyer type congruences for some truncated hypergeometric ${}_1F_0$ series, Acta Arith., 198 (2021), 169–186. doi: 10.4064/aa200405-8-8
[24]	Y. Zhang, Three supercongruences for Apéry numbers or Franel numbers, Publ. Math. Debrecen, 9023 (2021), 1–20.
[25]	W. Zudilin, Ramanujan-type supercongruences, J. Number Theory, 129 (2009), 1848–1857. doi: 10.1016/j.jnt.2009.01.013

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)