American Institute of Mathematical Sciences

Advanced
June  2020, 28(2): 1023-1030. doi: 10.3934/era.2020054

Proof of Sun's conjectural supercongruence involving Catalan numbers

Ji-Cai Liu

Department of Mathematics, Wenzhou University, Wenzhou 325035, China

Received  January 2020 Revised  April 2020 Published  June 2020

Fund Project: The author is supported by the National Natural Science Foundation of China (grant 11801417)

We confirm a conjectural supercongruence involving Catalan numbers, which is one of the 100 selected open conjectures on congruences of Sun. The proof makes use of hypergeometric series identities and symbolic summation method.

Keywords: Supercongruences, Catalan numbers, Fermat quotients.
Mathematics Subject Classification: Primary: 11A07, 11B65; Secondary: 05A19, 33C20.
Citation: Ji-Cai Liu. Proof of Sun's conjectural supercongruence involving Catalan numbers. Electronic Research Archive, 2020, 28 (2) : 1023-1030. doi: 10.3934/era.2020054
References:
[1]

V. J. W. Guo, Proof of a generalization of the (B.2) supercongruence of Van Hamme through a $q$-microscope, Adv. in Appl. Math., 116 (2020). doi: 10.1016/j.aam.2020.102016.  Google Scholar

[2]

V. J. W. Guo and J.-C. Liu, $q$-Analogues of two Ramanujan-type formulas for $1/\pi$, J. Difference Equ. Appl., 24 (2018), 1368-1373.  doi: 10.1080/10236198.2018.1485669.  Google Scholar

[3]

V. J. W. Guo, H. Pan and Y. Zhang, The Rodriguez-Villegas type congruences for truncated $q$-hypergeometric functions, J. Number Theory, 174 (2017), 358-368.  doi: 10.1016/j.jnt.2016.09.011.  Google Scholar

[4]

V. J. W. Guo and M. J. Schlosser, A family of $q$-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial, preprint, arXiv: 1909.10294. Google Scholar

[5]

V. J. W. Guo and M. J. Schlosser, Some new $q$-congruences for truncated basic hypergeometric series: Even powers, Results Math., 75 (2020), 15pp. doi: 10.1007/s00025-019-1126-4.  Google Scholar

[6] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, Oxford, 2008.   Google Scholar
[7]

E. Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of Math. (2), 39 (1938), 350-360.  doi: 10.2307/1968791.  Google Scholar

[8]

J.-C. Liu, On Van Hamme's (A.2) and (H.2) supercongruences, J. Math. Anal. Appl., 471 (2019), 613-622.  doi: 10.1016/j.jmaa.2018.10.095.  Google Scholar

[9]

J.-C. Liu, Semi-automated proof of supercongruences on partial sums of hypergeometric series, J. Symbolic Comput., 93 (2019), 221-229.  doi: 10.1016/j.jsc.2018.06.004.  Google Scholar

[10]

J.-C. Liu and F. Petrov, Congruences on sums of $q$-binomial coefficients, Adv. in Appl. Math., 116 (2020), 11pp. doi: 10.1016/j.aam.2020.102003.  Google Scholar

[11]

G.-S. Mao and Z.-W. Sun, New congruences involving products of two binomial coefficients, Ramanujan J., 49 (2019), 237-256.  doi: 10.1007/s11139-018-0089-5.  Google Scholar

[12]

E. Mortenson, A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function, J. Number Theory, 99 (2003), 139-147.  doi: 10.1016/S0022-314X(02)00052-5.  Google Scholar

[13]

E. Mortenson, Supercongruences between truncated ${}_2F_1$ hypergeometric functions and their Gaussian analogs, Trans. Amer. Math. Soc., 355 (2003), 987-1007.  doi: 10.1090/S0002-9947-02-03172-0.  Google Scholar

[14]

F. Rodriguez-Villegas, Hypergeometric families of Calabi-Yau manifolds, in Calabi-Yau Varieties and Mirror Symmetry, Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, 2003,223–231.  Google Scholar

[15]

C. Schneider, Symbolic summation assists combinatorics, Sém. Lothar. Combin., 56 (2006/07), 36pp.  Google Scholar

[16] L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.   Google Scholar
[17]

R. P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, 1999. doi: 10.1017/CBO9780511609589.  Google Scholar

[18]

Z.-H. Sun, Super congruences involving Bernoulli polynomials, Int. J. Number Theory, 12 (2016), 1259-1271.  doi: 10.1142/S1793042116500779.  Google Scholar

[19]

Z.-W. Sun, Super congruences and Euler numbers, Sci. China Math., 54 (2011), 2509-2535.  doi: 10.1007/s11425-011-4302-x.  Google Scholar

[20]

Z.-W. Sun, On congruences related to central binomial coefficients, J. Number Theory, 131 (2011), 2219-2238.  doi: 10.1016/j.jnt.2011.04.004.  Google Scholar

[21]

Z.-W. Sun, $p$-adic congruences motivated by series, J. Number Theory, 134 (2014), 181-196.  doi: 10.1016/j.jnt.2013.07.011.  Google Scholar

[22]

Z.-W. Sun, Open conjectures on congruences, Nanjing Univ. J. Math. Biquarterly, 36 (2019), 1-99.   Google Scholar

show all references

References:
[1]

V. J. W. Guo, Proof of a generalization of the (B.2) supercongruence of Van Hamme through a $q$-microscope, Adv. in Appl. Math., 116 (2020). doi: 10.1016/j.aam.2020.102016.  Google Scholar

[2]

V. J. W. Guo and J.-C. Liu, $q$-Analogues of two Ramanujan-type formulas for $1/\pi$, J. Difference Equ. Appl., 24 (2018), 1368-1373.  doi: 10.1080/10236198.2018.1485669.  Google Scholar

[3]

V. J. W. Guo, H. Pan and Y. Zhang, The Rodriguez-Villegas type congruences for truncated $q$-hypergeometric functions, J. Number Theory, 174 (2017), 358-368.  doi: 10.1016/j.jnt.2016.09.011.  Google Scholar

[4]

V. J. W. Guo and M. J. Schlosser, A family of $q$-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial, preprint, arXiv: 1909.10294. Google Scholar

[5]

V. J. W. Guo and M. J. Schlosser, Some new $q$-congruences for truncated basic hypergeometric series: Even powers, Results Math., 75 (2020), 15pp. doi: 10.1007/s00025-019-1126-4.  Google Scholar

[6] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, Oxford, 2008.   Google Scholar
[7]

E. Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of Math. (2), 39 (1938), 350-360.  doi: 10.2307/1968791.  Google Scholar

[8]

J.-C. Liu, On Van Hamme's (A.2) and (H.2) supercongruences, J. Math. Anal. Appl., 471 (2019), 613-622.  doi: 10.1016/j.jmaa.2018.10.095.  Google Scholar

[9]

J.-C. Liu, Semi-automated proof of supercongruences on partial sums of hypergeometric series, J. Symbolic Comput., 93 (2019), 221-229.  doi: 10.1016/j.jsc.2018.06.004.  Google Scholar

[10]

J.-C. Liu and F. Petrov, Congruences on sums of $q$-binomial coefficients, Adv. in Appl. Math., 116 (2020), 11pp. doi: 10.1016/j.aam.2020.102003.  Google Scholar

[11]

G.-S. Mao and Z.-W. Sun, New congruences involving products of two binomial coefficients, Ramanujan J., 49 (2019), 237-256.  doi: 10.1007/s11139-018-0089-5.  Google Scholar

[12]

E. Mortenson, A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function, J. Number Theory, 99 (2003), 139-147.  doi: 10.1016/S0022-314X(02)00052-5.  Google Scholar

[13]

E. Mortenson, Supercongruences between truncated ${}_2F_1$ hypergeometric functions and their Gaussian analogs, Trans. Amer. Math. Soc., 355 (2003), 987-1007.  doi: 10.1090/S0002-9947-02-03172-0.  Google Scholar

[14]

F. Rodriguez-Villegas, Hypergeometric families of Calabi-Yau manifolds, in Calabi-Yau Varieties and Mirror Symmetry, Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, 2003,223–231.  Google Scholar

[15]

C. Schneider, Symbolic summation assists combinatorics, Sém. Lothar. Combin., 56 (2006/07), 36pp.  Google Scholar

[16] L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.   Google Scholar
[17]

R. P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, 1999. doi: 10.1017/CBO9780511609589.  Google Scholar

[18]

Z.-H. Sun, Super congruences involving Bernoulli polynomials, Int. J. Number Theory, 12 (2016), 1259-1271.  doi: 10.1142/S1793042116500779.  Google Scholar

[19]

Z.-W. Sun, Super congruences and Euler numbers, Sci. China Math., 54 (2011), 2509-2535.  doi: 10.1007/s11425-011-4302-x.  Google Scholar

[20]

Z.-W. Sun, On congruences related to central binomial coefficients, J. Number Theory, 131 (2011), 2219-2238.  doi: 10.1016/j.jnt.2011.04.004.  Google Scholar

[21]

Z.-W. Sun, $p$-adic congruences motivated by series, J. Number Theory, 134 (2014), 181-196.  doi: 10.1016/j.jnt.2013.07.011.  Google Scholar

[22]

Z.-W. Sun, Open conjectures on congruences, Nanjing Univ. J. Math. Biquarterly, 36 (2019), 1-99.   Google Scholar

[1]

Xiaoni Du, Chenhuang Wu, Wanyin Wei. An extension of binary threshold sequences from Fermat quotients. Advances in Mathematics of Communications, 2016, 10 (4) : 743-752. doi: 10.3934/amc.2016038
[2]

Huaning Liu, Xi Liu. On the correlation measures of orders $ 3 $ and $ 4 $ of binary sequence of period $ p^2 $ derived from Fermat quotients. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021008
[3]

Grant Cairns, Barry Jessup, Marcel Nicolau. Topologically transitive homeomorphisms of quotients of tori. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 291-300. doi: 10.3934/dcds.1999.5.291
[4]

Peng Sun. Exponential decay of Lebesgue numbers. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3773-3785. doi: 10.3934/dcds.2012.32.3773
[5]

Danny Calegari, Alden Walker. Ziggurats and rotation numbers. Journal of Modern Dynamics, 2011, 5 (4) : 711-746. doi: 10.3934/jmd.2011.5.711
[6]

Xavier Buff, Nataliya Goncharuk. Complex rotation numbers. Journal of Modern Dynamics, 2015, 9: 169-190. doi: 10.3934/jmd.2015.9.169
[7]

Marc Deschamps, Olivier Poncelet. Complex ray in anisotropic solids: Extended Fermat's principle. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1623-1633. doi: 10.3934/dcdss.2019110
[8]

Eric Férard. On the irreducibility of the hyperplane sections of Fermat varieties in $\mathbb{P}^3$ in characteristic $2$. Advances in Mathematics of Communications, 2014, 8 (4) : 497-509. doi: 10.3934/amc.2014.8.497
[9]

Gilberto Bini, Margarida Melo, Filippo Viviani. On GIT quotients of Hilbert and Chow schemes of curves. Electronic Research Announcements, 2012, 19: 33-40. doi: 10.3934/era.2012.19.33
[10]

R. V. Gurjar, Mariusz Koras and Peter Russell. Two dimensional quotients of $CC^n$ by a reductive group. Electronic Research Announcements, 2008, 15: 62-64. doi: 10.3934/era.2008.15.62
[11]

Takao Komatsu, Bijan Kumar Patel, Claudio Pita-Ruiz. Several formulas for Bernoulli numbers and polynomials. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021006
[12]

Dmitry Krachun, Zhi-Wei Sun. On sums of four pentagonal numbers with coefficients. Electronic Research Archive, 2020, 28 (1) : 559-566. doi: 10.3934/era.2020029
[13]

Michael Björklund, Alexander Gorodnik. Central limit theorems in the geometry of numbers. Electronic Research Announcements, 2017, 24: 110-122. doi: 10.3934/era.2017.24.012
[14]

Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79
[15]

Stéphane Sabourau. Growth of quotients of groups acting by isometries on Gromov-hyperbolic spaces. Journal of Modern Dynamics, 2013, 7 (2) : 269-290. doi: 10.3934/jmd.2013.7.269
[16]

Marco Zambon, Chenchang Zhu. Distributions and quotients on degree $1$ NQ-manifolds and Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 469-485. doi: 10.3934/jgm.2012.4.469
[17]

Frédérique Oggier, B. A. Sethuraman. Quotients of orders in cyclic algebras and space-time codes. Advances in Mathematics of Communications, 2013, 7 (4) : 441-461. doi: 10.3934/amc.2013.7.441
[18]

Jifeng Chu, Meirong Zhang. Rotation numbers and Lyapunov stability of elliptic periodic solutions. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1071-1094. doi: 10.3934/dcds.2008.21.1071
[19]

Manfred G. Madritsch. Non-normal numbers with respect to Markov partitions. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 663-676. doi: 10.3934/dcds.2014.34.663
[20]

Dubi Kelmer. Quadratic irrationals and linking numbers of modular knots. Journal of Modern Dynamics, 2012, 6 (4) : 539-561. doi: 10.3934/jmd.2012.6.539

2020 Impact Factor: 1.833

Tools

Metrics

  • PDF downloads (124)
  • HTML views (272)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences