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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

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