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A family of $ q $-congruences modulo the square of a cyclotomic polynomial
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Proof of Sun's conjectural supercongruence involving Catalan numbers
Department of Mathematics, Wenzhou University, Wenzhou 325035, China |
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.
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. |
[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. |
[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. |
[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. |
[6] |
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, Oxford, 2008.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
C. Schneider, Symbolic summation assists combinatorics, Sém. Lothar. Combin., 56 (2006/07), 36pp. |
[16] |
L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.
![]() |
[17] |
R. P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, 1999.
doi: 10.1017/CBO9780511609589. |
[18] |
Z.-H. Sun,
Super congruences involving Bernoulli polynomials, Int. J. Number Theory, 12 (2016), 1259-1271.
doi: 10.1142/S1793042116500779. |
[19] |
Z.-W. Sun,
Super congruences and Euler numbers, Sci. China Math., 54 (2011), 2509-2535.
doi: 10.1007/s11425-011-4302-x. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[6] |
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, Oxford, 2008.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
C. Schneider, Symbolic summation assists combinatorics, Sém. Lothar. Combin., 56 (2006/07), 36pp. |
[16] |
L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.
![]() |
[17] |
R. P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, 1999.
doi: 10.1017/CBO9780511609589. |
[18] |
Z.-H. Sun,
Super congruences involving Bernoulli polynomials, Int. J. Number Theory, 12 (2016), 1259-1271.
doi: 10.1142/S1793042116500779. |
[19] |
Z.-W. Sun,
Super congruences and Euler numbers, Sci. China Math., 54 (2011), 2509-2535.
doi: 10.1007/s11425-011-4302-x. |
[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. |
[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. |
[22] |
Z.-W. Sun, Open conjectures on congruences, Nanjing Univ. J. Math. Biquarterly, 36 (2019), 1-99. Google Scholar |
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