Congruences concerning generalized central trinomial coefficients
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- by Jia-Yu Chen and Chen Wang
- Proc. Amer. Math. Soc. 150 (2022), 3725-3738
- DOI: https://doi.org/10.1090/proc/15985
- Published electronically: May 20, 2022
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Abstract:
For any $n\in \mathbb {N}=\{0,1,2,\ldots \}$ and $b,c\in \mathbb {Z}$, the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Let $p$ be an odd prime. In this paper, we determine the summations $\sum _{k=0}^{p-1}T_k(b,c)^2/m^k$ modulo $p^2$ for integers $m$ with certain restrictions. As applications, we confirm some conjectural congruences of Sun [Sci. China Math. 57 (2014), pp. 1375–1400].References
- George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
- Henry W. Gould, Combinatorial identities, Henry W. Gould, Morgantown, W. Va., 1972. A standardized set of tables listing 500 binomial coefficient summations. MR 0354401
- Cheng-Yang Gu and Victor J. W. Guo, Proof of two conjectures on supercongruences involving central binomial coefficients, Bull. Aust. Math. Soc. 102 (2020), no. 3, 360–364. MR 4176679, DOI 10.1017/s0004972720000118
- Victor J. W. Guo, Some $q$-congruences with parameters, Acta Arith. 190 (2019), no. 4, 381–393. MR 4016117, DOI 10.4064/aa180624-16-12
- Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716, DOI 10.1007/978-1-4757-2103-4
- Ji-Cai Liu, Proof of some divisibility results on sums involving binomial coefficients, J. Number Theory 180 (2017), 566–572. MR 3679814, DOI 10.1016/j.jnt.2017.05.006
- Guo-Shuai Mao and Hao Pan, Supercongruences on some binomial sums involving Lucas sequences, J. Math. Anal. Appl. 448 (2017), no. 2, 1061–1078. MR 3582273, DOI 10.1016/j.jmaa.2016.10.055
- Sandro Mattarei and Roberto Tauraso, Congruences for central binomial sums and finite polylogarithms, J. Number Theory 133 (2013), no. 1, 131–157. MR 2981405, DOI 10.1016/j.jnt.2012.05.036
- Sandro Mattarei and Roberto Tauraso, From generating series to polynomial congruences, J. Number Theory 182 (2018), 179–205. MR 3703936, DOI 10.1016/j.jnt.2017.06.007
- Eric Mortenson, A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function, J. Number Theory 99 (2003), no. 1, 139–147. MR 1957248, DOI 10.1016/S0022-314X(02)00052-5
- Tony D. Noe, On the divisibility of generalized central trinomial coefficients, J. Integer Seq. 9 (2006), no. 2, Article 06.2.7, 12. MR 2247938
- Hao Pan and Zhi-Wei Sun, A combinatorial identity with application to Catalan numbers, Discrete Math. 306 (2006), no. 16, 1921–1940. MR 2251572, DOI 10.1016/j.disc.2006.03.050
- Fernando Rodriguez-Villegas, Hypergeometric families of Calabi-Yau manifolds, Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001) Fields Inst. Commun., vol. 38, Amer. Math. Soc., Providence, RI, 2003, pp. 223–231. MR 2019156
- Carsten Schneider, Symbolic summation assists combinatorics, Sém. Lothar. Combin. 56 (2006/07), Art. B56b, 36. MR 2317679
- N. J. A. Sloane, On-line encyclopedia of integer sequences, http://oeis.org.
- Zhi-Hong Sun, Congruences concerning Legendre polynomials, Proc. Amer. Math. Soc. 139 (2011), no. 6, 1915–1929. MR 2775368, DOI 10.1090/S0002-9939-2010-10566-X
- Zhi-Wei Sun and Roberto Tauraso, New congruences for central binomial coefficients, Adv. in Appl. Math. 45 (2010), no. 1, 125–148. MR 2628791, DOI 10.1016/j.aam.2010.01.001
- Zhi-Wei Sun, Super congruences and Euler numbers, Sci. China Math. 54 (2011), no. 12, 2509–2535. MR 2861289, DOI 10.1007/s11425-011-4302-x
- Zhi-Wei Sun, On congruences related to central binomial coefficients, J. Number Theory 131 (2011), no. 11, 2219–2238. MR 2825123, DOI 10.1016/j.jnt.2011.04.004
- Zhi-Wei Sun, Supercongruences involving products of two binomial coefficients, Finite Fields Appl. 22 (2013), 24–44. MR 3044090, DOI 10.1016/j.ffa.2013.02.001
- Zhi-Wei Sun, On sums related to central binomial and trinomial coefficients, Combinatorial and additive number theory—CANT 2011 and 2012, Springer Proc. Math. Stat., vol. 101, Springer, New York, 2014, pp. 257–312. MR 3297084, DOI 10.1007/978-1-4939-1601-6_{1}8
- Zhi-Wei Sun, Congruences involving generalized central trinomial coefficients, Sci. China Math. 57 (2014), no. 7, 1375–1400. MR 3213876, DOI 10.1007/s11425-014-4809-z
- Zhi-Wei Sun, Supercongruences involving dual sequences, Finite Fields Appl. 46 (2017), 179–216. MR 3655755, DOI 10.1016/j.ffa.2017.03.007
- Z.-W. Sun, Open conjectures on congruences, Nanjing Univ. J. Biquarterly 36 (2019), 1–99.
- Zhi-Wei Sun, Congruences involving generalized central trinomial coefficients, Sci. China Math. 57 (2014), no. 7, 1375–1400. MR 3213876, DOI 10.1007/s11425-014-4809-z
- Chen Wang and Wei Xia, Proof of two congruences concerning Legendre polynomials, Results Math. 76 (2021), no. 2, Paper No. 90, 13. MR 4244322, DOI 10.1007/s00025-021-01389-3
- J. Wolstenholme, On certain properties of prime numbers, Quart. J. Appl. Math 5 (1862), 35–39.
Bibliographic Information
- Jia-Yu Chen
- Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
- Email: 563917224@qq.com
- Chen Wang
- Affiliation: Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, People’s Republic of China
- ORCID: 0000-0002-4214-0816
- Email: cwang@smail.nju.edu.cn
- Received by editor(s): June 3, 2021
- Received by editor(s) in revised form: November 17, 2021
- Published electronically: May 20, 2022
- Additional Notes: This work was supported by the National Natural Science Foundation of China (grant no. 11971222).
The second author is the corresponding author - Communicated by: Matthew A. Papanikolas
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 3725-3738
- MSC (2020): Primary 11A07, 11B75; Secondary 05A10, 11B65
- DOI: https://doi.org/10.1090/proc/15985
- MathSciNet review: 4446225