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Title: Congruences for certain families of Apéry-like sequences (English)
Author: Sun, Zhi-Hong
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 72
Issue: 3
Year: 2022
Pages: 875-912
Summary lang: English
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Category: math
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Summary: We systematically investigate the expressions and congruences for both a one-parameter family $\{G_n(x)\}$ as well as a two-parameter family $\{G_n(r,m)\}$ of sequences. (English)
Keyword: Apéry-like number
Keyword: congruence
Keyword: combinatorial identity
Keyword: Bernoulli polynomial
Keyword: binary quadratic form
MSC: 05A10
MSC: 05A19
MSC: 11A07
MSC: 11B68
MSC: 11E25
idZBL: Zbl 07584107
idMR: MR4467947
DOI: 10.21136/CMJ.2022.0224-21
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Date available: 2022-08-22T08:25:52Z
Last updated: 2024-10-04
Stable URL: http://hdl.handle.net/10338.dmlcz/150622
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Reference: [1] Ahlgren, S.: Gaussian hypergeometric series and combinatorial congruences.Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics Developments in Mathematics 4. Kluwer Academic, Dordrecht (2001), 1-12. Zbl 1037.33016, MR 1880076, 10.1007/978-1-4613-0257-5_1
Reference: [2] Almkvist, G., Zudilin, W.: Differential equations, mirror maps and zeta values.Mirror Symmetry V AMS/IP Studies in Advanced Mathematics 38. AMS, Providence (2006), 481-515. Zbl 1118.14043, MR 2282972
Reference: [3] Apéry, R.: Irrationalité de $\zeta(2)$ et $\zeta(3)$.Astérisque 61 (1979), 11-13 French. Zbl 0401.10049, MR 3363457
Reference: [4] Gould, H. W.: Combinatorial Identities. A Standardized Set of Tables Listing 500 Binomial Coefficient Summations.West Virginia University, Morgantown (1972). Zbl 0241.05011, MR 0354401
Reference: [5] Granville, A.: Arithmetic properties of binomial coefficients. I: Binomial coefficients modulo prime powers.Organic Mathematics CMS Conference Proceedings 20. AMS, Providence (1997), 253-276. Zbl 0903.11005, MR 1483922
Reference: [6] Guo, V. J. W.: Proof of Sun's conjectures on integer-valued polynomials.J. Math. Anal. Appl. 444 (2016), 182-191. Zbl 1383.11030, MR 3523373, 10.1016/j.jmaa.2016.06.028
Reference: [7] Liu, J.-C., Ni, H.-X.: On two supercongruences involving Almkvist-Zudilin sequences.Czech. Math. J. 71 (2021), 1211-1219. Zbl 07442486, MR 4339123, 10.21136/CMJ.2021.0384-20
Reference: [8] Mortenson, E.: Supercongruences for truncated $_{n+1}F_n$ hypergeometric series with applications to certain weight three newforms.Proc. Am. Math. Soc. 133 (2005), 321-330. Zbl 1152.11327, MR 2093051, 10.1090/S0002-9939-04-07697-X
Reference: [9] Petkovšek, M., Wilf, H. S., Zeilberger, D.: $A=B$.A. K. Peters, Wellesley (1996). Zbl 0848.05002, MR 1379802
Reference: [10] Sloane, N. J. A.: The On-Line Encyclopedia of Integer Sequences.Available at http://oeis.org. Zbl 1159.11327
Reference: [11] Sun, Z.-H.: Congruences concerning Bernoulli numbers and Bernoulli polynomials.Discrete Appl. Math. 105 (2000), 193-223. Zbl 0990.11008, MR 1780472, 10.1016/S0166-218X(00)00184-0
Reference: [12] Sun, Z.-H.: Congruences involving Bernoulli and Euler numbers.J. Number Theory 128 (2008), 280-312. Zbl 1154.11010, MR 2380322, 10.1016/j.jnt.2007.03.003
Reference: [13] Sun, Z.-H.: Congruences concerning Legendre polynomials.Proc. Am. Math. Soc. 139 (2011), 1915-1929. Zbl 1225.11006, MR 2775368, 10.1090/S0002-9939-2010-10566-X
Reference: [14] Sun, Z.-H.: Identities and congruences for a new sequence.Int. J. Number Theory 8 (2012), 207-225. Zbl 1290.11048, MR 2887891, 10.1142/S1793042112500121
Reference: [15] Sun, Z.-H.: Congruences concerning Legendre polynomials. II.J. Number Theory 133 (2013), 1950-1976. Zbl 1277.11002, MR 3027947, 10.1016/j.jnt.2012.11.004
Reference: [16] Sun, Z.-H.: Congruences involving ${2k\choose k}^{2}{3k\choose k}$.J. Number Theory 133 (2013), 1572-1595. Zbl 1300.11007, MR 3007123, 10.1016/j.jnt.2012.10.001
Reference: [17] Sun, Z.-H.: Legendre polynomials and supercongruences.Acta Arith. 159 (2013), 169-200. Zbl 1287.11004, MR 3062914, 10.4064/aa159-2-6
Reference: [18] Sun, Z.-H.: Generalized Legendre polynomials and related supercongruences.J. Number Theory 143 (2014), 293-319. Zbl 1353.11005, MR 3227350, 10.1016/j.jnt.2014.04.012
Reference: [19] Sun, Z.-H.: Super congruences concerning Bernoulli polynomials.Int. J. Number Theory 11 (2015), 2393-2404. Zbl 1388.11004, MR 3420752, 10.1142/S1793042115501110
Reference: [20] Sun, Z.-H.: Supercongruences involving Bernoulli polynomials.Int. J. Number Theory 12 (2016), 1259-1271. Zbl 1419.11005, MR 3498625, 10.1142/S1793042116500779
Reference: [21] Sun, Z.-H.: Supercongruences involving Euler polynomials.Proc. Am. Math. Soc. 144 (2016), 3295-3308. Zbl 1388.11005, MR 3503698, 10.1090/proc/13005
Reference: [22] Sun, Z.-H.: Super congruences for two Apéry-like sequences.J. Difference Equ. Appl. 24 (2018), 1685-1713. Zbl 1446.11007, MR 3867051, 10.1080/10236198.2018.1515930
Reference: [23] Sun, Z.-H.: Congruences involving binomial coefficients and Apéry-like numbers.Publ. Math. 96 (2020), 315-346. Zbl 1474.11002, MR 4108043, 10.5486/PMD.2020.8577
Reference: [24] Sun, Z.-H.: Supercongruences and binary quadratic forms.Acta Arith. 199 (2021), 1-32. Zbl 1472.11026, MR 4262886, 10.4064/aa200308-27-9
Reference: [25] Sun, Z.-W.: Super congruences and Euler numbers.Sci. China, Math. 54 (2011), 2509-2535. Zbl 1256.11011, MR 2861289, 10.1007/s11425-011-4302-x
Reference: [26] Sun, Z.-W.: On sums involving products of three binomial coefficients.Acta Arith. 156 (2012), 123-141. Zbl 1269.11019, MR 2997562, 10.4064/aa156-2-2
Reference: [27] Sun, Z.-W.: Conjectures and results on $x^2$ mod $p^2$ with $4p=x^2+dy^2$.Number Theory and Related Areas Advanced Lectures in Mathematics (ALM) 27. International Press, Somerville (2013), 149-197. Zbl 1317.11034, MR 3185874
Reference: [28] Sun, Z.-W.: Some new series for $1/\pi$ and related congruences.J. Nanjing Univ., Math. Biq. 31 (2014), 150-164. Zbl 1324.11008, MR 3362545, 10.3969/j.issn.0469-5097.2014.02.004
Reference: [29] Sun, Z.-W.: New series for some special values of $L$-functions.J. Nanjing Univ., Math. Biq. 32 (2015), 189-218. Zbl 1349.11127, MR 3616300, 10.3969/j.issn.0469-5097.2015.02.006
Reference: [30] Sun, Z.-W.: Supercongruences involving dual sequences.Finite Fields Appl. 46 (2017), 179-216. Zbl 1406.11007, MR 3655755, 10.1016/j.ffa.2017.03.007
Reference: [31] Tauraso, R.: Supercongruences for a truncated hypergeometric series.Integers 12 (2012), Article ID A45, 12 pages. Zbl 1301.11020, MR 3083418
Reference: [32] Tauraso, R.: Supercongruences related to $_3F_2(1)$ involving harmonic numbers.Int. J. Number Theory 14 (2018), 1093-1109. Zbl 1421.11008, MR 3801086, 10.1142/S1793042118500689
Reference: [33] Enckevort, C. van, Straten, D. van: Monodromy calculations of fourth order equations of Calabi-Yau type.Mirror Symmetry V AMS/IP Studies in Advanced Mathematics 38. AMS, Providence (2006), 539-559. Zbl 1117.14043, MR 2282974, 10.1090/amsip/038/23
Reference: [34] Wang, C.: On two conjectural supercongruences of Z.-W. Sun.Ramanujan J. 56 (2021), 1111-1121. Zbl 07438419, MR 4341113, 10.1007/s11139-020-00283-w
Reference: [35] Wang, C., Sun, Z.-W.: Proof of some conjectural hypergeometric supercongruences via curious identities.J. Math. Anal. Appl. 505 (2022), Article ID 125575, 20 pages. Zbl 07412972, MR 4302676, 10.1016/j.jmaa.2021.125575
Reference: [36] Zagier, D.: Integral solutions of Apéry-like recurrence equations.Groups and Symmetries CRM Proceedings and Lecture Notes 47. AMS, Providence (2009), 349-366. Zbl 1244.11042, MR 2500571, 10.1090/crmp/047
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