Abstract
In this paper, we prove some congruences involving the coefficients \(\{A_{n}\}_{n=0,1,2,\ldots }\) of the analytic solution \(y_0(z)=\sum _{n=0}^\infty A_nz^n\) of certian differential eqution \({\mathcal {D}}y=0\) normalized by the condition \(y_0(0)=A_0=1\), where \({\mathcal {D}}\) is a 4th-order linear differential operator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Almkvist, C. van Enckevort, D. van Straten and W. Zudilin, Tables of Calabi-Yau equations, arXiv:math/0507430v2.
G. Almkvist and W. Zudilin, Differential equations, mirror maps and zeta values, Mirror Symmetry V, (N. Yui, S.-T. Yau, and J. D. Lewis, eds.), Proceedings of BIRS workshop on Calabi-Yau Varieties and Mirror Symmetry (December 6-C11, 2003), AMS/IP Stud. Adv. Math. 38, Amer. Math. Soc. International Press, Providence, RI (2007), 481–515; math.NT/0402386 (2004).
L. Calitz, A theorem of Glaisher, Canadian J. Math. 5 (1953), 306–316.
H.-Q. Cao and H. Pan, Note on some congruences of Lehmer, J. Number Theory 129 (2009), no.8, 1813–1819.
M. E. Hoffman, Quasi-symmetric functions and mod\(p\)multiple hamonic sums, Kyushu. J. Math. 69 (2015), 345–366.
G.-S. Mao, Proof of some congruences conjectured by Z.-W. Sun, Int. J. Number Theory 13 (2017), No.8, 1983–1993.
G.-S. Mao, Proof of two conjectural supercongruences involving Catalan-Larcombe-French numbers, J. Number Theory 179 (2017), 88–96.
G.-S. Mao and Z.-W. Sun, Two congruences involving harmonic numbers with applications, Int. J. Number Theory 12 (2016), no.02, 527-539.
Z.-H. Sun, Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete. Appl. Math. 105 (2000), 193–223.
Z.-H. Sun, Congruences involving Bernoulli and Euler numbers, J. Number Theory 128 (2008), 280–312.
Z.-W. Sun and R. Tauraso, New congruences for central binomial cofficients, Adv. in Appl. Math. 45 (2010), no.1, 125–148
Z.-W. Sun, Super congruences and Euler numbers, Sci. China Math. 54 (2011), 2509–2535.
Z.-W. Sun, A new series for\(\pi ^3\)and related congruences, Internat. J. Math. 26 (2015), no.8, 1550055 (23 pages).
Z.-W. Sun and L.-L. Zhao, Arithmetic theory of harmonic numbers (II), Colloq. Math. 130 (2013), no.1, 67–78.
J. Wolstenholme, On certain properties of prime numbers, Quart. J. Math. 5 (1862), 35–39.
Acknowledgements
The authors would like to thank the anonymous referee for his/her useful suggestions on this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by B. Sury.
This research was supported by the National Natural Science Foundation of China (Grant Nos. 12001288, 12071208).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Mao, GS., Zhang, H. On some super-congruences for the coefficients of analytic solutions of certain differential equations. Indian J Pure Appl Math (2024). https://doi.org/10.1007/s13226-024-00582-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13226-024-00582-8