Home The ranks of (a,b)-Fibonacci sequences and congruences for certain partition functions and Ramanujan's mock theta functions
Article
Licensed
Unlicensed Requires Authentication

The ranks of (a,b)-Fibonacci sequences and congruences for certain partition functions and Ramanujan's mock theta functions

  • Ernest X. W. Xia EMAIL logo
Published/Copyright: January 13, 2025

Abstract

In this paper, employing some identities due to Newman, we present a new method for discovering infinite families of congruences and strange congruences for c ( n ) which is defined by

n = 0 c ( n ) q n = k = 1 ( 1 - q k ) r ( 1 - q k t ) s .

Here r , s , t are some integers with t > 1 . By our method, in order to discover infinite families of congruences modulo M for c ( n ) , it suffices to compute the ranks of ( a , b ) -Fibonacci sequences modulo M. As applications, we establish many infinite families of congruences and strange congruences for some Ramanujan’s mock theta functions and certain partition functions, such as, a partition function related to mock theta function ω ( q ) and broken k-diamond partitions.

MSC 2020: 05A17; 11P83

Communicated by Freydoon Shahidi


Award Identifier / Grant number: 12371334

Funding statement: This work was supported by the National Science Foundation of China (no. 12371334) and the Natural Science Foundation of Jiangsu Province of China (no. BK20221383).

Acknowledgements

The author would like to express his sincere gratitude to the anonymous referee for his/her careful reading of the manuscript and many constructive suggestions.

References

[1] G. E. Andrews, The Theory of Partitions, Encyclopedia Math. Appl. 2, Addison-Wesley, Reading, 1976. Search in Google Scholar

[2] G. E. Andrews, A. Dixit and A. J. Yee, Partitions associated with the Ramanujan/Watson mock theta functions ω ( q ) , ν ( q ) and ϕ ( q ) , Res. Number Theory 1 (2015), Paper No. 19. 10.1007/s40993-015-0020-8Search in Google Scholar

[3] G. E. Andrews and D. Hickerson, Ramanujan’s “lost” notebook. VII. The sixth order mock theta functions, Adv. Math. 89 (1991), no. 1, 60–105. 10.1016/0001-8708(91)90083-JSearch in Google Scholar

[4] G. E. Andrews, D. Passary, J. A. Sellers and A. J. Yee, Congruences related to the Ramanujan/Watson mock theta functions ω ( q ) and ν ( q ) , Ramanujan J. 43 (2017), no. 2, 347–357. 10.1007/s11139-016-9812-2Search in Google Scholar

[5] G. E. Andrews and P. Paule, MacMahon’s partition analysis. XI. Broken diamonds and modular forms, Acta Arith. 126 (2007), no. 3, 281–294. 10.4064/aa126-3-5Search in Google Scholar

[6] B. C. Berndt, Ramanujan’s Notebooks. Part III, Springer, New York, 1991. 10.1007/978-1-4612-0965-2Search in Google Scholar

[7] S. H. Chan and R. Mao, Two congruences for Appell–Lerch sums, Int. J. Number Theory 8 (2012), no. 1, 111–123. 10.1142/S1793042112500066Search in Google Scholar

[8] D. Chen and L. Wang, Representations of mock theta functions, Adv. Math. 365 (2020), Article ID 107037. 10.1016/j.aim.2020.107037Search in Google Scholar

[9] R. Chen and F. G. Garvan, A proof of the mod  4 unimodal sequence conjectures and related mock theta functions, Adv. Math. 398 (2022), Article ID 108235. 10.1016/j.aim.2022.108235Search in Google Scholar

[10] S.-P. Cui, N. S. S. Gu and A. X. Huang, Congruence properties for a certain kind of partition functions, Adv. Math. 290 (2016), 739–772. 10.1016/j.aim.2015.12.014Search in Google Scholar

[11] R. da Silva and J. A. Sellers, Congruences for the coefficients of the Gordon and McIntosh mock theta function ξ ( q ) , Ramanujan J. 58 (2022), no. 3, 815–834. 10.1007/s11139-021-00479-8Search in Google Scholar

[12] S. A. Garthwaite and D. Penniston, p-adic properties of Maass forms arising from theta series, Math. Res. Lett. 15 (2008), no. 3, 459–470. 10.4310/MRL.2008.v15.n3.a6Search in Google Scholar

[13] D. R. Hickerson and E. T. Mortenson, Hecke-type double sums, Appell–Lerch sums, and mock theta functions, I, Proc. Lond. Math. Soc. (3) 109 (2014), no. 2, 382–422. 10.1112/plms/pdu007Search in Google Scholar

[14] M. D. Hirschhorn and J. A. Sellers, On recent congruence results of Andrews and Paule for broken k-diamonds, Bull. Aust. Math. Soc. 75 (2007), no. 1, 121–126. 10.1017/S0004972700039010Search in Google Scholar

[15] M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with odd parts distinct, Ramanujan J. 22 (2010), no. 3, 273–284. 10.1007/s11139-010-9225-6Search in Google Scholar

[16] E. Lucas, Theorie des fonctions numeriques simplement periodiques, Amer. J. Math. 1 (1878), no. 4, 289–321. 10.2307/2369373Search in Google Scholar

[17] R. J. McIntosh, Second order mock theta functions, Canad. Math. Bull. 50 (2007), no. 2, 284–290. 10.4153/CMB-2007-028-9Search in Google Scholar

[18] E. T. Mortenson, On Ramanujan’s lost notebook and new tenth-order like identities for second-, sixth-, and eighth-order mock theta functions, Bull. Lond. Math. Soc. 56 (2024), no. 3, 1029–1053. 10.1112/blms.12980Search in Google Scholar

[19] M. Newman, Modular forms whose coefficients possess multiplicative properties, Ann. of Math. (2) 70 (1959), 478–489. 10.2307/1970326Search in Google Scholar

[20] M. Newman, Modular forms whose coefficients possess multiplicative properties. II, Ann. of Math. (2) 75 (1962), 242–250. 10.2307/1970172Search in Google Scholar

[21] P. Paule and S. Radu, Infinite families of strange partition congruences for broken 2-diamonds, Ramanujan J. 23 (2010), no. 1–3, 409–416. 10.1007/s11139-010-9283-9Search in Google Scholar

[22] S. Radu and J. A. Sellers, Parity results for broken k-diamond partitions and ( 2 k + 1 ) -cores, Acta Arith. 146 (2011), no. 1, 43–52. 10.4064/aa146-1-4Search in Google Scholar

[23] S. Radu and J. A. Sellers, An extensive analysis of the parity of broken 3-diamond partitions, J. Number Theory 133 (2013), no. 11, 3703–3716. 10.1016/j.jnt.2013.05.009Search in Google Scholar PubMed PubMed Central

[24] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Springer, Berlin, 1988. Search in Google Scholar

[25] M. Renault, The period, rank, and order of the ( a , b ) -Fibonacci sequence mod m , Math. Mag. 86 (2013), no. 5, 372–380. 10.4169/math.mag.86.5.372Search in Google Scholar

[26] M. Waldherr, On certain explicit congruences for mock theta functions, Proc. Amer. Math. Soc. 139 (2011), no. 3, 865–879. 10.1090/S0002-9939-2010-10538-5Search in Google Scholar

[27] L. Wang, New congruences for partitions related to mock theta functions, J. Number Theory 175 (2017), 51–65. 10.1016/j.jnt.2016.11.018Search in Google Scholar

[28] L. Wang, Parity of coefficients of mock theta functions, J. Number Theory 229 (2021), 53–99. 10.1016/j.jnt.2021.04.023Search in Google Scholar

[29] E. X. W. Xia, Arithmetic properties for a partition function related to the Ramanujan/Watson mock theta function ω ( q ) , Ramanujan J. 46 (2018), no. 2, 545–562. 10.1007/s11139-018-0004-0Search in Google Scholar

[30] E. X. W. Xia, New congruence properties for Ramanujan’s ϕ function, Proc. Amer. Math. Soc. 149 (2021), no. 12, 4985–4999. 10.1090/proc/15221Search in Google Scholar

[31] E. X. W. Xia and O. X. M. Yao, Analogues of Ramanujan’s partition identities, Ramanujan J. 31 (2013), no. 3, 373–396. 10.1007/s11139-012-9439-xSearch in Google Scholar

[32] X. Xiong, Two congruences involving Andrews–Paule’s broken 3-diamond partitions and 5-diamond partitions, Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 5, 65–68. 10.3792/pjaa.87.65Search in Google Scholar

[33] O. X. M. Yao, Proofs of Silva–Sellers’ conjectures on a mock theta function, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115 (2021), no. 4, Paper No. 170. 10.1007/s13398-021-01106-3Search in Google Scholar

[34] W. Zhang and J. Shi, Congruences for the coefficients of the mock theta function β ( q ) , Ramanujan J. 49 (2019), no. 2, 257–267. 10.1007/s11139-018-0056-1Search in Google Scholar

Received: 2024-05-22
Revised: 2024-09-08
Published Online: 2025-01-13
Published in Print: 2025-06-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 19.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2024-0240/html
Scroll to top button