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

References

[1] G. E. Andrews, The Theory of Partitions, Encyclopedia Math. Appl. 2, Addison-Wesley, Reading, 1976. Suche 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/s40993-015-0020-8Suche 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-JSuche 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/s11139-016-9812-2Suche 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/aa126-3-5Suche in Google Scholar

[6] B. C. Berndt, Ramanujan’s Notebooks. Part III, Springer, New York, 1991. 10.1007/978-1-4612-0965-2Suche 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/S1793042112500066Suche 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.2020.107037Suche 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.2022.108235Suche 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.2015.12.014Suche 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/s11139-021-00479-8Suche 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.a6Suche 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/pdu007Suche 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/S0004972700039010Suche 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/s11139-010-9225-6Suche in Google Scholar

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

[17] R. J. McIntosh, Second order mock theta functions, Canad. Math. Bull. 50 (2007), no. 2, 284–290. 10.4153/CMB-2007-028-9Suche 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.12980Suche in Google Scholar

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

[20] M. Newman, Modular forms whose coefficients possess multiplicative properties. II, Ann. of Math. (2) 75 (1962), 242–250. 10.2307/1970172Suche 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/s11139-010-9283-9Suche 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/aa146-1-4Suche 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.2013.05.009Suche in Google Scholar PubMed PubMed Central

[24] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Springer, Berlin, 1988. Suche 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.86.5.372Suche 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/S0002-9939-2010-10538-5Suche 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.2016.11.018Suche in Google Scholar

[28] L. Wang, Parity of coefficients of mock theta functions, J. Number Theory 229 (2021), 53–99. 10.1016/j.jnt.2021.04.023Suche 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/s11139-018-0004-0Suche 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/15221Suche 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-xSuche 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.87.65Suche 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/s13398-021-01106-3Suche 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/s11139-018-0056-1Suche in Google Scholar