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doi: 10.3934/era.2020092

Combinatorics of some fifth and sixth order mock theta functions

1. 

School of Mathematics, Thapar Institute of Engineering and Technology, Patiala-147004, India

2. 

Yadavindra College of Engineering, Punjabi University Guru Kashi Campus, Talwandi Sabo-151302, India

* Corresponding author: Meenakshi Rana

Received  February 2020 Revised  July 2020 Published  September 2020

Fund Project: The first author is supported by SERB project Ref no. MTR/2019/000123

The goal of this paper is to provide a new combinatorial meaning to two fifth order and four sixth order mock theta functions. Lattice paths of Agarwal and Bressoud with certain modifications are used as a tool to study these functions.

Citation: Meenakshi Rana, Shruti Sharma. Combinatorics of some fifth and sixth order mock theta functions. Electronic Research Archive, doi: 10.3934/era.2020092
References:
[1]

A. K. Agarwal, Partitions with $N$ copies $N$, Combinatoire énumérative (Montreal, Que., 1985/Quebec, Que., 1985), 1–4, Lecture Notes in Math., 1234, Springer, Berlin, 1986. doi: 10.1007/BFb0072504.  Google Scholar

[2]

A. K. Agarwal, $n$–color partition theoretic interpretations of some mock theta functions, Electron. J. combin., 11, (2004), Note 14, 6 pp. doi: 10.37236/1855.  Google Scholar

[3]

A. K. Agarwal, Lattice paths and mock theta functions, In: Proceedings of the Sixth International Conference of SSFA, (2005), 95–102. Google Scholar

[4]

A. K. Agarwal and G. E. Andrews, Rogers–Ramanujan identities for partitions with "$n$ copies of $n$", J. Combin. Theory Ser. A, 45 (1987), 40-49.  doi: 10.1016/0097-3165(87)90045-8.  Google Scholar

[5]

A. K. Agarwal and D. M. Bressoud, Lattice paths and multiple basic hypergeometric series, Pacific J. Math., 136 (1989), 209-228.  doi: 10.2140/pjm.1989.136.209.  Google Scholar

[6]

A. K. Agarwal and G. Narang, Generalized Frobenius partitions and mock-theta functions, Ars Combin., 99 (2011), 439-444.   Google Scholar

[7]

A. K. Agarwal and M. Rana, Two new combinatorial interpretations of a fifth order mock theta function, J. Indian Math. Soc. (N.S.), 2007 (2008), 11-24.   Google Scholar

[8]

G. E. Andrews, Enumerative proofs of certain $q$-identities, Glasg. Math. J., 8 (1967), 33-40.  doi: 10.1017/S0017089500000057.  Google Scholar

[9]

G. E. Andrews, Partitions with initial repetitions, Acta Math. Sin. (Engl. Ser.), 25 (2009), 1437–1442. doi: 10.1007/s10114-009-6292-y.  Google Scholar

[10]

G. E. Andrews, A. Dixit and A. J. Yee, Partitions associated with the Ramanujan/Watson mock theta functions $\omega (q)$, $\nu (q)$ and $\phi (q)$, Res. Number Theory, 1 (2015), Paper No. 19, 25 pp. doi: 10.1007/s40993-015-0020-8.  Google Scholar

[11]

G. E. Andrews and F. G. Garvan, Ramanujan's "lost" notebook Ⅵ: The mock theta conjectures, Adv. Math., 73 (1989), 242-255.  doi: 10.1016/0001-8708(89)90070-4.  Google Scholar

[12]

G. E. Andrews and A. J. Yee, Some identities associated with mock theta functions $\omega(q)$ and $\nu(q)$, Ramanujan J., 48 (2019), 613-622.  doi: 10.1007/s11139-018-0028-5.  Google Scholar

[13]

B. C. Berndt and S. H. Chan, Sixth order mock theta functions, Adv. Math., 216 (2007), 771-786.  doi: 10.1016/j.aim.2007.06.004.  Google Scholar

[14]

W. H. Burge, A correspondence between partitions related to generalizations of the Rogers–Ramanujan identities, Discrete Math., 34 (1981), 9-15.  doi: 10.1016/0012-365X(81)90017-0.  Google Scholar

[15]

W. H. Burge, A three-way correspondence between partitions, European J. Combin., 3 (1982), 195-213.  doi: 10.1016/S0195-6698(82)80032-2.  Google Scholar

[16]

Y.-S. Choi and B. Kim, Partition identities from third and sixth order mock theta functions, European J. Combin., 33 (2012), 1739-1754.  doi: 10.1016/j.ejc.2012.04.005.  Google Scholar

[17]

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc. Providence, RI, 1988. doi: 10.1090/surv/027.  Google Scholar

[18]

B. Gordon and R. J. McIntosh, A survey of classical mock theta functions, Partitions, $q$-series, and Modular Forms, Springer, New York. 23 (2012), 95–144. doi: 10.1007/978-1-4614-0028-8_9.  Google Scholar

[19]

D. Hickerson, A proof of the mock theta conjectures, Invent. Math., 94 (1988), 639-660.  doi: 10.1007/BF01394279.  Google Scholar

[20]

F. Z. K. Li and J. Y. X. Yang, Combinatorial proofs for identities related to generalizations of the mock theta functions $\omega(q)$ and $\nu(q)$, Ramanujan J., 50 (2019), 527-550.  doi: 10.1007/s11139-018-0094-8.  Google Scholar

[21]

R. J. McIntosh, Modular transformations of Ramanujan's sixth order mock theta functions, preprint. Google Scholar

[22]

S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988.  Google Scholar

[23]

J. K. Sareen and M. Rana, Combinatorics of tenth-order mock theta functions, Proc. Indian Acad. Sci. Math. Sci., 126 (2016), 549-556.  doi: 10.1007/s12044-016-0305-4.  Google Scholar

[24]

S. Sharma and M. Rana, Combinatorial interpretations of mock theta functions by attaching weights, Discrete Math., 341 (2018), 1903-1914.  doi: 10.1016/j.disc.2018.03.017.  Google Scholar

[25]

S. Sharma and M. Rana, On mock theta functions and weight-attached Frobenius partitions, Ramanujan J., 50 (2019), 289-303.  doi: 10.1007/s11139-018-0054-3.  Google Scholar

[26]

S. Sharma and M. Rana, Interperting some fifth and sixth order mock theta functions by attaching weights, J. Ramanujan Math. Soc., 34 (2019), 401-410.   Google Scholar

[27]

S. Sharma and M. Rana, A new approach in interpreting some mock theta functions, Int. J. Number Theory, 15 (2019), 1369-1383.  doi: 10.1142/S1793042119500763.  Google Scholar

show all references

References:
[1]

A. K. Agarwal, Partitions with $N$ copies $N$, Combinatoire énumérative (Montreal, Que., 1985/Quebec, Que., 1985), 1–4, Lecture Notes in Math., 1234, Springer, Berlin, 1986. doi: 10.1007/BFb0072504.  Google Scholar

[2]

A. K. Agarwal, $n$–color partition theoretic interpretations of some mock theta functions, Electron. J. combin., 11, (2004), Note 14, 6 pp. doi: 10.37236/1855.  Google Scholar

[3]

A. K. Agarwal, Lattice paths and mock theta functions, In: Proceedings of the Sixth International Conference of SSFA, (2005), 95–102. Google Scholar

[4]

A. K. Agarwal and G. E. Andrews, Rogers–Ramanujan identities for partitions with "$n$ copies of $n$", J. Combin. Theory Ser. A, 45 (1987), 40-49.  doi: 10.1016/0097-3165(87)90045-8.  Google Scholar

[5]

A. K. Agarwal and D. M. Bressoud, Lattice paths and multiple basic hypergeometric series, Pacific J. Math., 136 (1989), 209-228.  doi: 10.2140/pjm.1989.136.209.  Google Scholar

[6]

A. K. Agarwal and G. Narang, Generalized Frobenius partitions and mock-theta functions, Ars Combin., 99 (2011), 439-444.   Google Scholar

[7]

A. K. Agarwal and M. Rana, Two new combinatorial interpretations of a fifth order mock theta function, J. Indian Math. Soc. (N.S.), 2007 (2008), 11-24.   Google Scholar

[8]

G. E. Andrews, Enumerative proofs of certain $q$-identities, Glasg. Math. J., 8 (1967), 33-40.  doi: 10.1017/S0017089500000057.  Google Scholar

[9]

G. E. Andrews, Partitions with initial repetitions, Acta Math. Sin. (Engl. Ser.), 25 (2009), 1437–1442. doi: 10.1007/s10114-009-6292-y.  Google Scholar

[10]

G. E. Andrews, A. Dixit and A. J. Yee, Partitions associated with the Ramanujan/Watson mock theta functions $\omega (q)$, $\nu (q)$ and $\phi (q)$, Res. Number Theory, 1 (2015), Paper No. 19, 25 pp. doi: 10.1007/s40993-015-0020-8.  Google Scholar

[11]

G. E. Andrews and F. G. Garvan, Ramanujan's "lost" notebook Ⅵ: The mock theta conjectures, Adv. Math., 73 (1989), 242-255.  doi: 10.1016/0001-8708(89)90070-4.  Google Scholar

[12]

G. E. Andrews and A. J. Yee, Some identities associated with mock theta functions $\omega(q)$ and $\nu(q)$, Ramanujan J., 48 (2019), 613-622.  doi: 10.1007/s11139-018-0028-5.  Google Scholar

[13]

B. C. Berndt and S. H. Chan, Sixth order mock theta functions, Adv. Math., 216 (2007), 771-786.  doi: 10.1016/j.aim.2007.06.004.  Google Scholar

[14]

W. H. Burge, A correspondence between partitions related to generalizations of the Rogers–Ramanujan identities, Discrete Math., 34 (1981), 9-15.  doi: 10.1016/0012-365X(81)90017-0.  Google Scholar

[15]

W. H. Burge, A three-way correspondence between partitions, European J. Combin., 3 (1982), 195-213.  doi: 10.1016/S0195-6698(82)80032-2.  Google Scholar

[16]

Y.-S. Choi and B. Kim, Partition identities from third and sixth order mock theta functions, European J. Combin., 33 (2012), 1739-1754.  doi: 10.1016/j.ejc.2012.04.005.  Google Scholar

[17]

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc. Providence, RI, 1988. doi: 10.1090/surv/027.  Google Scholar

[18]

B. Gordon and R. J. McIntosh, A survey of classical mock theta functions, Partitions, $q$-series, and Modular Forms, Springer, New York. 23 (2012), 95–144. doi: 10.1007/978-1-4614-0028-8_9.  Google Scholar

[19]

D. Hickerson, A proof of the mock theta conjectures, Invent. Math., 94 (1988), 639-660.  doi: 10.1007/BF01394279.  Google Scholar

[20]

F. Z. K. Li and J. Y. X. Yang, Combinatorial proofs for identities related to generalizations of the mock theta functions $\omega(q)$ and $\nu(q)$, Ramanujan J., 50 (2019), 527-550.  doi: 10.1007/s11139-018-0094-8.  Google Scholar

[21]

R. J. McIntosh, Modular transformations of Ramanujan's sixth order mock theta functions, preprint. Google Scholar

[22]

S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988.  Google Scholar

[23]

J. K. Sareen and M. Rana, Combinatorics of tenth-order mock theta functions, Proc. Indian Acad. Sci. Math. Sci., 126 (2016), 549-556.  doi: 10.1007/s12044-016-0305-4.  Google Scholar

[24]

S. Sharma and M. Rana, Combinatorial interpretations of mock theta functions by attaching weights, Discrete Math., 341 (2018), 1903-1914.  doi: 10.1016/j.disc.2018.03.017.  Google Scholar

[25]

S. Sharma and M. Rana, On mock theta functions and weight-attached Frobenius partitions, Ramanujan J., 50 (2019), 289-303.  doi: 10.1007/s11139-018-0054-3.  Google Scholar

[26]

S. Sharma and M. Rana, Interperting some fifth and sixth order mock theta functions by attaching weights, J. Ramanujan Math. Soc., 34 (2019), 401-410.   Google Scholar

[27]

S. Sharma and M. Rana, A new approach in interpreting some mock theta functions, Int. J. Number Theory, 15 (2019), 1369-1383.  doi: 10.1142/S1793042119500763.  Google Scholar

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