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December  2021, 29(6): 4257-4268. doi: 10.3934/era.2021084

Congruences for sixth order mock theta functions $ \lambda(q) $ and $ \rho(q) $

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

* Corresponding author: Meenakshi Rana

Received  June 2021 Revised  September 2021 Published  December 2021 Early access  October 2021

Ramanujan introduced sixth order mock theta functions
$ \lambda(q) $
and
$ \rho(q) $
defined as:
$ \begin{align*} \lambda(q) & = \sum\limits_{n = 0}^{\infty}\frac{(-1)^n q^n (q;q^2)_n}{(-q;q)_n},\\ \rho(q) & = \sum\limits_{n = 0}^{\infty}\frac{ q^{n(n+1)/2} (-q;q)_n}{(q;q^2)_{n+1}}, \end{align*} $
listed in the Lost Notebook. In this paper, we present some Ramanujan-like congruences and also find their infinite families modulo 12 for the coefficients of mock theta functions mentioned above.
Citation: Harman Kaur, Meenakshi Rana. Congruences for sixth order mock theta functions $ \lambda(q) $ and $ \rho(q) $. Electronic Research Archive, 2021, 29 (6) : 4257-4268. doi: 10.3934/era.2021084
References:
[1]

G. E. AndrewsD. PassaryJ. A. Sellers and A. J. Yee, Congruences related to the Ramanujan/Watson mock theta functions $\omega(q)$ and $\nu(q)$, Ramanujan J., 43 (2017), 347-357.  doi: 10.1007/s11139-016-9812-2.  Google Scholar

[2]

N. D. Baruah and N. M. Begum, Generating functions and congruences for some partition functions related to mock theta functions, Int. J. Number Theory, 16 (2020), 423-446.  doi: 10.1142/S1793042120500220.  Google Scholar

[3]

E. H. M. BrietzkeR. da Silva and J. A. Sellers, Congruences related to an eighth order mock theta function of Gordon and McIntosh, J. Math. Anal. Appl., 479 (2019), 62-89.  doi: 10.1016/j.jmaa.2019.06.016.  Google Scholar

[4]

S. N. Fathima and U. Pore, Some congruences for partition functions related to mock theta functions $\omega(q)$ and $\nu(q)$, New Zealand J. Math., 47 (2017), 161-168.   Google Scholar

[5]

M. D. Hirschhorn, The Power of $q$, Developments in Mathematics, 49. Springer, Cham, 2017. doi: 10.1007/978-3-319-57762-3.  Google Scholar

[6]

M. D. Hirschhorn and J. A. Sellers, Arithmetic relations for overpartitions, J. Combin. Math. Combin. Comput., 53 (2005), 65-73.   Google Scholar

[7]

R. da Silva and J. A. Sellers, Congruences for the coefficients of the Gordon and McIntosh mock theta function $\xi(q)$, Ramanujan J., (2021), 1–20. doi: 10.1007/s11139-021-00479-8.  Google Scholar

[8]

W. Zhang and J. Shi, Congruences for the coefficients of the mock theta function $\beta(q)$, Ramanujan J., 49 (2019), 257-267.  doi: 10.1007/s11139-018-0056-1.  Google Scholar

show all references

References:
[1]

G. E. AndrewsD. PassaryJ. A. Sellers and A. J. Yee, Congruences related to the Ramanujan/Watson mock theta functions $\omega(q)$ and $\nu(q)$, Ramanujan J., 43 (2017), 347-357.  doi: 10.1007/s11139-016-9812-2.  Google Scholar

[2]

N. D. Baruah and N. M. Begum, Generating functions and congruences for some partition functions related to mock theta functions, Int. J. Number Theory, 16 (2020), 423-446.  doi: 10.1142/S1793042120500220.  Google Scholar

[3]

E. H. M. BrietzkeR. da Silva and J. A. Sellers, Congruences related to an eighth order mock theta function of Gordon and McIntosh, J. Math. Anal. Appl., 479 (2019), 62-89.  doi: 10.1016/j.jmaa.2019.06.016.  Google Scholar

[4]

S. N. Fathima and U. Pore, Some congruences for partition functions related to mock theta functions $\omega(q)$ and $\nu(q)$, New Zealand J. Math., 47 (2017), 161-168.   Google Scholar

[5]

M. D. Hirschhorn, The Power of $q$, Developments in Mathematics, 49. Springer, Cham, 2017. doi: 10.1007/978-3-319-57762-3.  Google Scholar

[6]

M. D. Hirschhorn and J. A. Sellers, Arithmetic relations for overpartitions, J. Combin. Math. Combin. Comput., 53 (2005), 65-73.   Google Scholar

[7]

R. da Silva and J. A. Sellers, Congruences for the coefficients of the Gordon and McIntosh mock theta function $\xi(q)$, Ramanujan J., (2021), 1–20. doi: 10.1007/s11139-021-00479-8.  Google Scholar

[8]

W. Zhang and J. Shi, Congruences for the coefficients of the mock theta function $\beta(q)$, Ramanujan J., 49 (2019), 257-267.  doi: 10.1007/s11139-018-0056-1.  Google Scholar

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