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

Harman Kaur; Meenakshi Rana; Harman Kaur; Meenakshi Rana

doi:10.3934/era.2021084

Electronic Research Archive

2021, Volume 29, Issue 6: 4257-4268. doi: 10.3934/era.2021084

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

Harman Kaur ,
Meenakshi Rana ^,

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

Received: 01 June 2021 Revised: 01 September 2021 Published: 26 October 2021
Primary: 11P83, 05A17

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.
- Mock theta function,
- congruence
Citation: Harman Kaur, Meenakshi Rana. Congruences for sixth order mock theta functions $ \lambda(q) $ and $ \rho(q) $[J]. Electronic Research Archive, 2021, 29(6): 4257-4268. doi: 10.3934/era.2021084

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Abstract

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.

References

[1]	Congruences related to the Ramanujan/Watson mock theta functions $\omega(q)$ and $\nu(q)$. Ramanujan J. (2017) 43: 347-357.
[2]	Generating functions and congruences for some partition functions related to mock theta functions. Int. J. Number Theory (2020) 16: 423-446.
[3]	Congruences related to an eighth order mock theta function of Gordon and McIntosh. J. Math. Anal. Appl. (2019) 479: 62-89.
[4]	Some congruences for partition functions related to mock theta functions $\omega(q)$ and $\nu(q)$. New Zealand J. Math. (2017) 47: 161-168.
[5]	M. D. Hirschhorn, The Power of $q$, Developments in Mathematics, 49. Springer, Cham, 2017. doi: 10.1007/978-3-319-57762-3
[6]	Arithmetic relations for overpartitions. J. Combin. Math. Combin. Comput. (2005) 53: 65-73.
[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
[8]	Congruences for the coefficients of the mock theta function $\beta(q)$. Ramanujan J. (2019) 49: 257-267.

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