### Electronic Research Archive

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

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

• 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.

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

### Related Papers:

• 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.

 [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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• 1.

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