Some new infinite families of congruences and recurrence relations for the coefficients of a third order mock theta function

In his last letter to Hardy, Ramanujan (1920) introduced the notion of mock theta function and gave examples of seventeen such functions, and classified them as of order three, five and seven. After Ramanujan, many new mock theta functions are introduced by different mathematicians and their arithmetic properties were studied. In 2003, Gordon and McIntosh introduced a new mock theta function of order three for which congruences modulo 2, 3, 5, 7 and 9 are established by Silva and Sellers (2021), Baruah and Das (2023) and Yao (2023). In this paper, we prove some new infinite families of congruences modulo 16, 24 and 32 for the third order mock theta function (due to Gordon and McIntosh) by employing theta function and q-series identities. We also establish some recurrence relations by connecting the function with certain restricted partition functions.