Proofs of Silva–Sellers’ conjectures on a mock theta function

Recently, Silva and Sellers proved a number of nice congruences for the third order mock theta function ξ(q)=1+2∑n=1∞q6n2-6n+1(q;q6)n(q5;q6)n,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \xi (q)=1+2\sum _{n=1}^\infty \frac{q^{6n^2-6n+1}}{(q;q^6)_n (q^5;q^6)_n }, \end{aligned}$$\end{document}which was introduced by Gordon and McIntosh. At the end of their paper, they posed two conjectures on congruences for the coefficients of ξ(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi (q)$$\end{document}. In this paper, we confirm the two conjectures due to Silva and Sellers by utilizing the (p, k) -parametrization of theta functions.