Congruences Modulo 2 for the Eighth-Order Mock Theta Function \(V_1(q)\)

Not many of the congruence properties of the eighth-order mock theta function \(V_1(q)\):

$$\begin{aligned} V_1(q):=\sum _{n=0}^\infty \dfrac{q^{(n+1)^2}\left( -q;q^2\right) _n}{\left( q;q^2\right) _{n+1}}=\sum _{n=1}^\infty v_1(n)q^n \end{aligned}$$

have been considered to date. We show that there are self-similarities of the coefficients of \(V_1(q)\). As consequences, we find congruences like the one below. For all \(n\ge 0\) and \(k\ge 1\), we have

$$\begin{aligned} v_1\left( 6\times 29^{2 k} n+ 6\times 29^{2 k-1} s+\dfrac{7\times 29^{2 k-1}+1}{4}\right) \equiv 0 \pmod {2} \end{aligned}$$

for \(0\le s< 29\), \(s\ne 13\).