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

IF 0.7 Q2 MATHEMATICS

Afrika Matematika Pub Date : 2025-09-30 DOI:10.1007/s13370-025-01380-z

Hirakjyoti Das

引用次数: 0

Abstract

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 $0\le s< 29$, $s\ne 13$.

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八阶模拟函数模2的同余式 $V_1(q)$

到目前为止，八阶模拟θ函数$V_1(q)$: $$\begin{aligned} V_1(q):=\sum _{n=0}^\infty \dfrac{q^{(n+1)^2}\left( -q;q^2\right) _n}{\left( q;q^2\right) _{n+1}}=\sum _{n=1}^\infty v_1(n)q^n \end{aligned}$$的同余性并不多。我们证明了$V_1(q)$的系数具有自相似性。作为结果，我们找到了如下所示的同余。对于所有$n\ge 0$和$k\ge 1$，我们有$$\begin{aligned} v_1\left( 6\times 29^{2 k} n+ 6\times 29^{2 k-1} s+\dfrac{7\times 29^{2 k-1}+1}{4}\right) \equiv 0 \pmod {2} \end{aligned}$$表示$0\le s< 29$, $s\ne 13$。

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