A mod 9 generating-function congruence for the Gordon–McIntosh mock theta coefficients

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Let pξ(n)p_{\xi}(n) denote the coefficients of the Gordon–McIntosh mock theta function ξ(q)\xi(q), so that ξ(q)=∑n≥0pξ(n)qn\xi(q)=\sum_{n\geq 0}p_{\xi}(n)q^n. Generating-function congruence. One has

∑n=0∞pξ(32n+12)qn≡6∑n=0∞q3n(n+1)/2(mod9).\sum_{n=0}^\infty p_{\xi}(32n+12)q^n \equiv 6\sum_{n=0}^\infty q^{3n(n+1)/2} \pmod{9}.

This is the second of two conjectured congruences motivated by computational evidence for the coefficients of ξ(q)\xi(q); its status is not resolved in the source.

References

Primary source

Robson da Silva and James A. Sellers, “Congruences for the coefficients of the Gordon and McIntosh mock theta function ξ(q)”, arXiv:2007.09819 (2020).

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