Hecke type formula for unifiedwitten-reshetikhin-turaev invariants as higher-order mock theta functions

We study the unified Witten-Reshetikhin-Turaev invariant for the Brieskorn homology sphere S(2, 3, 6 p-1) based on the cyclotomic expansion of the colored Jones polynomial for twist knot Kp. We discuss that the invariant has the same asymptotic expansion in N → 8 with the Ramanujan mock theta function when q is the root of unity q = exp(2 π i/N), and that it can be regarded as the (6 p - 1)-th order mock theta function. It is shown that it has the Hecke-type formula as in the case of the mock theta functions, though the quadratic form is positive definite while indefinite for almost all the Ramanujan mock theta functions.