Let K(m,p) denote the family of double twist knots where 2m−1 and 2p
are non-zero integers denoting the number of half-twists in each region. Using
a result of Takata, we prove a formula for the colored Jones polynomial of
K(−m,−p) and K(−m,p). The latter case leads to new families of
q-hypergeometric series generalizing the Kontsevich-Zagier series. We also
use Bailey pairs and formulas of Walsh to find cyclotomic-like expansions for
the colored Jones polynomials of K(m,p) and K(m,−p).Comment: 30 pages, to appear in the New York Journal of Mathematic