Generalized Fishburn numbers and torus knots

Andrews and Sellers recently initiated the study of arithmetic properties of Fishburn numbers. In this paper, we prove prime power congruences for generalized Fishburn numbers. These numbers are the coefficients in the $1-q$ expansion of the Kontsevich-Zagier series ${\mathcal{F}}_t(q)$ for the torus knots $T(3,2^t)$, $t\ge 2$. The proof uses a strong divisibility result of Ahlgren, Kim and Lovejoy and a new “strange identity” for ${\mathcal{F}}_t(q)$.