Generalized Fishburn numbers and torus knots

Abstract

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 ${\mathcal{F}}_t(q)$.

arXiv comment13 pages, to appear in the Journal of Combinatorial Theory, Series A

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Cited by in corpus (3)

• Quantum modularity of partial theta series with periodic coefficients
• Braid representatives minimizing the number of simple walks
• Asymptotics and sign patterns for coefficients in expansions of Habiro elements