Congruences for generalized Fishburn numbers at roots of unity

TL;DR

This paper establishes prime power congruences for generalized Fishburn numbers derived from special series related to Kontsevich-Zagier and torus knot series, at roots of unity, revealing new arithmetic properties.

Contribution

It proves prime power congruences for coefficients of generalized Fishburn numbers associated with specific series at roots of unity, extending known arithmetic results.

Findings

Proved prime power congruences for coefficients of generalized Fishburn numbers.

Established congruences for series related to Kontsevich-Zagier and torus knots.

Extended the understanding of arithmetic properties of these special series.

Abstract

There has been significant recent interest in the arithmetic properties of the coefficients of $F(1-q)$ and $\mathscr{F}_t(1-q)$ where $F(q)$ is the Kontsevich-Zagier strange series and $\mathscr{F}_t(q)$ is the strange series associated to a family of torus knots as studied by Bijaoui, Boden, Myers, Osburn, Rushworth, Tronsgard and Zhou. In this paper, we prove prime power congruences for two families of generalized Fishburn numbers, namely, for the coefficients of $(\zeta_N - q)^s F((\zeta_N - q)^r)$ and $(\zeta_N - q)^s \mathscr{F}_t((\zeta_N - q)^r)$, where $\zeta_N$ is an $N$th root of unity and $r$, $s$ are certain integers.