Congruences for generalized Fishburn numbers at roots of unity

2020/06/17 by Goswami, Ankush
#05A19 #11B65 #11P83 #33D15 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2006.09659

Abstract

There has been significant recent interest in the arithmetic properties of the coefficients of F(1-q) and \mathscrF_t(1-q) where F(q) is the Kontsevich-Zagier strange series and \mathscrF_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 (ζ_N - q)^s F((ζ_N - q)^r) and (ζ_N - q)^s \mathscrF_t((ζ_N - q)^r), where ζ_N is an Nth root of unity and r, s are certain integers.

Related

• Dissections of a "strange" function
• Dissections of strange q-series
• Congruences and relations for r-Fishburn numbers
• Congruences for Taylor expansions of quantum modular forms
• Bailey pairs and strange identities