Congruences for generalized Fishburn numbers at roots of unity
Tom 199 / 2021
Streszczenie
There has been significant recent interest in the arithmetic properties of the coefficients of 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 Nth root of unity and r, s are certain integers.