On the existence of the Legendre constants for some complex continued fraction expansions over imaginary quadratic fields

Abstract We consider some types of complex continued fraction expansions and their associated maps of the Euclidean fields Q ( − d ) , d = 1 , 2 , 3 , 7 and 11 introduced by Hurwitz (1887) [7] , Lakein (1973) [10] and Shiokawa et al. (1975) [20] . We show that there exists the Legendre constant for every type of continued fraction expansions introduced by them improving the proof by Ei et al. (2019) [4] in the case of the one by A. Hurwitz with Q ( − 1 ) .