An element g in a group G is called reciprocal if there exists h in G such that
g^-1=hgh^-1. The reciprocal elements are also known as `real elements' or `reversible elements' in the literature. In this paper, we consider the Hecke groups, which are Fuchsian groups of the first kind and generalizations of the modular group. We have classified and parametrized the reciprocal classes in the Hecke groups. This generalizes a result by Sarnak on the reciprocal elements in the modular group.

The authors thank the referee for many comments and suggestions to improve the exposition of this paper. Das acknowledges full support from a UGC-NFSC research fellowship during the course of this work. Gongopadhyay acknowledges partial support from the SERB core research grant CRG/2022/003680.