Reciprocity theorems involving the q-gamma function

Abstract

In one of his notebooks, Ramanujan stated but did not prove an identity for a ratio of gamma functions involving two separate variables. Such an identity was referred to as a reciprocity theorem due to the symmetry relations between the variables. The two variable result was later proved and generalized to several variables by Berndt and Koukoulopoulos (Proc Am Math Soc 601:2369–2373, 2007). In this paper, we state and prove similar reciprocity theorems for ratios of q-gamma functions involving two and more variables.