Regularizations and finite ladders in multiple trigonometry

Abstract

We provide an extended interpretation of the zeta regularized product in \cite{D}. This allows us to get regularized product expressions of Holder's double sine function and its companion, i.e. the double and triple trigonometric functions. The expressions may reasonably explain the ladder structure among these multiple trigonometric functions. We also introduce the notion of finite ladders of functions which helps us understand the meaning behind these regularizations.