On Certain Series Expansions of the Sine Function Containing Embedded Catalan Numbers: A Complete Analytic Formulation

This article continues the study of a class of non-terminating expansions of sin\( (m\alpha) \) (even $m \geq 2 \)) which in each case possesses embedded Catalan numbers. A known series form of the sine function (said to be associated with Euler) is taken here as our basic representation, the coefficient of the general term being developed analytically in an interesting fashion and shown to be dependent on the Catalan sequence in the manner expected.
The work, which has a historical backdrop to it, is discussed in the context of prior results by the author and others.