Annamalai's Binomial Coefficient, Identities, and Generating Functions

This paper explores the combinatorial system developed by Chinnaraji Annamalai, focusing on his definition of a generalized binomial coefficient and its application in deriving the Combinatorial Geometric Series (CGS). The CGS serves as the generating function for this sequence of coefficients, successfully confirming a fundamental, known result in a compact, closed-form expression. This framework is significant for its emphasis on the intrinsic recursive and product relationships of the coefficients, offering a valuable alternative perspective on established principles of combinatorial enumeration.