Combinatorial Geometric Series and Generating Functions

This paper provides a critical examination of 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 is established 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 and details the application of the framework in expressing the negative binomial theorem and the generating functions for both finite and infinite sums. Annamalai's methodology offers a valuable, alternative perspective on established principles of combinatorial enumeration.