Annamalai’s Combinatorial System and Generating Functions

This research paper rigorously examines a novel combinatorial framework proposed by Chinnaraji Annamalai, centered on a specialized binomial coefficient. It formally defines this coefficient, establishes its equivalence to a corresponding standard binomial coefficient, and derives the Combinatorial Geometric Series (CGS) through multiple summation techniques. The primary contribution of this framework is the elegant, closed-form expression for the generating function of these coefficients, which directly links the CGS to the negative binomial theorem. The paper details key combinatorial identities, recursive relationships, and series representations developed by Annamalai, showcasing an alternative, structured methodology for addressing fundamental problems in combinatorial enumeration and computational mathematics.