Recursive Relationships and Closed-Form Expressions in Annamalai’s Combinatorial System: A Framework for Large-Scale Data and Stochastic Modeling

Traditional methods for calculating binomial coefficients and modeling discrete probability distributions often face computational bottlenecks when applied to large-scale data, such as genomic sequencing or network reliability models. This paper presents a consolidated framework using Annamalai’s coefficients. It demonstrates how the Combinatorial Geometric Series (CGS) provides a superior alternative for expressing the Negative Binomial Distribution (NBD) and simplifies high-dimensional combinatorial data analysis and stochastic modeling through Recursive Relationships and Closed-Form Expressions.