Generalized Closed-Form Finite Sum Identities and Derivations within the Annamalai Combinatorial System for Stochastic Network Optimization

This paper presents a generalized algebraic derivation for finite sum identities within the Annamalai Combinatorial System, extending the conventional boundary of combinatorial geometric series (CGS) to accommodate arbitrary starting indices. While traditional methods rely on mathematical induction, this approach utilizes series decomposition and the Cauchy product of generating functions to establish a robust closed-form expression. The derivation rigorously accounts for truncation by evaluating the remainder term through index shifting and the substitution of sub-generating functions. By providing a precise mathematical bridge between truncated segments and infinite series, this generalized identity offers significant computational efficiency for stochastic network optimization, high-dimensional traffic analysis, and real-time algorithm execution in FPGA-based architectures.