Binomial coefficients are among the oldest tools in discrete mathematics; Annamalai's identity introduces an elegant algebraic transformation that simplifies higher-order combinatorial summations.
For centuries, mathematicians have explored the symmetrical properties of Pascal's triangle and the binomial theorem, utilizing binomial identities as the structural scaffolding for number theory, probability, and calculus.
Evaluating complex combinatorial summations involving alternating signs and factorial fractions frequently demands cumbersome generating functions or contour integration, obscuring underlying arithmetic symmetries.
This mathematical paper presents Annamalai's binomial identity and theorem, deriving novel algebraic identities that transform nested binomial sums into compact, closed-form polynomial expressions.
These combinatorial identities provide powerful analytical shortcuts for algorithm complexity analysis, statistical distribution moments, and discrete algebraic geometry.
Annamalai's Binomial Identity and Theorem
: This paper presents Annamalai’s binomial theorem, coefficient, identity, and binomial expansion developed by Chinnaraji Annamalai of the Indian Institute of Technology Kharagpur. Also, an extended geometric series is introduced with innovative summation of single terms and more successive terms of the series in this article.
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