Annamalai's Binomial Coefficient, Identities, and Generating Functions

Page | 1 Annamalai's Binomial Coefficient, Identities, and Generating Functions Chinnaraji Annamalai Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur, India Email: [email protected] https://orcid.org/0000-0002-0992-2584 Abstract: This paper explores 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 serves 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, offering a valuable alternative perspective on established principles of combinatorial enumeration. MSC Classification codes: 05A10, 11B65, 40A05 (65B10) Keywords: computation, combinatorial identities, binomial series, multiple summations 1. Introduction Combinatorics, the study of discrete structures, relies heavily on fundamental tools such as binomial coefficients {1-5] and generating functions. The standard binomial coefficient (𝑛 𝑘) is central to counting subsets and coefficients in the binomial expansion. Similarly, the geometric series ∑𝑥𝑛=(1−𝑥)−1 is the generating function for the sequence of all ones. This paper explores the structure proposed by Annamalai, which defines a new binomial coefficient, 𝑉𝑛𝑟, and derives a related power series, the Combinatorial Geometric Series [6-9]. The significance of this framework lies in emphasizing the recursive and product relationships of these coefficients, providing an alternative but equivalent pathway to known results in combinatorial enumeration [21, 22]. 2. Binomial Coefficient Annamalai defines a binomial coefficient 𝑉𝑛𝑟 for non-negative integers 𝑛 and 𝑟. The definition for 𝑉𝑛𝑟 is given by the product form: 𝑉𝑛𝑟=(𝑛+1)(𝑛+2)(𝑛+3)⋯(𝑛+𝑟) 𝑟! =∏𝑛+𝑖 𝑖 𝑟 𝑖=1 The identity between Annamalai's coefficient [10-16] and the standard binomial coefficient is established as follows: 𝑉𝑛𝑟=(𝑛+𝑟)! 𝑛!𝑟! =(𝑛+𝑟 𝑟) Initial conditions are defined as 𝑉0𝑟=𝑉𝑛0=𝑉00=1. A key property of this coefficient is symmetry: 𝑉𝑛𝑟=𝑉𝑟𝑛⇒(𝑛+𝑟 𝑟)=(𝑛+𝑟 𝑛) Page | 2 3. Binomial Identities Annamalai's binomial identity is the binomial series developed by the multiple summations of the extended geometric series. The 𝑟𝑡ℎ order Combinatorial Geometric Series (CGS) is defined by the result of 𝑟+1 iterative summations [17-20]: ∑𝑉𝑖𝑟𝑥𝑖 𝑛 𝑖=0 =∑∑∑⋯ 𝑛 𝑖=0 ⋯⋯∑𝑥𝑖𝑟 𝑛 𝑖=0 𝑛 𝑖=0 𝑛 𝑖=0 ⏟ 𝑟+1 summations Specifically, the 𝑟𝑡ℎ order CGS is defined the following summation structure: ∑𝑉𝑖𝑟𝑥𝑖 𝑛 𝑖=0 =∑ ∑ ⋯⋯⋯∑∑∑𝑥𝑖0 𝑛 𝑖0=0 𝑛 𝑖1=0 𝑛 𝑖2=0 𝑛 𝑖𝑟−1=0 𝑛 𝑖𝑟=0 The 0𝑡ℎ order CGS is the standard geometric series: ∑𝑉𝑖0𝑥𝑖 𝑛 𝑖=0 =∑𝑥𝑖 𝑛 𝑖=0 ,where 𝑉𝑖0=1 Annamalai's binomial theorem states that multiple summations of extended geometric series with binomial coefficients form a binomial series: ∑𝑉𝑖𝑟+1𝑥𝑖= 𝑛 𝑖=0 ∑𝑉𝑖𝑟𝑥𝑖 𝑛 𝑖=0 +∑𝑉𝑖−1 𝑟𝑥𝑖 𝑛 𝑖=1 +∑𝑉𝑖−2 𝑟𝑥𝑖 𝑛 𝑖=2 +⋯⋯⋯+ ∑ 𝑉𝑖−(𝑛−1) 𝑟𝑥𝑖 𝑛 𝑖=𝑛−1 +∑𝑉𝑖−𝑛 𝑟𝑥𝑖 𝑛 𝑖=𝑛 The sum of successive coefficients is equal to the next higher-order coefficient: ∑𝑉𝑖𝑟 𝑛 𝑖=0 = 𝑉0𝑟+𝑉1𝑟+𝑉2𝑟+𝑉3𝑟+⋯+𝑉𝑛−1 𝑟+𝑉𝑛𝑟=𝑉𝑛𝑟+1 By grouping terms based on the coefficient 𝑉𝑘𝑟 and re-expressing the sums as geometric series, we obtain the following identity: ∑𝑉𝑖𝑟+1𝑥𝑖= 𝑛 𝑖=0 𝑉0𝑟∑𝑥𝑖 𝑛 𝑖=0 +𝑉1𝑟∑𝑥𝑖 𝑛 𝑖=1 +𝑉2𝑟∑𝑥𝑖 𝑛 𝑖=2 +𝑉3𝑟∑𝑥𝑖 𝑛 𝑖=3 +⋯⋯⋯+𝑉𝑛−1 𝑟∑ 𝑥𝑖 𝑛 𝑖=𝑛−1 +𝑉𝑛𝑟∑𝑥𝑖 𝑛 𝑖=𝑛 Substituting the standard formula for the geometric summation,∑𝑥𝑖 𝑛−1 𝑖=𝑘 =𝑥𝑛−𝑥𝑘 𝑥−1 , yields the final series form: ∑𝑉𝑖𝑟+1𝑥𝑖=1 𝑥−1∑𝑉𝑖𝑟𝑥𝑖(𝑥𝑛−𝑖−1) 𝑛−1 𝑖=0 , 𝑛−1 𝑖=0 ∀ 𝑥≠1 Another identity involving a weighted sum of coefficients is given by: ∑(𝑖+1)𝑉𝑖𝑟 𝑛 𝑖=0 =∑𝑉𝑖𝑟+1 𝑛−1 𝑖=0 ⇒(𝑛+1)𝑉0𝑟+𝑛𝑉1𝑟+(𝑛−1)𝑉2𝑟+⋯+2𝑉𝑛−1 𝑟+𝑉𝑛𝑟=𝑉𝑛𝑟+2 4. Multiple Identical Finite Geometric Series The product of the sum of multiple finite geometric series, specifically, the product of 𝑟 identical finite geometric series, is given by: Page | 3 (∑𝑥𝑖 𝑛−1 𝑖=0 )(∑𝑥𝑖 𝑛−1 𝑖=0 )(∑𝑥𝑖 𝑛−1 𝑖=0 )⋯⋯⋯(∑𝑥𝑖 𝑛−1 𝑖=0 ) ⏟ 𝑟 times =(∑𝑥𝑖 𝑛−1 𝑖=0 )𝑟=(1−𝑥𝑛 1−𝑥)𝑟=(1−𝑥𝑛)𝑟 (1−𝑥)𝑟 The binomial expansion for (1+𝑥)𝑛 is given by the binomial theorem: (1+𝑥)𝑛=∑(𝑛 𝑘)𝑥𝑘 𝑛 𝑘=0 Similarly, the binomial expansion of (1−𝑥𝑛)𝑟 is presented as: (1−𝑥𝑛)𝑟= ∑(𝑟 𝑘) 𝑟 𝑘=0 (−𝑥𝑛)𝑘=∑(𝑟 𝑘) 𝑟 𝑘=0 ((−1)𝑥𝑛)𝑘=∑(𝑟 𝑘) 𝑟 𝑘=0 (−1)𝑘𝑥𝑘𝑛 A generalized form of the infinite geometric series is also noted: ∑𝑉𝑖𝑟−1𝑥𝑖= 1 (1−𝑥)𝑟 ∞ 𝑖=0 5. Generating Function The generating function [21, 22] provides the closed-form expression for the infinite sum of Combinatorial Geometric Series (CGS). 𝐺𝑟(𝑥)=∑𝑉𝑖𝑟−1𝑥𝑖= ∑(𝑖+𝑟 𝑖)𝑥𝑖 ∞ 𝑖=0 ∞ 𝑖=0 This is the generating function for the sequence of coefficients (𝑟 𝑟), (𝑟+1 𝑟), (𝑟+2 𝑟), ⋯⋯ It is a well-known result that the generating function for these coefficients is denoted by: ∑𝑉𝑖𝑟𝑥𝑖= 1 (1−𝑥)𝑟+1 ∞ 𝑖=0 This identity holds as a convergent power series for |𝑥|<1. The generating function for the closed-form expression for the finite sum of CGS is given by: 𝑁(𝑥)= ∑(𝑟 𝑘) 𝑟 𝑘=0 (−1)𝑘𝑥𝑘𝑛 =(1−𝑥𝑛)𝑟 6. Conclusion Annamalai's work provides a significant extension to classical combinatorial and geometric series theory. By introducing the extended geometric series and the optimized binomial coefficient 𝑉𝑛𝑟, the framework establishes new identities, theorems, and series representations. The derived results, including the Annamalai's Binomial Theorem and the Annamalai Series, along with the understanding of the underlying generative function, are foundational tools that will be useful for future research and development in computational mathematics. 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