Libro

This book offers a comprehensive and unified exploration of summation formulae for classical figurate numbers—triangular, pentagonal, hexagonal, and tetrahedral numbers via complex integration and the hockey-stick identity. By bringing complex analysis and discrete mathematics, the text reveals elegant structural connections that are often studied within a coherent and accessible framework. This book is designed for undergraduate students, early-stage researchers, and motivated self-learners. The book assumes only a basic background in high school mathematics. Familiarity with elementary combinatorics and introductory calculus may be helpful, but all essential ideas and techniques are carefully developed within the text. Each chapter is devoted to a specific class of figurate numbers, beginning with clear definitions and fundamental properties, followed by the derivation of classical summation formulae. In the first five chapters, Cauchy’s integral formulae regarding contour integration are used to derive these results in a novel and insightful manner. These analytic techniques not only provide elegant proofs but also deepen the reader’s understanding of the interplay between algebraic structures and analytic methods. The final chapter brings all five types of figurate numbers together under a unifying combinatorial lens, using the hockey-stick identity to derive summation relations in a purely discrete setting. This dual perspective highlights the profound connections between continuous and discrete mathematics, offering readers a richer and more integrated view of the subject. With its clear exposition, systematic organization, and innovative methodology, this book serves as both an introduction and an invitation to explore the beauty and unity of mathematics across different domains.