BERNOULLI NUMBERS

This book explores Bernoulli numbers and their role in combinatorics, number theory, and related areas of discrete mathematics. It develops the subject from classical problems on sums of integer powers, a topic that has been studied since antiquity, leading to the discovery of a unified sequence of constants by Jakob Bernoulli.The early chapters introduce the necessary combinatorial and analytical tools, including partitions, permutations, generating functions, recurrence relations, and finite sums. These foundations are used to study classical summation problems and key techniques such as finite differences, asymptotic formulas, and the Euler–Maclaurin formula.The main part of the book focuses on Bernoulli numbers, including their construction, generating functions, closed formulas, and fundamental properties. It further discusses their connections with other special numbers and extends the discussion to second Bernoulli numbers and their generalizations, including Euler numbers, Stirling numbers, Bernoulli and Euler polynomials, and poly-Bernoulli numbers, highlighting the broader structure of this class of special numbers.