On Cauchy Numbers and Their Polynomials

Abstract

Cauchy numbers are a special sequence of numbers used in mathematics, especially in analysis and combinatorics. They are defined using a generating function related to the natural logarithm. These numbers appear in formulas for approximations, finite differences, and expansions of functions. They are also linked to other well-known number sequences, such as Bernoulli and Stirling numbers. Because of these connections, Cauchy numbers are useful in studying relationships between discrete and continuous mathematics. Their properties make them helpful in areas like numerical methods and theoretical research. This thesis generalizes these results by introducing p-Cauchy numbers and Cauchy polynomials of both kinds, establishing their integral representations, hypergeometric generating series, recurrence relations, as well as their connections with r-Stirling numbers and Bernoulli polynomials. The study highlights the Appell-type nature of these polynomials and opens perspectives toward q-analogues and multivariate generalizations