Review on the Beauty and Interplay Chebyshev Polynomials: Some Properties and Identities of the First Four Kinds

Abstract

This paper present a sysmatic treatment of the four classical kinds of Chebyshev polynomials within a common analytic form. Although the first two kinds are widely recognised and routinly applied in approximation theory and numericall analysis, the third and fourth kinds have received comparatively less attention, despite their distintive properties. We consolidate and extends known results, derive some identities, and present new generating functions, both ordinary and exponential that reveal deeper connection among these families. Additionally, we provide systematic derivations of monomial expansion, and explicit formulas for derivatives. A unified framework for the interconnecting identities among all four kinds is established, facilitating their application in spectral methods and numerical analysis.