Euler Polynomials are a sequence of polynomials that arise in number theory and are closely related to the Bernoulli numbers. They are defined using a generating function and can be expressed in terms of binomial coefficients. These polynomials play a significant role in various mathematical fields, including combinatorics and numerical analysis. The n -th Euler polynomial, denoted as E_n(x) , can be evaluated at specific points, such as E_n(0) and E_n(1) , which yield interesting results related to sums of powers. Euler polynomials are also used in approximating functions and in the study of special functions.