Novel Identities and Representations for Euler Polynomials of the Second Kind

Abstract

We investigate the Euler polynomials of the second kind (EPs2) and develop a collection of structural identities for this modified Euler family. The analysis begins with an explicit expansion and a compact inverse relation, followed by connections with the classical Euler basis. We then obtain formulas governing differentiated moments and products, together with linearization relations. Higher-order derivatives are expanded in symmetric and non-symmetric bases, including Bernoulli, Laguerre, Jacobi, and generalized Fibonacci and Lucas polynomials. Converse expansions are established as well. We then evaluate several definite integrals to illustrate how the derived identities can be applied.