New Results of Generalized Jacobsthal–Lucas Polynomials with Some Integral Applications

We study a generalized class of Jacobsthal–Lucas polynomials that depends on two parameters. First, we introduce essential formulas for these polynomials, involving their series representation, inverse formula, and moment formula. These formulas allow us to investigate this generalized class of polynomials further and to develop novel formulations. The essential standard linearization problem of these polynomials is solved, and the linearization coefficients are given in simple forms. In addition, some mixed linearization formulas with other classes of polynomials are presented. The derivative formulas of these polynomials, expressed as combinations of different polynomials, are given. By employing symbolic algebra methods—most notably Zeilberger’s algorithm and other well-known identities from the literature—many hypergeometric functions appearing in the coefficients can be reduced, resulting in simpler expressions. In addition, some definite integrals are evaluated using the newly introduced formulas.