Some Orthogonal Combinations of Legendre Polynomials

The main purpose of this study is to introduce and study certain orthogonal polynomials (OPs) that are written as combinations of Legendre polynomials. These polynomials can be viewed as generalized Jacobi polynomials (GJPs) since they are Jacobi polynomials (JPs) of certain negative parameters. The analytic and inversion formulas of the GJPs are established. New expressions of the derivatives of these polynomials are derived in detail as combinations of their original ones. Other derivative expressions for these polynomials are found, but as combinations of some orthogonal and non-orthogonal polynomials. Some product formulas with some other polynomials are also obtained. Certain definite and weighted definite integrals are obtained using the newly introduced connection and product formulas.