Some Orthogonal Combinations of Legendre Polynomials

The principle objective of this article is to introduce and investigate a type of orthogonal polynomials that are written as combinations of Legendre polynomials. This kind of polynomials can be viewed as generalized Jacobi polynomials (GJPs), since they are written in terms of Jacobi polynomials (JPs) with certain negative parameters. The analytic and inversion formulas of these polynomials are established. New derivative expressions for these polynomials are derived in detail in terms of their original polynomials. Other derivative expressions for these polynomials are found but in terms 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.