Novel Formulae of Certain Generalized Jacobi Polynomials

The main goal of this article is to investigate theoretically a kind of orthogonal polynomials, namely, generalized Jacobi polynomials (GJPs). These polynomials can be expressed as certain combinations of Legendre polynomials. Some basic formulas of these polynomials such as the power form representation and inversion formula of these polynomials are first introduced, and after that, some interesting formulas concerned with these polynomials are established. The formula of the derivatives of the moments of these polynomials is developed. As special cases of this formula, the moment and high-order derivative formulas of the GJPs are deduced. New expressions for the high-order derivatives of the GJPs, but in terms of different symmetric and non-symmetric polynomials, are also established. These expressions lead to some interesting connection formulas between the GJPs and some various polynomials.