Some properties of the Generalized Jacobi polynomial

Abstract

An attempt is made to find recurrence relation and further properties of the generalized Jacobi polynomial $$$P_{n,\tau }^{\left( {\alpha ,\gamma ,\beta } \right)}(x)$$$ . Yet another generalization $$$P_{n,\tau ,\lambda }^{\left( {\alpha ,\gamma ,\beta } \right)}(x)$$$ of Jacobi polynomial $$$P_n^{\left( {\alpha ,\beta } \right)}(x)$$$ has been introduced and its properties have been studied. For $$$\lambda  = \tau$$$ we recover $$$P_{n,\tau }^{\left( {\alpha ,\gamma ,\beta } \right)}(x)$$$. The results so obtained may be useful in the theory of special functions, where Jacobi polynomials occur naturally.