Some properties of the Generalized Jacobi polynomial

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.