Certain Generalizations of Jacobi Polynomial and Their Properties

Jacobi polynomial $$P_{n}^{{\left( {\alpha ,\beta } \right)}}(x)$$ is a well-known orthogonal polynomial. In the present work, several new properties of generalized Jacobi polynomial $$P_{{n,\tau }}^{{\left( {\alpha ,\gamma ,\beta } \right)}}(x)$$ (Waghela D., Rao S.B. A note on sequence of functions associated with the generalized Jacobi polynomial, Researchers in Mathematics 31 (2), 1–18 (2023)) and its special case $$P_{n}^{{\left( {\alpha ,\gamma ,\beta } \right)}}(x)$$ , have been studied, which along with different representations of the said generalization includes crucial orthogonality property, generating function, results involving integral representation, differentiation of generalized Jacobi polynomial; also many well known transformations of this generalized polynomial have been obtained. Thus the main aim of the work is to study various basic properties of recently developed above generalized Jacobi polynomial $$P_{{n,\tau }}^{{\left( {\alpha ,\gamma ,\beta } \right)}}(x)$$ and its special case.