Appears in collection : Analytical properties and applications of orthogonal polynomials and special functions

This course is dedicated to introducing the properties and application of orthogonal polynomials and special functions. At first, we present some families of hypergeometric orthogonal polynomials belonging to the Askey scheme. We list the most important properties of the polynomials such as a representation as hypergeometric function, recurrence relations, orthogonality relations, the secondorder differential equation, generating functions, the Laplace transform and a Rodrigues-type formula. Next, we derive the properties of the Eta functions. To show the advantages of the Eta functions in the computational method, we develop a new numerical method to solve the differential equations based on the Eta functions. We introduce the operational matrices of integration and derivative for the Eta functions to develop the new numerical method. Finally, we derive the error bounds of the numerical method and demonstrate the precision of the new numerical technique by solving some numerical tests.