Bernoulli and Hermite Polynomials Associated with Jacobi Polynomials

Abstract

In 2017 N.U.Khan, T.Usman∗ and M.Aman [21] introduced the generating function for a Legendre based poly-Bernoulli polynomials and give some identities of these polynomials related to the stirling numbers of the second kind by making use of the generating function method and some functional equations. They conduct a further investigation in order to obtain some implicit summation formula for Legendre based poly-Bernoulli number and polynomials. In this thesis we study the relationship between Bernoulli and Hermite polynomial associated with Jacobi polynomials. The main purpose of this thesis is to investigate some identities of Bernoulli and Hermite polynomials associated with Jacobi polynomial arising from the orthogonality of Jacobi polynomials in the inner product space Pn. Finally the inner product space Pn of orthogonal Jacobi polynomial is an effective and used as a definitive modeling method to discuss relationship between Bernoulli and Hermite polynomials associated with Jacobi polynomials.