Hermite Polynomials and their Applications Associated with Bernoulli and Euler numbers

Abstract

In this thesis, we consider Bernoulli polynomial, Euler polynomial, Bernoulli number and Euler number, the squares of Bernoulli polynomial, Euler polynomial. We also provide the concept of exponential function, explicit formulas for higher order derivatives of the generating functions of the Bernoulli polynomials and their square which can be associated with Hermite polynomial. Exponential function that the generating functions of the Bernoulli and their square satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials. This study focused on the Hermite polynomials and their applications associated with square of Bernoulli and Euler numbers. The concept of Hermite polynomials can be found in extensive range of subject areas. For this reason, the concept of Hermite polynomial associated with square of Bernoulli number and square of Euler number should be examined.