Identities for the Classical Polynomials Through Sums of Liouville Type

Polynomials defined recursively over the integers such as Dickson polynomials, Chebychev polynomials, Fibonacci polynomials, Lucas polynomials, Bernoulli polynomials, Euler polynomials, and many others have been extensively studied in the past. Most of these polynomials have some type of relationship between them and share a large number of interesting properties. They have been also found to be topics of interest in many different areas of pure and applied sciences. Most recently, some of these families of polynomials have been found to be useful in cryptography and related topics, which keep making them a very interesting area of research for many people in this era of communication. In this paper, we use a sum of Liouiville type to prove new properties concerning many of these families of polynomials. The following notations and definitions will be adopted throughout this paper. The sets of positive integers, nonnegative integers, integers, real numbers, and complex numbers are respectively denoted by N, N0, Z, R, and C. The sets of even positive integers and the set of odd positive integers will be written 2N and 2N − 1 respectively and the sets of even integers and odd integers will be written 2Z and 2Z − 1 respectively. The Euler phi