SOME CLASSES OF GENERALIZED CYCLOTOMIC POLYNOMIALS

Abstract

For a positive integer n the nth cyclotomic polynomial can be written as Φn(x)=∏︂(x−e^{2pi k/n} ). k∈(Z/nZ)^{×} When n = p is an odd prime, the nth cyclotomic polynomial has the special form

Φp(x)=∑︂x^{k} =x^{p−1} +x^{p−2} +...+x+1.

These two representations of the cyclotomic polynomials highlight the roots of Φn(x) and the coefficients of Φn(x), respectively. Continuing with the work of Kwon, J. Lee, and K. Lee and Harrington we investigate the generalization of the cyclotomic polynomials in two distinct ways; one affecting the roots of Φn(x) and the other affecting the coefficients of Φn(x).

In the final chapter of the thesis we discuss congruences for particular binomial sums and use those congruences to prove results concerning two special cases of Jacobi polynomials, the Chebyshev polynomials and the Legendre polynomials.