Higher-Order Lucas Sequences and Dickson Polynomials

In this paper we determine when there exists a matrix M in PGL2(F), and its form, such that L_k = D^M_k where D^M_k is a higher-order Dickson polynomial. We first examine the cases where M has projective orders 3, 4, and 6. For the order 3 case, we find that M has entries in, at worst, a quadratic extension of F. This is also true for the orders 4 and 6, but requires a restriction on the coefficients of h(x), the characteristic polynomial of L. In all cases, an explicit formula for M is given, and in the order 4 case the meaning of the extension is interpreted in terms of the Galois group of h. Lastly, we examine the case where F is finite, and offer a formula for M of order 5.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Campus Only Dissertation

Discipline thesis:degree_discipline

Mathematics

Year

2013

Author dc:creator

• Boone, Joshua Daniel

Contributors dc:contributor

• Fitzgerald, Robert

• Algebra
• Dickson
• Lucas
• Polynomials
• Sequences

Repository record dc:identifier

https://opensiuc.lib.siu.edu/dissertations/770

OAI identifier oai:identifier

oai:opensiuc.lib.siu.edu:dissertations-1773