## UNF Graduate Theses and Dissertations

2023

Spring

Master's Thesis

#### College

College of Arts and Sciences

#### Degree Name

Master of Science in Mathematical Sciences (MS)

#### Department

Mathematics & Statistics

#### NACO controlled Corporate Body

University of North Florida. Department of Mathematics and Statistics

Dr. Jae-Ho Lee

Dr. Ognjen Milatovic

Dr. Mei-Qin Zhan

#### Department Chair

Dr. Richard Patterson

#### College Dean

Kaveri Subrahmanyam

#### Abstract

The Lie algebra L = sl2(C) consists of the 2 × 2 complex matrices that have trace zero, together with the Lie bracket [y, z] = yz − zy. In this thesis we study a relationship between L and Krawtchouk polynomials. We consider a type of element in L said to be normalized semisimple. Let a, a^∗ be normalized semisimple elements that generate L. We show that a, a^∗ satisfy a pair of relations, called the Askey-Wilson relations. For a positive integer N, we consider an (N + 1)-dimensional irreducible L-module V consisting of the homogeneous polynomials in two variables that have total degree N. We define a certain nondegenerate symmetric bilinear form ⟨ , ⟩ on V . We display two bases for V , denoted {v_i} for i ≤ 1 ≤ N and {(v_i) ^ *} for i ≤ 1 ≤ N, each basis diagonalizes a and a^∗ , respectively. We show that each of these bases is orthogonal with respect to ⟨ , ⟩ and also show that ⟨vi , (v_j)^*⟩ = K_i(j; p, N), i, j = 0, 1, 2, . . . , N, where K_i(j; p, N) is the ith Krawtchouk polynomial with parameters N and p. Using these results we find some well-known facts about Krawtchouk polynomials including the three-term recurrence, the orthogonality, the difference equation, and the generating function.

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