#### Graduation Year

2006

#### Document Type

Dissertation

#### Degree

Ph.D.

#### Degree Granting Department

Mathematics and Statistics

#### Major Professor

Mourad E. H. Ismail, Ph.D.

#### Keywords

Continuous q-Jacobi polynomials, Lowering operator, Generating function, Weight function, Rodrigues formula, Discriminant

#### Abstract

Two problems related to orthogonal polynomials and special functions are considered. For q greater than 1 it is known that continuous q-Jacobi polynomials are orthogonal on the imaginary axis. The first problem is to find proper normalization to form a system of polynomials that are orthogonal on the real line. By introducing a degree reducing operator and a scalar product one can show that the normalized continuous q-Jacobi polynomials satisfies an eigenvalue equation. This implies orthogonality of the normalized continuous q-Jacobi polynomials. As a byproduct, different results related to the normalized system of polynomials, such as its closed form,three-term recurrence relation, eigenvalue equation, Rodrigues formula and generating function will be computed. A discriminant related to the normalized system is also obtained. The second problem is related to recent results of Dilcher and Stolarky on resultants of Chebyshev polynomials. They used algebraic methods to evaluate the resultant of two combinations of Chebyshev polynomials of the second kind. This work provides an alternative method of computing the same resultant and also enables one to compute resultants of more general combinations of Chebyshev polynomials of the second kind. Resultants related to combinations of Chebyshev polynomials of the first kind are also considered.

#### Scholar Commons Citation

Gishe, Jemal Emina, "A finite family of q-orthogonal polynomials and resultants of Chebyshev polynomials" (2006). *Graduate Theses and Dissertations.*

https://scholarcommons.usf.edu/etd/2533