# Classical Orthogonal Polynomials (MATH484) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Classical Orthogonal Polynomials MATH484 Area Elective 3 0 0 3 6
Pre-requisite Course(s)
Math 262 or Math 276 or Consent of the instructor
Course Language English Elective Courses Bachelor’s Degree (First Cycle) Face To Face Lecture, Question and Answer, Team/Group. This course is intended primarily for the student of mathematics, physics or engineering who wishes to use the “orthogonal” polynomials associated with the names of Legendre, Hermite and Languerre. It aims at providing in a compact form most of the properties of these polynomials in the simplest possible way. The students who succeeded in this course; understand and apply the special functions understand the basic properties of the orthogonal polynomials know the basic notions of generating functions understand the Legendre, Hermite and Laguerre functions understand the Chebyshev and Gegenbauer polynomials Generating functions; orthogonal polynomials; Legendre polynomials; Hermite polynomials; Laguerre polynomials; Tchebicheff polynomials; Gegenbauer polynomials.

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Generating Functions: The generating function concept, Generating functions of the form G(2xt-t^2), Sets generated by exp(t)f(xt), The generating functions A(t)exp[-xt/(1-t)] pp. 129-137
2 Orthogonal Polynomials: Simple sets of polynomials, Orthogonality, An equivalent condition for orthogonality, Zeros of orthogonal polynomials pp. 147-150
3 Expansion of polynomials, The three-term recurrence relation, The Cristoffel-Darboux formula, Normalization; Bessel's inequality pp. 150-155
4 Legendre Polynomials: A generating function, Differential recurrence relation, the pure recurrence relation, Legendre's differential equation pp. 157-161
5 The Rodrigues formula, Hypergeometric forms of P_n(x), Special properties of P_n(x), Laplace's first integral form, Some bounds on P_n(x) pp. 165-181
6 Orthogonality, An expansion theorem, Expansion of analytic functions pp. 187-190
7 Midterm
8 Hermite Polynomials: Definition, Recurrence relations, The Rodrigues formula, Other generating functions pp. 191-196
9 Integrals, The Hermite polynomials as a 2_F_0, Differential equation, Orthogonality, Expansion of polynomials, More generating functions pp. 200-203
10 Laguerre Polynomials: The Laguerre polynomials, Generating functions, Recurrence relations, The Rodrigues formula pp. 204-213
11 Christffel-Darboux Formula, The differential equation, Orthogonality, Expansion of polynomials, Special properties, Other generating functions, The simple Laguerre polynomials pp.254-260
12 Chebyshev polynomials: A generating function relation, Recurrence relation, Some other representations, Differential equation, Orthogonality pp. 261-269
13 Gegenbauer Polynomials: A generating function relation, Recurrence relation, Some other representations s. 276-283
14 Differential equation, Orthogonality, Expansion of polynomials, Rodrigues formula p. 285, pp. 299-301
15 Review
16 Final Exam

### Sources

Course Book 1. Earl D. Rainville, Special Functions, MacMillan, New York, 1960. 2. Z. X. Wang, D. R. Guo, Special Functions, World Scientific, 1989 3. N. N. Lebedev, Special Functions and Their Applications, Prentice-Hall, 1965

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 5 10
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 50
Final Exam/Final Jury 1 40
Toplam 8 100
 Percentage of Semester Work 60 40 100

### Course Category

Core Courses X

### The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area
2 Can apply gained knowledge and problem solving abilities in inter-disciplinary research
3 Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary
4 Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study
5 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework
6 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility
7 Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation
8 To have an oral and written communication ability in at least one of the common foreign languages ("European Language Portfolio Global Scale", Level B2)
9 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge
10 Follows scientific, cultural, and ethical criteria in collecting, interpreting and announcing data in the research area and has the ability to teach.
11 Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications.

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Laboratory
Application
Special Course Internship
Field Work
Study Hours Out of Class 16 3 48
Presentation/Seminar Prepration
Project
Report
Homework Assignments 5 8 40
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 2 12 24
Prepration of Final Exams/Final Jury 1 18 18