Fachbereich 10 Institut für Mathematik
Wintersemester 2017/18
Prof. Dr. Wolfram Koepf

Computeralgebra
und
orthogonale Polynome

Veranstaltung SWS Tag Zeit Ort/Raum Dozent/ Übungsleiter
Beginn
Vorlesung 2
Mittwoch

11-13

HPS / R. 2404
Prof. Dr. Wolfram Koepf 25.10.17
Vorlesung
2
Donnerstag
9-11
HPS / R. 1409
Prof. Dr. Wolfram Koepf
19.10.17
Übung 
2
Freitag
9-11
HPS / R. 2421
Dr. Daniel Tcheutia
20.10.17


Geignet für:
Bachelor, Master Mathematik, Lehramt Gymnasien, Master Informatik


Orthogonal polynomials are very important mathematical structures used in applications, in particular in Physics and the numerical analysis of ordinary and partial differential equations. In this lecture the most important properties of orthogonal polynomials are studied. Then the so-called classical orthogonal polynomials, that are named after Hermite, Laguerre, Bessel, Jacobi, Gegenbauer, Chebyshev and Legendre, are classified. Next we classify the discrete classical orthogonal polynomials named after Charlier, Meixner, Krawchouk and Hahn. In both classical cases hypergeometric representations pray a prominent role. In the whole lecture algorithmic methods are used which play a natural role through the use of computer algebra.    


Topics of the lecture:

  1. The Gamma Function
  2. Hypergeometric Functions
  3. Hypergeometric Database
  4. Holonomic Recurrence Equations
  5. Systems of Orthogonal Functions and Polynomials and their Properties
  6. Connection Coefficients
  7. Rodrigues Representations and Generating Functions
  8. Efficient computation of Orthogonal Polynomials
  9. The Askey-Wilson Scheme

              






Literatur
[1] Koepf, W.: Hypergeometric Summation. Vieweg, Braunschweig-Wiesbaden, 1998.
[2] Koepf, W. and Schmersau, D.: Representations of orthogonal polynomials. J. Comput. Appl. Math 90, 1998, 57-95
[3] Chihara, T. S.: An Introduction to Orthogonal Polynomials. Gordon and Breach Publ., New York, 1978.
[4] Tricomi, F. G.: Vorlesungen über Orthogonalreihen. Grundlehren der Mathematischen Wissenschaften 76, Springer, Berlin-Göttigen-Heidelberg, 1955.
Bemerkung
The lecture will be in English language. A German language manuscript of the lecture will be available. The two references by the lecturer can also be used as manuscript.
Voraussetzung
Computeralgebra I bzw. Kenntnisse über Mathematica und Maple
Leistungsnachweis
Bearbeitung von 50 % der Übungsaufgaben sowie mündliche oder schriftliche Prüfung