Special Functions and Their Symmetries
Postgraduate Course in Applied Analysis

Lecturers:	Vadim Kuznetsov (Room 9.18h) and Vladimir Kisil (Room 8.18l)
Web page:	http://maths.leeds.ac.uk/~kisilv/courses/special.html

Description: This course is suitable for postgraduate students in both applied and pure mathematics. It presents fundamentals of special functions theory and its applications in partial differential equations of mathematical physics. The course covers topics in harmonic, classical and functional analysis, and combinatorics. It consists of the two parts: the first part gives the classic analytical approach and the second links the theory with groups of symmetries. The main objective of the course is to learn how:

• to determine types of PDEs which may be solved by application of special functions.

• to analyze properties of special functions by their integral representations and symmetries.

• to classify differential equations by their singularities; to obtain properties of solutions of PDE by their symmetries.

Part I Algebraic and analytic methods, The PDF version is recommended. Lecturer: Vadim Kuznetsov
1. Gamma and Beta functions.

2. Hypergeometric series.

3. Orthogonal polynomials.

4. Separation of variables and special functions.

5. Integrable systems and special functions.


Part II Algebraic and symmetry methods. The PDF version is recommended. Lecturer: Vladimir Kisil
1. Groups and Homogeneous Spaces .

2. Representation of Groups and Their Decompositions .

3. Real Harmonic Analysis .

4. Harmonic Analysis on Spheres .

5. Hermit Polynomials, Heisenberg Group, and Segal-Bargmann Spaces .

References

[1]

George E. Andrews, Richard Askey, and Ranjan Roy. Special functions. Cambridge University Press, Cambridge, 1999.

[2]

Willard Miller, Jr. Lie Theory and Special Functions. Academic Press, New York, 1968. Mathematics in Science and Engineering, Vol. 43.

[3]

N. Ja. Vilenkin. Special Functions and the Theory of Group Representations. American Mathematical Society, Providence, R. I., 1968. Translated from the Russian by V. N. Singh. Translations of Mathematical Monographs, Vol. 22.

• If you experienced a problem reading the course HTML pages then an easy solution could be found in Help on Browser Problems or (even better) use the PDF files (see bellow).

• PDF version is provided online and preferable in many respects. Acrobat Reader for your computer platform could be obtained free of charge.

• It is optimal to use AcroRead plug-in for your Web browser. If you use AcroRead as a stand-alone application, then for hyperlinks between different PDF files you need to download all PDF files to the same directory on your computer.

• It is easier to read PDF links if you set the following option in your copy of AcroReader:
  File | Preferences | General | Default Page Layout: Continuous

• These pages are created using free software from L^AT_EX  source, there are also few advises on Web publishing.

• Send comments and remarks by email kisilv@maths.leeds.ac.uk.