Special Functions and Their Symmetries \
Postgraduate Course in Applied Analysis
Special Functions and Their Symmetries
Postgraduate Course in Applied Analysis
University of Leeds,
School of Mathematics
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.
To print lecture notes use PostScript
files of the first and the
second parts. See Technical notes on viewing
online materials.
Contents:
Part I Algebraic and analytic methods,
The PDF version is recommended.
Lecturer: Vadim Kuznetsov
 Gamma and Beta functions.
 Hypergeometric series.
 Orthogonal polynomials.
 Separation of variables and special functions.
 Integrable systems and special functions.
Part II Algebraic and symmetry methods.
The PDF version is recommended.
Lecturer: Vladimir Kisil
 Groups and
Homogeneous Spaces .
 Representation of
Groups and Their Decompositions .
 Real Harmonic Analysis .
 Harmonic Analysis on Spheres .
 Hermit Polynomials,
Heisenberg Group, and SegalBargmann Spaces .
Prerequisites: Basic real and complex analysis,
rudimentary algebra.
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.
Tecnical Notes:
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(see bellow).
 PDF version is provided online and preferable in many
respects.
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On 22 May 2003, 14:49.