Introduction to special functions
In the first part of the course held in September - October 2024, we observed classical theory of Gauss hypergeometric function, including related topics on Euler integrals, Riemann zeta function, Riemann differential equation and Barnes integrals.
The second part starts from the survey of degenerations of Gauss hypergeometric function, including Bessel and Whittaker functions and their applications in mathematical physics. We investigate further the emergence of special functions in representation theory of classical groups as specific matrix elements. The next subject is the theory of basic hypergeometric series – the q -difference analog of the theory of hypergeometric function.
The multivariable generalizations originate from the theory of spherical functions on symmetric spaces. Here we touch the geometry of symmetric spaces, constructions , classification and duality between compact and noncompact cases. We touch the Heckman-Opdam development of the theory of special functions and finish, once have time left, with Calogero-Sutherland hyperbolic systems and Ruijsenaar’s relativistic generalization.
The second part starts from the survey of degenerations of Gauss hypergeometric function, including Bessel and Whittaker functions and their applications in mathematical physics. We investigate further the emergence of special functions in representation theory of classical groups as specific matrix elements. The next subject is the theory of basic hypergeometric series – the q -difference analog of the theory of hypergeometric function.
The multivariable generalizations originate from the theory of spherical functions on symmetric spaces. Here we touch the geometry of symmetric spaces, constructions , classification and duality between compact and noncompact cases. We touch the Heckman-Opdam development of the theory of special functions and finish, once have time left, with Calogero-Sutherland hyperbolic systems and Ruijsenaar’s relativistic generalization.
Lecturer
Date
3rd April, 2025 ~ -
Syllabus
Part II
1. Confluent hypergeometric functions. Bessel and Whitakker functions.
2. Special functions in representation theory and in mathematical physics
3. Basic hypergeometric theories
4. Symmetric spaces and spherical functions
5. Dunkl operators and Heckman-Opdam hypergeometric functions
6. Ruijsenaars hyperbolic function
1. Confluent hypergeometric functions. Bessel and Whitakker functions.
2. Special functions in representation theory and in mathematical physics
3. Basic hypergeometric theories
4. Symmetric spaces and spherical functions
5. Dunkl operators and Heckman-Opdam hypergeometric functions
6. Ruijsenaars hyperbolic function
Reference
1. Whittaker, E. T., and G. N. Watson. A Course of Modern Analysis, I, II. University press, 1920.
2. Andrews, George E., et al. Special functions. Vol. 71. Cambridge: Cambridge university press, 1999.
3. Gasper, George, and Mizan Rahman. Basic hypergeometric series. Vol. 96. Cambridge university press, 2011.
4.Vilenkin N. I. A. Special functions and the theory of group representations. American Mathematical Soc., 1978. – Т. 22.
5. Helgason, Sigurdur. Differential geometry and symmetric spaces. Vol. 341. American Mathematical Society, 2024.
6. Helgason, Sigurdur. Groups and geometric analysis: integral geometry, invariant differential operators, and spherical functions. Vol. 83. American Mathematical Society, 2022.
7. Ruijsenaars, S. N. M. "A Generalized Hypergeometric Function Satisfying Four Analytic Difference Equations of Askey--Wilson Type." Communications in mathematical physics 206, no. 3 (1999): 639-690.
2. Andrews, George E., et al. Special functions. Vol. 71. Cambridge: Cambridge university press, 1999.
3. Gasper, George, and Mizan Rahman. Basic hypergeometric series. Vol. 96. Cambridge university press, 2011.
4.Vilenkin N. I. A. Special functions and the theory of group representations. American Mathematical Soc., 1978. – Т. 22.
5. Helgason, Sigurdur. Differential geometry and symmetric spaces. Vol. 341. American Mathematical Society, 2024.
6. Helgason, Sigurdur. Groups and geometric analysis: integral geometry, invariant differential operators, and spherical functions. Vol. 83. American Mathematical Society, 2022.
7. Ruijsenaars, S. N. M. "A Generalized Hypergeometric Function Satisfying Four Analytic Difference Equations of Askey--Wilson Type." Communications in mathematical physics 206, no. 3 (1999): 639-690.
Video Public
Yes
Notes Public
Yes
Language
English