Exact Results in Quantum Theory

Properties of hypergeometric type equations become quite transparent if they are derived from appropriate 2nd order partial differential equations with constant coefficients. In particular, symmetries of the hypergeometric and Gegenbauer equation follow from conformal symmetries of the 4- and 3-dimensional Laplace equation. The symmetries of the confluent and Hermite equation follow from the so-called Schroedinger symmetries of the heat equation in 2 and 1 dimension. Finally, the properties of the Bessel equation follow from the Helmholtz equation in 2 dimensions. The lectures are based on [1, 2]. References:[1] Jan Derezinski, Hypergeometric type functions and their symmetries, Annales Henri Poincare 15 (2014), 1569– 1653. [2] Jan Derezinski, Przemysław Majewski, From conformal group to symmetries of hypergeometric type equations, SIGMA 12 (2016).