### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 108, 69 pages      arXiv:1505.02271      https://doi.org/10.3842/SIGMA.2016.108

### From Conformal Group to Symmetries of Hypergeometric Type Equations

Jan Dereziński a and Przemysław Majewski ab
a) Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, Poland
b) Bureau of Air Defence and Anti-missile Defence Systems, PIT-RADWAR S.A., Poligonowa 30, 04-025 Warszawa, Poland

Received February 24, 2016, in final form October 20, 2016; Published online November 05, 2016

Abstract
We show that properties of hypergeometric type equations become transparent if they are derived from appropriate 2nd order partial differential equations with constant coefficients. In particular, we deduce the symmetries of the hypergeometric and Gegenbauer equation from conformal symmetries of the 4- and 3-dimensional Laplace equation. We also derive the symmetries of the confluent and Hermite equation from the so-called Schrödinger symmetries of the heat equation in 2 and 1 dimension. Finally, we also describe how properties of the ${}_0F_1$ equation follow from the Helmholtz equation in 2 dimensions.

Key words: Laplace equation; hypergeometric equation; confluent equation; Kummer's table; Lie algebra; conformal group.

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