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Group Theory and Special Functions Theory: a two way street

It is well known the role of symmetries in Science and, in particular, in the Physical Sciences. Almost all physical laws are consequence of symmetries. Take as an example the conservation laws of  Classical Mechanics and the the Noether's theorem. Another interesting example is related to Quantum Mechanics, where the rotation group SO(3) plays an essential role. In fact, the Representation theory of S0(3) (in general the representation theory of groups) is a very powerful tool for solving problems in Quantum Physics.

Interestingly when one works on spaces functions (as it is the case of Quantum Mechanics, where the space of square integrable functions is used) a very elegant connection appears between different concepts of the group representation theory and the special functions and polynomials orthogonal theory. In this talk we will briefly show this connection starting with some simple examples that allow us to conclude the talk extrapolating those results to the case of q-algebras and q-orthogonal polynomials on non-uniform lattices. In particular, we will show how this relationship can translate results of one theory to the other one and vice versa.
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