Connection between representations of Heisenberg and su(1, 1) algebras

The expression of Hermite polynomials via Laguerre ones allows to connect the standard representations of the Heisenberg algebra with representations of generalized oscillator algebra, related to the Laguerre polynomials. We give the connection between the measures participating in the orthogonality relations for these polynomials and the connection between energy levels of related oscillators. Such considerations can be extended to a general case in which Hermite polynomials are replaced by polynomials orthogonal with respect to a symmetric measure.