  We discuss as a fundamental characteristic of orthogonal polynomials like the
existence of a Lie algebra behind them, can be added to their other relevant
aspects. At the basis of the complete framework for orthogonal polynomials we
put thus --in addition to differential equations, recurrence relations, Hilbert
spaces and square integrable functions-- Lie algebra theory.
  We start here from the square integrable functions on the open connected
subset of the real line whose bases are related to orthogonal polynomials. All
these one-dimensional continuous spaces allow, besides the standard uncountable
basis ${|x>}$, for an alternative countable basis ${|n>}$. The matrix elements
that relate these two bases are essentially the orthogonal polynomials: Hermite
polynomials for the line and Laguerre and Legendre polynomials for the
half-line and the line interval, respectively.
  Differential recurrence relations of orthogonal polynomials allow us to
realize that they determine a unitary representation of a non-compact Lie
algebra, whose second order Casimir ${\cal C}$ gives rise to the second order
differential equation that defines the corresponding family of orthogonal
polynomials. Thus, the Weyl-Heisenberg algebra $h(1)$ with ${\cal C}=0$ for
Hermite polynomials and $su(1,1)$ with ${\cal C}=-1/4$ for Laguerre and
Legendre polynomials are obtained.
  Starting from the orthogonal polynomials the Lie algebra is extended both to
the whole space of the ${\cal L}^2$ functions and to the corresponding
Universal Enveloping Algebra and transformation group. Generalized coherent
states from each vector in the space ${\cal L}^2$ and, in particular,
generalized coherent polynomials are thus obtained.
