Coherent Orthogonal Polynomials

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.