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Laguerre polynomials, restriction principle, and holomorphic representations of SL\((2\mathbb R)\). (English) Zbl 0993.22012

For \(\alpha>0\), let \(H_\alpha\) be the Bergman space of holomorphic functions in the upper half-plane \(\mathcal H\) square-integrable with respect to the measure \((\text{Im} z)^{\alpha-1} dz\), where \(dz\) is the Lebesgue measure; the spaces \(H_\alpha\) can, in fact, be “analytically continued” to all \(\alpha>-1\). The Cayley transform induces an isomorphism \(c:H_\alpha(\mathcal D)\to H_\alpha\), \(cf(z):=f(\frac{z-i}{z+i})(z+i)^ {-\alpha-1}\), between \(H_\alpha\) and the similar Bergman space on the unit disc \(\mathcal D\) with respect to the measure const\(\cdot(1-|z|^2)^ {\alpha-1}\).
The authors consider the restriction map \(R\) on \(H_\alpha\) given by \(Rf(t):=f(it)\), \(t>0\). It turns out that \(R\) is a densely defined, closed, and injective operator from \(H_\alpha\) into \(L^2(\mathbb R^+,t^\alpha dt)\) with dense range; hence, the partial isometry component \(U\) in the polar decomposition \(R^*=U\sqrt{RR^*}\) is unitary. They identify \(U\) explicitly, and show that \(U^* c (z^n)=\text{const}\cdot e^{-t} L^\alpha_n(2t)\), where \(L^\alpha_n\) are the Laguerre polynomials. This fact can then be used to obtain various recurrence formulas for \(L^\alpha_n\).
The main idea underlying all these developments is the so-called “restriction principle” of B.Ørsted and the third author. For more information and other applications of this principle, see the papers of B. Ørsted and G. Zhang [Indiana Univ. Math. J. 43, 551-583 (1994; Zbl 0805.46053)] and G. Ólafsson and B. Ørsted [in: Lie theory and its applications in physics (Clausthal, 1995; Zbl 0916.22006)].

MSC:

22E46 Semisimple Lie groups and their representations
43A85 Harmonic analysis on homogeneous spaces
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
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