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Clebsch-Gordan-type linearisation relations for the products of Laguerre polynomials and hydrogen-like functions. (English) Zbl 0582.33008
Two series of Clebsch-Gordan type are derived for the most general product of the Laguerre polynomials, $$L_{n_ 1}^{\alpha_ 1}(u_ 1x)L_{n_ 2}^{\alpha_ 2}(u_ 2x)$$, which differ in orders, n, weights, $$\alpha$$, and scaling multipliers, u. The general form and particular cases of coefficients in the expansion of the polynomial $$x^ kL_{n_ 1}^{\alpha_ 1}(u_ 1x)...L_{n_ N}^{\alpha_ N}(u_ Nx)$$ in terms of the Laguerre polynomials are established. The applications to hydrogen-like functions and Morse oscillators are indicated. Connection with an earlier Carlitz expansion, the technical links with the hyperspherical harmonics formalism and different approaches to the important Koornwinder’s positivity theorems are discussed briefly.

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 33C05 Classical hypergeometric functions, $${}_2F_1$$
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