Laguerre Polynomials and singular differential operators
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This paper is concerned with the connections between the orthogonal polynomials and the differential operators generated by the Laguerre differential equation of Ten in the space L' (0, w ). However, for the left definite case, a suitably w determined resolvent function Q is used to define a bounded self-adjoint operator A, whose inverse is the required self-adjoint “differential” operator 5m in the space H 2M1 (0, 00). In both cases the spectra of Tat and Sm are shown to be discrete and the corresponding eigenvectors turn out to be the orthogonal polynomials of Laguerre. These results provide an alternative proof of the completeness of the Laguerre polynomials in the spaces L‘: (0,00) and ).