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Some functions that generalize the Krall-Laguerre polynomials. (English) Zbl 0926.33007
The paper deals with at most \(\alpha\) successive Darboux transformations of \(L(\alpha)\), where \(L(\alpha)\) denotes the (semi-infinite) tridiagonal matrix associated with the three-term recursion relation satisfied by the Laguerre polynomials with weight function \(\frac{1}{\Gamma(\alpha+1)} z^\alpha e^{-z}\), \(\alpha>-1\), on the interval \([0,\infty[\). It is shown that the resulting (bi-infinite) tridiagonal matrix \(\widetilde{L} (\alpha)\) is bispectral, i.e. the corresponding function, called Krall-Laguerre functions, are orthogonal polynomials on \([0,\infty[\) with respect to some weight distribution \(w(k,\alpha)\) with \(1\leq k\leq\alpha\). Furthermore, as a consequence of the rational character of the Darboux factorization, these polynomials are eigenfunctions of a (finite order) differential operator. The concept is enlarged to the two-parameters bi-infinite extension \(L(\alpha, \varepsilon)\) of the matrix \(L(\alpha)\), where \(L(\alpha,0)= L(\alpha)\).

MSC:
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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