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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
##### Keywords:
Krall-Laguerre polynomials; Darboux transformations
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##### References:
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