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The orthogonality of the Laguerre polynomials \(\{ L_ n^{-k}(x)\}\) for positive integers \(k\). (English) Zbl 0831.33003

The generalized Laguerre polynomials \(L_n^\alpha (x)\) are orthogonal with respect to the weight \(x^\alpha e^{-x}\) on the positive half axis for \(\alpha> -1\). The standard orthogonality fails for \(\alpha\leq -1\), although the polynomials \(L_n^\alpha (x)\) are well defined for all real values of the parameter \(\alpha\). The authors show that for \(\alpha= -k\), where \(k= \mathbb{N}\), the polynomials \(L_n^{-k} (x)\) are orthogonal with respect to a so called Sobolev positive definite inner product defined on the set of all polynomials and involving the first \(k-1\) derivatives. This inner product is given explicitly.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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