Kwon, K. H.; Littlejohn, L. L. The orthogonality of the Laguerre polynomials \(\{ L_ n^{-k}(x)\}\) for positive integers \(k\). (English) Zbl 0831.33003 Ann. Numer. Math. 2, No. 1-4, 289-303 (1995). 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. Reviewer: R.Szwarc (Wrocław) Cited in 34 Documents MSC: 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) Keywords:weighted Sobolev inner product; Laguerre polynomials PDFBibTeX XMLCite \textit{K. H. Kwon} and \textit{L. L. Littlejohn}, Ann. Numer. Math. 2, No. 1--4, 289--303 (1995; Zbl 0831.33003)