An asymptotic result for Laguerre-Sobolev orthogonal polynomials
Consider the Sobolev inner product \[ (f,g)_S= \int^\infty_0 f(x)g(x) x^\alpha e^{-x}dx +\lambda \int^\infty_0 f'(x)g'(x) x^\alpha e^{-x} dx, \] where \(\alpha>-1\) and \(\lambda >0\). Let \(\{S_n(x)\}^\infty_{n=0}\) denote the sequence of polynomials which are orthogonal with respect to this inner product, normalized by the condition that \(S_n(x)\) and the Laguerre polynomial \(L_n^{(\alpha)} (x)\) have the same leading coefficient. These polynomials are called the (continuous) Laguerre-Sobolev orthogonal polynomials. The algebraic properties of these polynomials were intensively studied by three of the authors in [\textit{F. Marcellán}, \textit{T. E. Pérez} and \textit{M. A. Piñar}, J. Comput. Appl. Math. 71, 245-265 (1996; Zbl 0855.33015)]. The authors now study the asymptotic behaviour of these polynomials. The main result of the paper is \[ S_n(x)= {2\over \sqrt {\lambda^2+4 \lambda}- \lambda} L_n^{(\alpha-1)} (x) \bigl\{1+O (n^{-1/2}) \bigr\}, \quad x\in C\setminus [0,\infty), \quad n\to\infty. \] The asymptotic behaviour of the classical Laguerre polynomial \(L_n^{(\alpha)} (x)\) for arbitrary real \(\alpha\) is well-known (due to Perron) and can be found, for instance, in the book [\textit{G. Szegő}, Orthogonal polynomials (1939; Zbl 0023.21505)].
scientific article; zbMATH DE number 1115953