Asymptotics of Sobolev orthogonal polynomials for Hermite (1,1)-coherent pairs
The purpose of this article is to study asympotics of monic polynomials (SMOP), orthogonal with respect to a certain Sobolev inner product. There are many references in this paper and the paper starts with a historical introduction on generalized Laguerre functions and Hermite functions [\textit{Y. Liu} et al., ``Hermite-Sobolev orthogonal functions and spectral methods for second and fourth order problems on unbounded domains'', preprint]. However, in the paper, mostly Hermite polynomials are used. Several crucial formulas contain this Szegö Hermite polynomial in the denominator. In order to obtain an algebraic relation between the sequences \(\{Q_n(x)\}\) and \(\{\tilde{Q}_n(x)\}\), several lemmas are proved. Reviewer's remark: It is not an easy task to determine the validity of the given formulas at the beginning of the paper, since references are not accompanied by page numbers, mathematical definitions are not given and it is not clear if the authors know that there are two types of Hermite polynomials, one of them monic (i.e., the probabilistic Hermite polynomials). A few examples: Formula (7) for so-called monic Hermite polynomials is false. The reference to formula (8) is false. Formula (10) is not identical with 22.15.3 and 22.15.4 in reference [1]. Formula (9) is false since it is not identical with [14](23) and the variable \(b\) is not defined. A `shadow' to formula (23) is found in [p. 299](23). Accordingly, the variable \(b\) appears again in formulas (21) and (23). The statement between formulas (13) and (14) on so-called monic Laguerre polynomials is false. It would be advisable to tell what \(b\) and \(H_n(x)\) are in the similar formulas (48) and (51). Probably the factor \(\sqrt{\frac n2}\) in formulas (9), (23), (48) and (51) could have been avoided by choosing monic Hermite polynomials instead, which clearly shows the advantage of Appell polynomials. It is also not clear why the polynomials in (42) are monic. Finally, Mehler-Heine type formulas for scaled polynomials are found. Conclusion: it is not possible to `mend' this article unless Appell polynomials are used and/or proper polynomial definitions are given.
scientific article; zbMATH DE number 6918469
scientific article