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Orthogonal polynomials associated with \(\exp (-x^ 4)\). (English) Zbl 0561.42013

Approximation theory, 2nd Conf., Edmonton/Albert 1982, CMS Conf. Proc. 3, 263-285 (1983).
[For the entire collection see Zbl 0537.00008.]
The paper is a report of the work done by the author and several other workers on the polynomials, orthogonal on the whole real line with respect to the weight function exp(-x\({}^ 4)\). According to the author, the main goal is to find out estimates and asymptotic expansions of the above polynomials. For this aim, the recursive coefficients of a recurrence relation the above polynomials have been investigated. The differential equations satisfied by them have been obtained. For the cases when the weight function is \(| x|^ b\exp (-| x|^ a),\) b, a, reals \(>0\), the lack of results is pointed out. In this connection, the papers of the reviewer and M. Kanchan Probha [Int. J. Math. Math. Sci. 6, 171-180 (1983; Zbl 0521.33008)] and papers cited in the bibliography of this paper may be seen.
Reviewer: M.Dutta

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
26D15 Inequalities for sums, series and integrals
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)