Some remarks on a paper by L. Carlitz. (English) Zbl 1115.33007
In this paper the author considers the following problem (which is a particular case of the problem by [I. M. Sheffer, “Some properties of polynomial sets of type zero.” Duke Math. J. 5, 590-622 (1939; Zbl 0022.01502)]: For which functions \(f(z)\), will the polynomials \(\Psi_n(x)\), generated by
\[
exp[xf(z)]=\sum_{n=0}^\infty \Psi_n(x) \frac{z^n}{n!},
\]
form an orthogonal set? For solving this problem he first shows that the solution requires the function \(f(z)\) to be of the \(\arctan\) type. He shows that the solution of the problem are polynomials that are not orthogonal with respect to the standard inner-products, but with respect to some new inner-products involving differential or difference operators: the so-called Sobolev orthogonal polynomials. In particular, he shows that the obtaining polynomials are limiting cases of the Laguerre, Meixner–Pollaczek and Meixner polynomials.
Reviewer: Renato Alvarez-Nodarse (Sevilla)
MSC:
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 11B83 | Special sequences and polynomials |
Citations:
Zbl 0022.01502References:
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