## A new approach to the theory of classical hypergeometric polynomials

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- by José Manuel Marco and Javier Parcet PDF
- Trans. Amer. Math. Soc.
**358**(2006), 183-214 Request permission

## Abstract:

In this paper we present a unified approach to the spectral analysis of a hypergeometric type operator whose eigenfunctions include the classical orthogonal polynomials. We write the eigenfunctions of this operator by means of a new Taylor formula for operators of Askey-Wilson type. This gives rise to some expressions for the eigenfunctions, which are unknown in such a general setting. Our methods also give a general Rodrigues formula from which several well-known formulas of Rodrigues-type can be obtained directly. Moreover, other new Rodrigues-type formulas come out when seeking for regular solutions of the associated functional equations. The main difference here is that, in contrast with the formulas appearing in the literature, we get non-ramified solutions which are useful for applications in combinatorics. Another fact, that becomes clear in this paper, is the role played by the theory of elliptic functions in the connection between ramified and non-ramified solutions.## References

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## Additional Information

**José Manuel Marco**- Affiliation: Department of Mathematics, Universidad Autónoma de Madrid, Madrid 28049, Spain
**Javier Parcet**- Affiliation: Department of Mathematics, Universidad Autónoma de Madrid, Madrid 28049, Spain
- Email: javier.parcet@uam.es
- Received by editor(s): July 17, 2003
- Received by editor(s) in revised form: January 8, 2004
- Published electronically: December 28, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**358**(2006), 183-214 - MSC (2000): Primary 33D15, 33D45
- DOI: https://doi.org/10.1090/S0002-9947-04-03620-7
- MathSciNet review: 2171229