Multiple orthogonal polynomials, irrationality and transcendence. (English) Zbl 0952.42014
Berndt, Bruce C. (ed.) et al., Continued fractions: from analytic number theory to constructive approximation. A volume in honor of L. J. Lange. Proceedings of the conference, University of Missouri–Columbia, Columbia, MO, USA, May 20-23, 1998. Providence, RI: American Mathematical Society. Contemp. Math. 236, 325-342 (1999).
This survey is a quick and nice introduction to some basic properties of multiple orthogonal polynomials related to the Hermite-Padé approximation of a system of functions. Some examples of such polynomials are given explicitly. Finally, the author shows how this construction can be used in the study of the irrationality and transcendence of some constants. In particular, the proof of the classical Apery’s result on irrationality of \(\zeta(3)\) is produced.
For the entire collection see [Zbl 0922.00015].
For the entire collection see [Zbl 0922.00015].
Reviewer: Andrei Martínez Finkelshtein (Almeria)
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 11J72 | Irrationality; linear independence over a field |
| 11J81 | Transcendence (general theory) |
| 41A28 | Simultaneous approximation |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
