### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 068, 11 pages      arXiv:1104.3773      https://doi.org/10.3842/SIGMA.2011.068
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

### Recurrence Coefficients of a New Generalization of the Meixner Polynomials

Galina Filipuk a and Walter Van Assche b
a) Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, Warsaw, 02-097, Poland
b) Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium

Received April 18, 2011, in final form July 07, 2011; Published online July 13, 2011

Abstract
We investigate new generalizations of the Meixner polynomials on the lattice N, on the shifted lattice N+1−β and on the bi-lattice N∪(N+1−β). We show that the coefficients of the three-term recurrence relation for the orthogonal polynomials are related to the solutions of the fifth Painlevé equation PV. Initial conditions for different lattices can be transformed to the classical solutions of PV with special values of the parameters. We also study one property of the Bäcklund transformation of PV.

Key words: Painlevé equations; Bäcklund transformations; classical solutions; orthogonal polynomials; recurrence coefficients.

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References

1. Abramowitz M., Stegun I., Handbook of mathematical functions, Dover Publications, New York, 1965.
2. Boelen L., Filipuk G., Van Assche W., Recurrence coefficients of generalized Meixner polynomials and Painlevé equations, J. Phys. A: Math. Theor. 44 (2011), 035202, 19 pages.
3. Chihara T.S., An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York - London - Paris, 1978.
4. Filipuk G., Van Assche W., Zhang L., On the recurrence coefficients of semiclassical Laguerre polynomials, arXiv:1105.5229.
5. Filipuk G., Van Assche W., Recurrence coefficients of the generalized Charlier polynomials and the fifth Painlevé equation, arXiv:1106.2959.
6. Gromak V., Filipuk G., On functional relations between solutions of the fifth Painlevé equation, Differ. Equ. 37 (2001), 614-620.
7. Gromak V., Filipuk G., The Bäcklund transformations of the fifth Painlevé equation and their applications, Math. Model. Anal. 6 (2001), 221-230.
8. Gromak V., Filipuk G., Bäcklund transformations of the fifth Painlevé equation and their applications, in Proceedings of the Summer School "Complex Differential and Functional Equations" (Mekrijärvi, 2000), Univ. Joensuu Dept. Math. Rep. Ser., Vol. 5, Univ. Joensuu, Joensuu, 2003, 9-20.
9. Gromak V.I., Laine I., Shimomura S., Painlevé differential equations in the complex plane, de Gruyter Studies in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin, 2002.
10. Magnus A.P., Painlevé type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials, J. Comput. Appl. Math. 57 (1995), 215-237, math.CA/9307218.
11. Noumi M., Painlevé equations through symmetry, Translations of Mathematical Monographs, Vol. 223, American Mathematical Society, Providence, RI, 2004.
12. Smet C., Van Assche W., Orthogonal polynomials on a bi-lattice, Constr. Approx., to appear, arXiv:1101.1817.