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

SIGMA 11 (2015), 039, 12 pages      arXiv:1501.03880      https://doi.org/10.3842/SIGMA.2015.039
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

### The Combinatorics of Associated Laguerre Polynomials

Jang Soo Kim a and Dennis Stanton b
a) Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea
b) School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

Received January 30, 2015, in final form May 06, 2015; Published online May 11, 2015

Abstract
The explicit double sum for the associated Laguerre polynomials is derived combinatorially. The moments are described using certain statistics on permutations and permutation tableaux. Another derivation of the double sum is provided using only the moment generating function.

Key words: associated Laguerre polynomial; moment of orthogonal polynomials, permutation tableau.

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References

1. Chihara T.S., An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York - London - Paris, 1978.
2. Corteel S., Josuat-Vergès M., Personal communication.
3. Corteel S., Kim J.S., Combinatorics on permutation tableaux of type A and type B, European J. Combin. 32 (2011), 563-579, arXiv:1006.3812.
4. Corteel S., Nadeau P., Bijections for permutation tableaux, European J. Combin. 30 (2009), 295-310.
5. Corteel S., Williams L.K., Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials, Duke Math. J. 159 (2011), 385-415, arXiv:0910.1858.
6. Drake D., The combinatorics of associated Hermite polynomials, European J. Combin. 30 (2009), 1005-1021, arXiv:0709.0987.
7. Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2005.
8. Ismail M.E.H., Rahman M., The associated Askey-Wilson polynomials, Trans. Amer. Math. Soc. 328 (1991), 201-237.
9. Jones W.B., Thron W.J., Continued fractions, Encyclopedia of Mathematics and its Applications, Vol. 11, Addison-Wesley Publishing Co., Reading, Mass., 1980.
10. Nadeau P., The structure of alternative tableaux, J. Combin. Theory Ser. A 118 (2011), 1638-1660.
11. Postnikov A., Total positivity, Grassmannians, and networks, math.CO/0609764.
12. Simion R., Stanton D., Specializations of generalized Laguerre polynomials, SIAM J. Math. Anal. 25 (1994), 712-719, math.CA/9307219.
13. Simion R., Stanton D., Octabasic Laguerre polynomials and permutation statistics, J. Comput. Appl. Math. 68 (1996), 297-329.
14. Stanley R.P., Enumerative combinatorics, Vol. 1, Cambridge Studies in Advanced Mathematics, Vol. 49, 2nd ed., Cambridge University Press, Cambridge, 2012.
15. Viennot G., A combinatorial theory for general orthogonal polynomials with extensions and applications, in Orthogonal Polynomials and Applications (Bar-le-Duc, 1984), Lecture Notes in Math., Vol. 1171, Springer, Berlin, 1985, 139-157.
16. Viennot X., Alternative tableaux, permutations and partially asymmetric exclusion process, available at http://www.newton.ac.uk/webseminars/pg+ws/2008/csm/csmw04/0423/viennot/.
17. Wimp J., Explicit formulas for the associated Jacobi polynomials and some applications, Canad. J. Math. 39 (1987), 983-1000.