On explicit probability densities associated with Fuss-Catalan numbers
Authors:
Dang-Zheng Liu, Chunwei Song and Zheng-Dong Wang
Journal:
Proc. Amer. Math. Soc. 139 (2011), 3735-3738
MSC (2010):
Primary 60B20, 46L54; Secondary 05A99
DOI:
https://doi.org/10.1090/S0002-9939-2011-11015-3
Published electronically:
May 24, 2011
MathSciNet review:
2813403
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper we give explicitly a family of probability densities, the moments of which are Fuss-Catalan numbers. The densities appear naturally in random matrices, free probability and other contexts.
- 1. Alexeev, N., Götze, F., Tikhomirov, A.: Asymptotic distribution of singular values of powers of random matrices, Lithuanian Math. J. 50, no. 2 (2010), 121-132.
- 2. George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958
- 3. Banica, T., Belinschi, S., Capitaine, M., Collins, B.: Free Bessel laws, Canad. J. Math. 63 (2011), 3-37.
- 4. Benoît Collins, Ion Nechita, and Karol Życzkowski, Random graph states, maximal flow and Fuss-Catalan distributions, J. Phys. A 43 (2010), no. 27, 275303, 39. MR 2658283, https://doi.org/10.1088/1751-8113/43/27/275303
- 5. Dang-Zheng Liu, Zheng-Dong Wang, and Kui-Hua Yan, Density of eigenvalues and its perturbation invariance in unitary ensembles of random matrices, J. Approx. Theory 162 (2010), no. 9, 1588–1606. MR 2718887, https://doi.org/10.1016/j.jat.2010.04.005
- 6. Młtkowski, W.: Fuss-Catalan numbers in noncommutative probability, Documenta Mathematica 15 (2010), 939-955.
- 7. Alexandru Nica and Roland Speicher, Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, vol. 335, Cambridge University Press, Cambridge, 2006. MR 2266879
- 8. Penson, K. A., Solomon, A. I.: Coherent states from combinatorial sequences, arXiv:quant-ph/0111151v1
- 9. Eugene P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. of Math. (2) 62 (1955), 548–564. MR 0077805, https://doi.org/10.2307/1970079
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Additional Information
Dang-Zheng Liu
Affiliation:
School of Mathematical Sciences, LMAM, Peking University, Beijing, 100871, People’s Republic of China
Address at time of publication:
Instituto Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile
Email:
dzliu@inst-mat.utalca.cl
Chunwei Song
Affiliation:
School of Mathematical Sciences, LMAM, Peking University, Beijing, 100871, People’s Republic of China
Email:
csong@math.pku.edu.cn
Zheng-Dong Wang
Affiliation:
School of Mathematical Sciences, LMAM, Peking University, Beijing, 100871, People’s Republic of China
Email:
zdwang@pku.edu.cn
DOI:
https://doi.org/10.1090/S0002-9939-2011-11015-3
Keywords:
Fuss-Catalan numbers,
moments,
free Bessel laws
Received by editor(s):
August 4, 2010
Published electronically:
May 24, 2011
Communicated by:
Jim Haglund
Article copyright:
© Copyright 2011
American Mathematical Society
