Abstract
We pursue the study started in Demni and Hmidi (Colloq Math 137(2):271–296, 2014) of the dynamics of the spectral distribution of the free Jacobi process associated with one orthogonal projection. More precisely, we use Lagrange inversion formula in order to compute the Taylor coefficients of the local inverse around \(z=0\) of the flow determined in Demni and Hmidi (Colloq Math 137(2):271–296, 2014). When the rank of the projection equals 1/2, the obtained sequence reduces to the moment sequence of the free unitary Brownian motion. For general ranks in (0, 1), we derive a contour integral representation for the first derivative of the Taylor series.
Similar content being viewed by others
Notes
In this introductory part, we ommit the dependence of our notations on \(\{P,Q\}\).
The principal branch of the square root is taken.
References
Biane, P.: Free Brownian motion, free stochastic calculus and random matrices. Fields Inst. Commun., 12, Amer. Math. Soc. Providence, RI, pp. 1–19 (1997)
Biane, P.: Segal–Bargmann transform, functional calculus on matrix spaces and the theory of semi-circular and circular systems. J. Funct. Anal. 144(1), 232–286 (1997)
Cohen, M.E.: Some classes of generating functions for the Laguerre and Hermite polynomials. Math. Comput. 31(238), 511–518 (1977)
Collins, B., Kemp, T.: Liberation of projections. J. Funct. Anal. 266, 1988–2052 (2014)
Demni, N.: Free Jacobi process. J. Theor. Probab. 21(1), 118–143 (2008)
Demni, N., Hamdi, T., Hmidi, T.: Spectral distribution of the free Jacobi process. Indiana Univ. Math. J. 61(3), 1351–1368 (2012)
Demni, N., Hmidi, T.: Spectral distribution of the free Jacobi process associated with one projection. Colloq. Math. 137(2), 271–296 (2014)
Doumerc, Y.: Matrix Jacobi Process. Ph. D. Thesis. Paul Sabatier Univ. May (2005)
Ebenfelt, P., Khavinson, D., Shapiro, H.S.: Analytic continuation of Jacobi polynomial expansions. Indag. Math. 8(1), 19–31 (1997)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. I. McGraw-Hill, New York (1981)
Gross, D.J., Matytsin, A.: Some properties of large-\(N\) two dimensional Yang–Mills theory. Nucl. Phys. B. 437(3), 541–584 (1995)
Izumi, M., Ueda, Y.: Remarks on free mutual information and orbital free entropy. Available on arXiv
Manocha, H.L., Srivastava, H.M.: A treatise on generating functions. Mathematics and Its Applications, Ellis Horwood Series (1984)
Rainville, E.D.: Special Functions. The Macmillan Co., New York (1960)
Riordan, J.: Combinatorial Identities. Wiley series in probability and mathematical statistics. Wiley (1968)
Szego, G.: Orthogonal Polynomials. American Mathematical Society, Colloquium Publications, Providence (1975)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hari Bercovici.
Rights and permissions
About this article
Cite this article
Demni, N. Free Jacobi Process Associated with One Projection: Local Inverse of the Flow. Complex Anal. Oper. Theory 10, 527–543 (2016). https://doi.org/10.1007/s11785-015-0475-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11785-015-0475-6