On the Variance of the Index for the Gaussian Unitary Ensemble

@article{Witte2011OnTV,
  title={On the Variance of the Index for the Gaussian Unitary Ensemble},
  author={Nicholas S. Witte and Peter J. Forrester},
  journal={arXiv: Classical Analysis and ODEs},
  year={2011}
}
We derive simple linear, inhomogeneous recurrences for the variance of the index by utilising the fact that the generating function for the distribution of the number of positive eigenvalues of a Gaussian unitary ensemble is a $\tau$-function of the fourth Painlev\'e equation. From this we deduce a simple summation formula, several integral representations and finally an exact hypergeometric function evaluation for the variance. 

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