PaulOlivier Dehaye

A note on moments of derivatives of characteristic polynomials
dmtcs:2823 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2010,
DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)

https://doi.org/10.46298/dmtcs.2823
A note on moments of derivatives of characteristic polynomials
Authors: PaulOlivier Dehaye ^{1}
0000000209071927
PaulOlivier Dehaye
1 Department of Mathematics  ETH
We present a simple technique to compute moments of derivatives of unitary characteristic polynomials. The first part of the technique relies on an idea of Bump and Gamburd: it uses orthonormality of Schur functions over unitary groups to compute matrix averages of characteristic polynomials. In order to consider derivatives of those polynomials, we here need the added strength of the Generalized Binomial Theorem of Okounkov and Olshanski. This result is very natural as it provides coefficients for the Taylor expansions of Schur functions, in terms of shifted Schur functions. The answer is finally given as a sum over partitions of functions of the contents. One can also obtain alternative expressions involving hypergeometric functions of matrix arguments.
Assiotis, Theodoros; Gunes, Mustafa Alper; Soor, Arun, 2022, Convergence And An Explicit Formula For The Joint Moments Of The Circular Jacobi $$\Beta $$Ensemble Characteristic Polynomial, Mathematical Physics, Analysis And Geometry, 25, 2, 10.1007/s11040022094274.
Forrester, Peter J., 2022, Joint Moments Of A Characteristic Polynomialand Its Derivative For The Circular ÎEnsemble, Probability And Mathematical Physics, 3, 1, pp. 145170, 10.2140/pmp.2022.3.145.