A note on moments of derivatives of characteristic polynomials

TL;DR

This paper introduces a straightforward method to compute moments of derivatives of unitary characteristic polynomials, combining Schur function orthonormality and the Generalized Binomial Theorem to derive explicit formulas.

Contribution

It develops a novel technique that extends existing methods to include derivatives of characteristic polynomials using advanced symmetric function theory.

Findings

Provides explicit formulas for moments of derivatives of characteristic polynomials.

Utilizes Schur functions and the Generalized Binomial Theorem for computations.

Derives alternative expressions involving hypergeometric functions.

Abstract

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.