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Haagerup formulas

Let $(T_n)_{n\geq0}$ be the Chebyshev polynomials of the first kind given by

$\displaystyle T_n(\cos(x))=\cos(nx).$

These polynomials are orthogonal with respect to the arcsine probability distribution

$\displaystyle x\mapsto \frac{1}{\pi\sqrt{1-x^2}}\mathbf{1}_{[-1,1]}(x).$

They satisfy to the recurrence relation $T_0=1$, $T_1(x)=x$ and

$\displaystyle T_{n+1}(x)=2xT_n(x)-T_{n-1}(x).$

First Haagerup formula: if $-2\leq x\neq y\leq 2$ then (the series is convergent)

$\displaystyle \log\left|x-y\right|=-\sum_{n=1}^{\infty}\frac{2}{n}T_n\left(\frac{x}{2}\right)T_n\left(\frac{y}{2}\right).$

Second Haagerup formula: if $x>2$ and $-2\leq y\leq 2$ then (absolutely convergent series)

$\displaystyle \log\left|x-y\right|=\log\left|\frac{x+\sqrt{x^2-4}}{2}\right|-\sum_{n=1}^\infty\frac{2}{n}\left(\frac{x-\sqrt{x^2-4}}{2}\right)^nT_n\left(\frac{y}{2}\right).$

I have learnt these beautiful formulas in a talk given by Ionel Popescu during the Workshop Probability and Geometry in High Dimensions held at Marne-la-Vallée. The proofs are elementary. These formulas are deeply related to the fact that the arcsine distribution  on $[-a,a]$ is the maximum of the Voiculescu entropy (i.e. minimum of logarithmic energy) over the set of probability distributions supported in $[-a,a]$. This fact is quite classical, and goes back at least to the works of  Erdős and Turán , and Szegő, on the equilibrium measure of the roots of orthogonal polynomials. You may take a look at the books by Saff and Totik and by van Assche.

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Note: Uffe Haagerup is a Danish mathematician. His MR number is 78865.

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