$\\displaystyle \\log\\left|x-y\\right|=-\\sum_{n=1}^{\\infty}\\frac{2}{n}T_n\\left(\\frac{x}{2}\\right)T_n\\left(\\frac{y}{2}\\right).$<\/p>\n
Second Haagerup formula:<\/strong> if $x>2$ and $-2\\leq y\\leq 2$ then (absolutely convergent series)<\/p>\n $\\displaystyle \\log\\left|x-y\\right|=\\log\\left|\\frac{x+\\sqrt{x^2-4}}{2}\\right|-\\sum_{n=1}^\\infty\\frac{2}{n}\\left(\\frac{x-\\sqrt{x^2-4}}{2}\\right)^nT_n\\left(\\frac{y}{2}\\right).$<\/p>\n I have learnt these beautiful formulas in a talk given by Ionel Popescu<\/a> during the Workshop Probability and Geometry in High Dimensions<\/em><\/a> held at Marne-la-Vall\u00e9e. The proofs are elementary. These formulas are deeply related to the fact that the arcsine distribution\u00a0 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\u00a0 Erd\u0151s and Tur\u00e1n , and Szeg\u0151, on the equilibrium measure of the roots of orthogonal polynomials. You may take a look at the books by Saff and Totik<\/a> and by van Assche<\/a>.<\/p>\n Note: Uffe Haagerup<\/a> is a Danish mathematician. His MR number is 78865<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":" 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…<\/p>\n![\"Danemark](\"\/blog\/wp-content\/uploads\/Danemark_flag.gif\" "\\"Danemark")
<\/p>\n