Complete Minimal Logarithmic Energy Asymptotics for Points in a Compact Interval: A Consequence of the Discriminant of Jacobi Polynomials

Abstract

The electrostatic interpretation of zeros of Jacobi polynomials, due to Stieltjes and Schur, enables us to obtain the complete asymptotic expansion as $$n \rightarrow \infty $$ n → ∞ of the minimal logarithmic potential energy of n point charges restricted to move in the interval $$[-1,1]$$ [ - 1 , 1 ] in the presence of an external field generated by endpoint charges. By the same methods, we determine the complete asymptotic expansion as $$N \rightarrow \infty $$ N → ∞ of the logarithmic energy $$\sum _{j\ne k} \log (1/| x_j - x_k |)$$ ∑ j ≠ k log ( 1 / | x j - x k | ) of Fekete points, which, by definition, maximize the product of all mutual distances $$\prod _{j\ne k} | x_j - x_k |$$ ∏ j ≠ k | x j - x k | of N points in $$[-1,1]$$ [ - 1 , 1 ] . The results for other compact intervals differ only in the quadratic and linear term of the asymptotics. Explicit formulas and their asymptotics follow from the discriminant, leading coefficient, and special values at $$\pm 1$$ ± 1 of Jacobi polynomials. For all these quantities we derive complete Poincaré-type asymptotics.