Though it ended in a tragic manner, Thomas Johannes Stieltjes' stay in Toulouse left its mark on the mathematics in this city. Stieltjes played a prominent role in the early years of the Annales de la Faculté des Sciences de Toulouse. A sincere tribute has been paid to him in 1994 at the occasion of the centenary of his death. Among the many results published by Stieltjes in the AFST, an article of 1890  is devoted to the Legendre polynomials which are the orthogonal polynomials for the Lebesgue measure on [-1,1]. Stieltjes obtains an accurate description of the asymptotic behavior of these polynomials on this interval and deduces approximate values for the roots of the polynomials (some of which with accuracy 10-6 !).

The Legendre polynomials arise in quadrature formulas, approximation theory, but also in the construction of transparent boundary conditions for partial differential equations. They also display fascinating arithmetic properties.