Bar-le-Duc paper copy etc.: BarlDhtm

Software: Freud machine by Alphonse P. Magnus math. , Univ. of Louvain (Belgium) Aug. 1996 See here FORTRAN programs associated to the numerical approximate computation of the recurrence coefficients a1, a2, ... , aN of the orthonormal polynomials p0, p1, ... : a_{n+1} p_{n+1}(x) = x p_n(x) -a_n p_{n-1}(x) , n=0,1,... (a0=0), related to the (even) weight function |x|^\rho \exp(-P(x^2)) on the whole real line. Elementary recurrence transformations also show that a_{2n+2} a_{2n+1} q_{n+1}(y) = (y- a_{2n+1}^2 -a_{2n}^2 ) q_n(y) -a_{2n} a_{2n-1} q_{n-1}(y) is the recurrence relation of the orthonormal polynomials q_n(y) related to the weight y^{(\rho -1)/2} \exp(-P(y)) on the positive real line (take y=x^2). -rw-r--r-- 17505 Aug 30 14:49 freud.txt
-rw-r--r-- 88319 Aug 30 09:48 freud008.dat
-rw-r--r-- 2194 Aug 30 09:48 freud1.f
-rw-r--r-- 1766 Aug 30 14:49 freud2.f
-rw-r--r-- 4179 Aug 30 14:49 freud3.f

  A recent preprint:

A.P. MAGNUS, Freud's equations for orthogonal polynomials as discrete Painlevé equations, pp. 228-243 in Symmetries and Integrability of Difference Equations, Edited by Peter A. Clarkson & Frank W. Nijhoff, Cambridge U.P., Lond. Math. Soc. Lect. Note Ser. 255, 1999. <br> See here the pdf preprint (complete, with the Chaucer lines!) freudpain.pdf (217K)