Software: Freud machine
by Alphonse P. Magnus
math. , Univ. of Louvain (Belgium)
magnus@anma.ucl.ac.be http://www.math.ucl.ac.be/~magnus/
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)