In some recent work I found myself having to investigate the asymptotics of the Debye polynomials, which I will here take as *defined* by the following integrodifferential recurrence:
$D_{n+1}(x)= \frac{1}{2}x^2(1-x)^2D'_n(x) + \frac{1}{8}\int_0^x(1-5t^2)D_n(t)dt$
with $D_0(x)=1$
These are polynomials of degree 3n. I mostly care about the leading term as n grows, but I'd also like to know how to compute these polynomials to high degree assuming only this recurrence. I have a horribly slow and ugly GP script which I am too embarrassed by to inflict on this list, and it doesn't go high enough to get the digits to guess at asymptotics.
More generally, I can't find from the usual sources how to accelerate computing families of *functions* given by recurrences. If for example all one had was
this recurrence for Bernoulli polynomials, could one easily compute B_1000(x) (or at least an arbitrary coefficient) in PARI/GP?
As usual any help, comments or references are appreciated.