Re: Solving a functional equation

[Pierre]

Le lundi 8 octobre 2012 18:37:45 UTC+2, Pierre a écrit :

Consider the functional equation

      (f*z + z * L(z,f)^2 + z * L(z,f+1) - L(z,f) == 0)

This defines implicitly a bivariate function L(z,f).  My goal is to find the Taylor development of L(z,0).  I know from

     Bernhard Gittenberger Enumeration of generalized BCI λ terms - AofA'11

that the result is

z^2 + 2z^3 + 4z^4 + 13z^5 + 42z^6 + 139z^7 + 506z^8 +1915z^9 + 7558z^10 + · · ·

How do I do that in SAGE?

Thank you in advance.

Pierre

Actually the idea (thanks to Bernhard and Philippe Flajolet) is to compute a fix point:

zero(z,u) = 0

# nb of terms of size n with exactly m variables and |x| = 1
def Phi(L):
    return z*L(z,u)^2 + z*u + z*L(z,u+1)
   
def iter(n):
    result = zero
    for i in [1..n]:
        result(z,u) = Phi(result)
    return result
   
L11 = iter(11)
taylor(L11(z,u=0), z, 0, 10)


Pierre