generatingfunctions

[Martin Klazar]

As for the possible representations of sparse ("lacunary")
power series, such as

F(z)=z+z^2+z^4+z^8+...,

the following theorem of Lipshitz and Rubel (1986) applies
and shows that F(z) satisfies no algebraic differential
equation over C(x). (Theta functions, mentioned in
this discussion, do satisfy alg. dif. equations).

Thm.
If a_0.x^{n_0}+a_1.x^{n_1}+..., where n_0<n_1<... and
all a_i are nonzero, is dif. algebraic
then lim inf n_{i+1}/n_i=1.

(Their actual result is somewhat stronger.)


Martin Klazar