Martin Klazar klazar at
Tue Apr 29 14:49:24 CEST 2003

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


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).

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

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