Split-denominator recurrence for the modular invariant

Construct a recurrence of the form (1.5), with a denominator polynomial M split over the rationals, for the Taylor coefficients of the elliptic modular invariant F(q)=J(q^2), or determine whether such a recurrence exists.

Background

The elliptic modular invariant F(q)=J(q^2) is shown to satisfy a third-order algebraic differential equation. The associated indicial polynomial is P_0(X)=X^3+8X^2−10X+64, which has no rational root.

Consequently, the general recurrence supplied by the cited Sibuya–Sperber procedure has a denominator polynomial M that is not split over the rationals, so the paper’s main denominator theorems cannot be applied. The authors leave unresolved whether a different recurrence of the required split form exists, despite the coefficients of F being integers.