Prime divisors of binary holonomic sequences

We prove that if (un)n⩾0(un)n⩾0 is a sequence of rational numbers satisfying a recurrence of the typef0(n)un+2+f1(n)un+1+f2(n)un=0,f0(n)un+2+f1(n)un+1+f2(n)un=0, where fi(X)∈Q[X]fi(X)∈Q[X] are not all zero for i=0,1,2i=0,1,2, which is not binary recurrent for all sufficiently large n, then there exists a positive constant c   depending on the sequence (un)n⩾0(un)n⩾0 such that the product of the numerators and denominators of the nonzero rational numbers unun for all n⩽Nn⩽N has at least clogNclogN prime factors as N→∞N→∞.