Complex Moment Problems and Recursive Relations of Fibonacci Type

The complex moment problem for a sequence γ(2n) = {γij}0≤i,j≤n has been studied by Curto-Fialkow, where positivity and extension properties of the moment matrix M(n) = M(n)(γ) (γ ≡ γ(2n)) are involved, for guaranteeing the existence of representing measure. But it was showed that positivity and recursiveness are not sufficient in order to have a representing measure for γ. Here we combine our techniques based on the Fibonacci sequences’s properties with some Curto-Fialkow’s results to obtain sufficient conditions for insuring that γ is a truncated moment sequence. We focus ourself on the case when rank M(n) ≤ n + 1, and finally we stretch our exploration to the finite-rank infinite positive moment matrix.