We introduce a new methodology to solve the truncated complex moment problem. To this aim we investigate recursive doubly indexed sequences and their characteristic polynomials. A characterization of recursive doubly indexed moment sequences is given. A simple application gives a computable solution to the complex moment problem for cubic harmonic characteristic polynomials of the form <$Ez sup 3 ~+~az~+~b z bar>, where a and b are arbitrary real numbers. We also recapture a recent result due to Curto - Yoo given for cubic column relations in M(3) of the form <$EZ sup 3 ~=~itZ~+~u Z Bar> with t, u real numbers satisfying some suitable inequalities. Furthermore, we solve the truncated complex moment problem with column dependence relations of the form <$EZ sup k+1 ~=~sum from {0~symbol Г~n~+~m~symbol Г~k} ~a sub nm {Z Bar } sup n ~Z sup m> (<$Ea sub nm ~symbol <174>~roman bold C>).

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