We give in this paper a new method for solving the generalized truncated power moment problem. To this aim we use r-generalized Fibonacci sequences and their associated minimal polynomials. We provide an algorithm of construction of solutions in a short method. This method allows us to avoid any appeal to Hankel matrices or any positive definiteness conditions as in the Flessas-Burton-Whitehead (FBW) approach. Examples and general cases are discussed, (C) 2001 Academic Press. [References: 19]
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