A specific form of the Hamburger Problem

Question

I'm having a hard time trying to solve a moment-related problem. Let M be a random variable, I denote its k-th moment $m_{k}(X)=E({X^k})$ . How can i prove that the sequence $ u_{k}=\frac{m_{k+2n}(X)}{m_{2n}(X)} $ defines the k-th moment of another random variable, Y ? I shall not use any tool from measure theory, and the variable X doesn't necesserily have a density. I tried to use the algebric characterization using Henkel matrices, which proved to be a deadlock. Carlman's criterium isn't helpful either.