On moments and symmetrical sequences

In this article we consider questions related to the behavior of the moments $M_m\left(\left(z_j\right)\right)$ when the indices are restricted to specific subsequences of integers, such as the even or odd moments. If $n\ge 2$ we introduce the notion of symmetrical series of order $n,$ showing that if $\left(z_j\right)$ is symmetrical then $M_m\left(\left(z_j\right)\right)=0$ whenever $n\nmid m;$ in particular, the odd moments of a symmetrical series of order 2 vanish. We prove that when $\left(z_j\right)\in l^p$ for some $p$ then several results characterizing the sequence from its moments hold. We show, in particular, that if $M_m\left(\left(z_j\right)\right)=0$ whenever $n\nmid m$ then $\left(z_j\right)$ is a rearrangement of a symmetrical series of order $n.$ We then construct examples of sequences whose moments vanish with required density. Lastly, we construct counterexamples of several of the results valid in the $l^p$ case if we allow the moment series to be all conditionally convergent. We show that for each arbitrary sequence of real numbers ${\left({\mu }_m\right)}_{m=0}^{\infty }$ there are real sequences ${\left(u_j\right)}_{j=0}^{\infty }$ such that $\sum _{j=0}^{\infty }{u}_j^{2m+1}={\mu }_{m}m\ge 0.$