L\'evy processes, martingales, reversed martingales and orthogonal polynomials

We study class of L'{e}vy processes having distributions being indentifiable by moments. We define system of polynomial martingales \newline $\left{ M_{n}(X_{t},t),$\mathcal{F}${\leq t}\right} {n\geq 1},$where$% $\mathcal{F}${\leq t}$ is a suitable filtration defined below. We present several properties of these martingales. Among others we show that $% M{1}(X_{t},t)/t$ is a reversed martingale as well as a harness. Main results of the paper concern the question if martingale say $M_i$ multiplied by suitable determinstic function ${\mu }_i(t)$ is a reversed martingale. We show that for $n\ge 3$ $M_n(X_t,t)$ is a reversed martingale (or orthogonal polynomial) only when the L'{e}vy process in question is Gaussian (i.e. is a Wiener process). We study also a more general question if there are chances for a linear combination (with coefficients depending on $t)$ of martingales $M_i,$ $i=1,\dots ,n$ to be reversed martingales. We analyze case $$ in detail listing all possible cases.