Maps Preserving Moment Sequences

For any sequence s of real numbers, we consider the class \(\mathcal {L}\) of maps (from \(\mathbb {R}^{\mathbb {N}_0}\) to \(\mathbb {R}^{\mathbb {N}_0}\)) that linearly combine a finite or infinite number of elements of s to obtain the new values of the transformed sequence. We characterize those maps in \(\mathcal {L}\) that transform moment sequences into moment sequences in terms of the existence of a stochastic process fulfilling appropriate requirements. Then, well-known stochastic processes are used to construct significant examples of such preserving mappings. As application, we also show that some celebrated numerical sequences (including several important combinatorial sequences) are actually transformed moment sequences.