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The characterization of the solvability of matrix versions of truncatedStieltjes-type moment problems led to the class of $\alpha$-Stieltjesnon-negative definite sequences of complex $q \times q$ matrices. In [21], aparametrization of this class was introduced, the so-called $\alpha$-Stieltjesparametrization. The main topic of this first part of the paper is theconstruction of a Schur-type algorithm which produces exactly the$\alpha$-Stieltjes parametrization.

On a Simultaneous Approach to the Even and Odd Truncated Matricial  Stieltjes Moment Problem I: An $\alpha$-Schur-Stieltjes-type algorithm for  sequences of complex matrices