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The degeneracy of the lowest weight representations of the quantumsuperalgebra $osp_q(1|2)$ and their tensor products at exceptional values of%when deformation parameter $q$ takes exceptional values is studied. The mainfeatures of the structures of the finite dimensional lowest weightrepresentations and their fusion rules are illustrated using realization ofgroup generators as finite-difference operators acting in the space of thepolynomials. The complete fusion rules for the decompositions of the tensorproducts at roots of unity are presented. The appearance of indecomposablerepresentations in the fusions is described using Clebsh-Gordan coefficientsderived for general values of $q$ and at roots of unity.Comment: 28 pages, 3 figures; completed (section 5, section 6); version  published in JP

Fusion rules of the lowest weight representations of {\rm {osp}}_q(1|2) at roots of unity: polynomial realization