Enveloping Superalgebra U(osp(1|2)) and Orthogonal Polynomials in Discrete Indeterminate

Let A be an associative simple (central) superalgebra over C and L an invariant linear functional on it (trace). Let a 7→ a be an antiautomorphism of A such that (a) = (−1)p(a)a, where p(a) is the parity of a, and let L(a) = L(a). Then A admits a nondegenerate supersymmetric invariant bilinear form 〈a, b〉 = L(ab). For A = U(sl(2))/m, where m is any maximal ideal of U(sl(2)), Leites and I have constructed orthogonal basis in A whose elements turned out to be, essentially, Chebyshev (Hahn) polynomials in one discrete variable. Here I take A = U(osp(1|2))/m for any maximal ideal m and apply a similar procedure. As a result we obtain either Hahn polynomials over C[τ ], where τ ∈ C, or a particular case of Meixner polynomials, or — when A = Mat(n+1|n) — dual Hahn polynomials of even degree, or their (hopefully, new) analogs of odd degree. Observe that the nondegenerate bilinear forms we consider for orthogonality are, as a rule, not sign definite.