Random walk polynomial sequences arise from three-term recurrences where the coefficients sum to one — they encode transition probabilities. When the product of any two polynomials decomposes as a convex combination of others in the sequence, you get nonnegative linearization: the product structure is probabilistic, not just algebraic. This gives rise to hypergroups and sophisticated harmonic analysis.
Now consider switching the roles of the forward and backward coefficients. If the original sequence has nonnegative linearization and so does the switched sequence, you have a double symmetry — the random walk and its reversal are both well-behaved. Lasser and Obermaier recently asked whether any nontrivial example exists beyond the Chebyshev polynomials of the first kind, where the coefficients are identically 1/2 and the symmetry is trivially forced.
This paper constructs explicit nontrivial examples and provides a sufficient criterion for their existence. The Chebyshev polynomials are characterized by additionally imposing properties on the duals and Haar measures — they are the unique solution when you demand everything to be symmetric. Relax that demand slightly and a new class of orthogonal polynomials emerges: random walks whose forward-backward symmetry in the product structure isn't forced by coefficient symmetry but by a subtler algebraic constraint. The question “are there any?” is answered by construction.