Quantum polynomials from deformed quantum algebras: Probability distributions, generating functions and difference equations

In this paper, we provide a novel generalization of quantum orthogonal polynomials from [Formula: see text]-deformed quantum algebras introduced in earlier works. We construct related quantum Jacobi polynomials and their probability distribution, factorial moments, recurrence relation, and governing difference equation. Surprisingly, these polynomials obey non-conventional recurrence relations. Particular cases of generalized quantum little Legendre, little Laguerre, Laguerre, Bessel, Rogers–Szegö, Stieltjes–Wigert and Kemp binomial polynomials are derived and discussed.