The factorization of a q-difference equation for continuous q-Hermite polynomials

We argue that a customary q-difference equation for the continuous q-Hermite polynomials H-n(x vertical bar q) can be written in the factorized form as [ (D-x(q))(2) - 1] H-n(x vertical bar q) = (q(-)n - 1)H-n(x vertical bar q), where D-x(q) is some explicitly known q-difference operator. This means that the polynomials H-n(x vertical bar q) are in fact governed by the q-difference equation D-x(q) H-n(x vertical bar q) = q(-n/)2 H-n(x vertical bar q), which is simpler than the conventional one. It is shown that a similar factorization holds for the continuous q(-1)-Hermite polynomials h(n)(x vertical bar q).