  We prove that for |x|,|t|<1, -1 <q \leq1 and n\geq0:
\Sigma_{i\geq0}((t^{i})/((q)_{i}))h_{n+i}(x|q) = h_{n}(x|t,q)
\Sigma_{i\geq0}((t^{i})/((q)_{i}))h_{i}(x|q), where h_{n}(x|q) and h_{n}(x|t,q)
are respectively the so called q-Hermite and the big q-Hermite polynomials and
(q)_{n} denotes the so called q-Pochhammer symbol. We prove similar equalities
involving big q-Hermite and Al-Salam---Chihara (ASC) polynomials and ASC and
the so called continuous dual q-Hahn (c2h) polynomials. Moreover we are able to
relate in this way some other 'ordinary ' orthogonal polynomials such as e.g.
Hermite, Chebyshev or Laguerre. These equalities give new interpretation of the
polynomials involved and moreover can give rise to a simple method of
generating more and more general (i.e. involving more and more parameters)
families of orthogonal polynomials. We pose some conjectures concerning
Askey--Wilson polynomials and their possible generalizations. We prove that
these conjectures are true for the cases q = 1 (classical case) and q = 0 (free
case) thus paving the way to generalization of AW polynomials at least in these
two cases.
