We prove that for |x|≤|t| < 1, -1 < q ≤ 1 and n ≥0: Σ _(i≥0)t ~i/(q) ih _(n+i)(x|q)=h _n(x|t,q)Σ _(i≥0)t ~i/(q) _ih _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 polynomials, and Al-Salam-Chihara and the so-called continuous dual q-Hahn 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 a 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 Askey-Wilson polynomials at least in these two cases.
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