Table of Contents 1. Introduction 1 Recurrence relation 5 2. The orthogonality relation 9 A q-beta integral 9 The absolute continuous orthogonality 11 3. A quadratic transformation 18 4. Some interesting special cases 20 Continuous q-ultraspherical polynomials 20 Continuous q-Jacobi polynomials 23 Continuous q-Laguerre polynomials 24 Al-Salam-Chihara polynomials 24 Generalized TchebychefF polynomials 27 Continuous q-symmetric Hahn polynomials 29 q-Meixner-Pollaczek polynomials 30 5. Divided difference operators 32 Second order difference equation 35 Rodrigues type formula 35 6. Connection coefficients 36 7. Zeros and inequalitites 40 8. Remarks 42 Appendix 45 References 50 iii
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