Askey–Wilson polynomials

This result can be proven since it is known that :p_n(\cos{\theta}) = p_n(\cos{\theta};a,b,c,d\mid q) and using the definition of the q-Pochhammer symbol :p_n(\cos{\theta})= a^{-n}\sum_{\ell=0}^{n}q^{\ell}\left(abq^{\ell},acq^{\ell},adq^{\ell};q\right)_{n-\ell}\times\frac{\left(q^{-n},abcdq^{n-1};q\right)_{\ell}}{(q;q)_{\ell}}\prod_{j=0}^{\ell-1}\left(1-2aq^j\cos{\theta}+a^2q^{2j}\right) which leads to the conclusion that it equals :a^{-n}(ab,ac,ad;q)_n\;_{4}\phi_3 \left[\begin{matrix} q^{-n}&abcdq^{n-1}&ae^{i\theta}&ae^{-i\theta} \\ ab&ac&ad \end{matrix} ; q,q \right] ==See also==