CAOP

\[ \begin{align} P_n^{(a,b)}(x|q) &= \frac{(q^{a+1};q,n)}{(q;q,n)} \sum_{k=0}^\infty \frac{(q^{-n};q)_k (q^{n+a+b+1};q)_k (q^{1/2a+1/4}e^{it};q)_k (q^{1/2a+1/4}e^{-it};q)_k}{(q^{a+1};q)_k (-q^{1/2(a+b+1)};q)_k (-q^{1/2(a+b+2)};q)_k (q;q)_k} q^k\\ &= \frac{(q^{a+1};q,n)}{(q;q,n)} {}_{3}\phi_{2}\!\left(\left. {q^{-n}, q^{n+a+b+1}, q^{1/2a+1/4}e^{it}, q^{1/2a+1/4}e^{-it} \atop q^{a+1}, -q^{1/2(a+b+1)}, -q^{1/2(a+b+2)}} \; \right| q ; q \right),\quad x=\cos(t) \end{align} \]