Poisson kernel and multilinear generating functions of some orthogonal polynomials (Q914010)

Many of the basic orthogonal polynomials are specializations of \[ \Omega_ n(x;a,b| q)=\sum \left[ \begin{matrix} n\\ r\end{matrix} \right](a;q)_ r(b;q)_{n-r}x^ r. \] The authors find an identity for the multilinear generating function \[ \sum \frac{\Omega_{N+s}(x;a,b| q)}{(ab)_{N+s}}\prod^{k}_{j=1}\frac{\Omega_{n_ j+m_ j}(y_ j;c_ j,d_ j| q)z_ j^{n_ j}}{(q)_{n_ j}}, \] where the sum is over \(n_ 1,...,n_ k\geq 0\) and \(N=n_ 1+...+n_ k\). Many classical results on generating functions are then derived as special cases.