| \begin{array}{l}{\displaystyle\Psi_{1}\left(\left.\begin{array}{l}{1,\beta}\\{2,1}\end{array}\right|t,-s\right)\,=\,\sum_{k=0}^{\infty}\,\sum_{\ell=0}^{\infty}\,\frac{(1)_{k+\ell}\,(\beta)_{k}}{(2)_{k}(1)_{\ell}}\,\frac{t^{k}}{k!}\,\frac{(-s)^{l}}{\ell!}\,=\,}\\{\,=\,\displaystyle\sum_{k=0}^{\infty}\,\frac{(1)_{k}\,(\beta)_{k}}{(2)_{k}}\,\frac{t^{k}}{k!}\sum_{\ell=0}^{\infty}\,\frac{(1+k)_{\ell}}{(1)_{\ell}}\,\,\frac{(-s)^{l}}{\ell!}\,=\,}\\{\,=\,\displaystyle\sum_{k=0}^{\infty}\,\frac{(1)_{k}\,(\beta)_{k}}{(2)_{k}}\,\frac{t^{k}}{k!}_{1}F_{1}(1+k;1;-s)\,=\,}\\{\,=\,\displaystyle\exp(-s)\,\sum_{k=0}^{\infty}\,\frac{(\beta)_{k}}{(2)_{k}}\,t^{k}L_{k}(s)\,.}\end{array} |
