A family of summation formulas on the Fox-Wright function (Q432374)

The Fox-Wright function \(_p\Psi_q\) is defined by NEWLINE\[NEWLINE _p\Psi_q\left[\left.\begin{matrix} (\alpha_1,A_1), \dotsc, (\alpha_p,A_p)\\ (\beta_1,B_1), \dotsc, (\beta_q,B_q)\end{matrix} \right|z\right]=\sum_{n=0}^\infty\frac{\prod_{i=1}^p\Gamma(\alpha_i+A_ik)}{\prod_{j=1}^q\Gamma(\beta_j+B_jk)}\frac{z^k}{k!}.NEWLINE\]NEWLINE The authors prove dozens of summation formulas for these functions for special values of \(p\) and \(q\) and for special arguments. To show some of the results, we list two formulas:NEWLINENEWLINENEWLINENEWLINE\[NEWLINE_2\Psi_3\left[\left.\begin{matrix} (1+a+n,b-1),(3a,3b-1)\\ (1+3n,-1),(a-n,b),(1+3a,3b-2)\end{matrix}\right|-\frac13\right]=\frac{(-1)^n}{3n!27^n},NEWLINE\]NEWLINE NEWLINE\[NEWLINE_2\Psi_3\left[\left.\begin{matrix}(1+a+n,b-1),(3a,3b-1)\\ (2+3n,-1),(a-n-1,b),(1+3a,3b-2)\end{matrix}\right|-\frac13\right]=(-b)\frac{(-1)^n}{3n!27^n}.NEWLINE\]