Transcendental values of the incomplete gamma function and related questions (Q501161)

The authors prove that under the special conditions the sums of the series \[ \sum_{n_1,\dots ,n_r\geq 0}\frac{\prod_{j=1}^rA_j(n_j)z_j^{n_j}}{\prod_{j=1}^rB_j(n_j)n_j!}, \quad \sum_{n_1,\dots ,n_r\geq 0}\frac{\prod_{j=1}^rA_j(n_j)z^{n_j}}{\prod_{j=1}^rB_j(n_j)n_j!}\quad \text{and}\quad \sum_{n\geq 0}\frac{A(n)f(n)z^n}{B(n)n!} \] are transcendental numbers where \(z_1,\dots ,z_r\), \(z\) are algebraic numbers, \(f(n)\) is periodic function and \(A_1(x),\dots ,A_r(x)\), \(A(x)\), \(B_1(x),\dots ,B_r(x)\), \(B(x)\) are polynomials with algebraic coefficients.