Research projects

As for the multiple hypergeometric type generating functions for the values of Lerch zeta-functions, the head investigator has succeeded in formulating the expected generating functions (of several complex variables) for the values of Lerch zeta-functions, in the form of Lauricella (type A) multiple hypergeometric series. The major achievements of the present research include complete asymptotic expansions for these multiple generating functions when the variables $(z_1,\ldots,z_n)$ tend to $0$ and to $\infty$, while suitable mutual order conditions on $z_j$'s are imposed, through an appropriate poly-sector. These asymptotic expansions further yield: 1) asymptotics for higher derivatives of the generating functions when the variable $s$ is at any integer point; 2) closed form evaluation of the generating functions when $s$ is at any non-positive integer point; 3) asymptotics for two variable analogues of the classical trigonometric sums treated in [Hardy-Littlewood (1936)].