An asymptotic expansion of (k,  c)-functions for \(\textrm{GL}_{n}(F)\)

Given the important role an asymptotic expansion of Whittaker functions or Shalika functions plays in the study of L-functions via the Rankin–Selberg method, we first establish an asymptotic expansion of (k,  c)-functions for \(\textrm{GL}_{n}(F)\) ${\text{GL}}_n(F)$ over a p-adic field F via Casselman–Shalika’s argument. As an application, we then show the holomorphy of Ginzburg–Kaplan’s local “tensor product” L-functions for \(\textrm{GL}_c(F)\times \textrm{GL}_k(F)\) ${\text{GL}}_c(F)\times {\text{GL}}_k(F)$ . In the appendix, following Wallach–Soudry’s argument, we also provide an analogous asymptotic expansion of (k,  c)-functions for \(\textrm{GL}_n(F)\) ${\text{GL}}_n(F)$ with F an Archimedean local field.