Calculus of archimedean Rankin--Selberg integrals with recurrence relations

Let n and n’ be positive integers such that n-n’\in \{0,1\}. Let F be either \mathbb {R} or \mathbb {C}. Let K_n and K_{n’} be maximal compact subgroups of \mathrm {GL}(n,F) and \mathrm {GL}(n’,F), respectively. We give the explicit descriptions of archimedean Rankin–Selberg integrals at the minimal K_n- and K_{n’}-types for pairs of principal series representations of \mathrm {GL}(n,F) and \mathrm {GL}(n’,F), using their recurrence relations. Our results for F=\mathbb {C} can be applied to the arithmetic study of critical values of automorphic L-functions.