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.