  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.
