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