We survey many old and new results on solutions of the following pair of adjoint differential-difference equations: (1)
\[
u
p
′
(
u
)
=
−
a
p
(
u
)
−
b
p
(
u
−
1
)
,
up’(u) = - ap(u) - bp(u - 1),
\]
(2) (
\[
(
u
q
(
u
)
)
′
=
a
q
(
u
)
+
b
q
(
u
+
1
)
.
(uq(u))’ = aq(u) + bq(u + 1).
\]
We bring together scattered results usually proved only for specific
(
a
,
b
)
(a,b)
pairs, while emphasizing the connections between the two equations. We also point out some of the ways these two equations are used in number theory. We giv s several new integral relationships between (1) and (2) and use them to prove a new application of (2) in number theory, namy el
\[
∑
1
>
n
⩽
x
P
2
(
n
)
⩽
P
1
(
n
)
1
/
u
(
log
P
1
(
n
)
)
α
∼
u
α
f
(
u
)
x
(
log
x
)
α
(
x
→
∞
,
u
⩾
1
,
α
∈
R
)
\sum \limits _{\begin {array}{*{20}{c}} {1 > n \leqslant x} \\ {{P_2}(n) \leqslant {P_1}{{(n)}^{1/u}}} \\ \end {array} } {{{(\log {P_1}(n))}^\alpha } \sim {u^\alpha }f(u)x{{(\log x)}^\alpha }} \qquad (x \to \infty ,\;u \geqslant 1,\;\alpha \in {\mathbf {R}})
\]
where
P
1
(
n
)
{P_1}(n)
and
P
2
(
n
)
{P_2}(n)
are the first and second largest prime divisors of
n
n
and
f
(
u
)
f(u)
satisfies (2) with
(
a
,
b
)
=
(
1
−
α
,
−
1
)
(a,b) = (1 - \alpha , - 1)
.