For a projective algebraic surface
X
X
with an ample line bundle
H
H
, let
M
H
X
(
c
)
M_H^X(c)
be the moduli space
H
H
-semistable sheaves
E
\mathcal {E}
of class
c
c
in the Grothendieck group
K
(
X
)
K(X)
. We write
c
=
(
r
,
c
1
,
c
2
)
c=(r,c_1,c_2)
or
c
=
(
r
,
c
1
,
χ
)
c=(r,c_1,\chi )
with
r
r
the rank,
c
1
,
c
2
c_1,c_2
the Chern classes, and
χ
\chi
the holomorphic Euler characteristic. We also write
M
H
X
(
2
,
c
1
,
c
2
)
=
M
X
X
(
c
1
,
d
)
M_H^X(2,c_1,c_2)=M_X^X(c_1,d)
, with
d
=
4
c
2
−
c
1
2
d=4c_2-c_1^2
. The
K
K
-theoretic Donaldson invariants are the holomorphic Euler characteristics
χ
(
M
H
X
(
c
1
,
d
)
,
μ
(
L
)
)
\chi (M_H^X(c_1,d),\mu (L))
, where
μ
(
L
)
\mu (L)
is the determinant line bundle associated to a line bundle on
X
X
. More generally for suitable classes
c
∗
∈
K
(
X
)
c^*\in K(X)
there is a determinant line bundle
D
c
,
c
∗
\mathcal {D}_{c,c^*}
on
M
H
X
(
c
)
M^X_H(c)
. We first compute some generating functions for
K
K
-theoretic Donaldson invariants on
P
2
\mathbb {P}^2
and rational ruled surfaces, using the wallcrossing formula of [Pure Appl. Math. Q. 5 (2009), pp. 1029–1111].
Then we show that Le Potier’s strange duality conjecture relating
H
0
(
M
H
X
(
c
)
,
D
c
,
c
∗
)
H^0(M^X_H(c),\mathcal {D}_{c,c^*})
and
H
0
(
M
H
X
(
c
∗
)
,
D
c
∗
,
c
)
H^0(M^X_H(c^*),\mathcal {D}_{c^*,c})
holds for the cases
c
=
(
2
,
c
1
=
0
,
c
2
>
2
)
c=(2,c_1=0,c_2>2)
and
c
∗
=
(
0
,
L
,
χ
=
0
)
c^{*}=(0,L,\chi =0)
with
L
=
−
K
X
L=-K_X
on
P
2
\mathbb {P}^2
, and
L
=
−
K
X
L=-K_X
or
−
K
X
+
F
-K_X+F
on
P
1
×
P
1
\mathbb {P}^1\times \mathbb {P}^1
and
P
2
^
\widehat {\mathbb {P}^2}
with
F
F
the fibre class of the ruling, and also the case
c
=
(
2
,
H
,
c
2
)
c=(2,H,c_2)
and
c
∗
=
(
0
,
2
H
,
χ
=
−
1
)
c^*=(0,2H,\chi =-1)
on
P
2
\mathbb {P}^2
.