Let
a
1
,
a
2
,
…
,
a
p
{a_1},{a_2}, \ldots ,{a_p}
be given real numbers ordered by size, and let
[
α
,
β
]
[\alpha ,\beta ]
be a real interval disjoint from the set
{
a
1
,
a
2
,
…
,
a
p
}
\{ {a_1},{a_2}, \ldots ,{a_p}\}
. Let
{
c
j
(
i
)
:
j
=
1
,
2
,
…
}
\{ c_j^{(i)}:j = 1,2, \ldots \}
, be sequences of real numbers and
c
0
{c_0}
be a real number. The extended Stieltjes moment problem is to find a distribution function
ψ
\psi
with all its points of increase in
[
α
,
β
]
[\alpha ,\beta ]
such that
\[
∫
α
β
d
ψ
(
t
)
=
c
0
,
∫
α
β
d
ψ
(
t
)
(
t
−
a
i
)
j
=
c
j
(
i
)
,
i
=
1
,
…
,
p
,
j
=
1
,
2
,
…
.
\int _\alpha ^\beta {d\psi (t) = {c_0},\quad \int _\alpha ^\beta {\frac {{d\psi (t)}}{{{{(t - {a_i})}^j}}} = c_j^{(i)},\quad i = 1, \ldots ,p,\;j = 1,2, \ldots .} }
\]
Necessary and sufficient conditions for the existence of a unique solution of the problem are given. Orthogonal
R
R
-functions and Gaussian quadrature formulas play important roles in the proof.