The problem of the existence and construction of a table of osculating rational functions
r
1
,
m
{r_{1,m}}
for
1
,
m
⩾
0
1,m \geqslant 0
is considered. First, a survey is given of some results from the theory of osculatory rational interpolation of order
s
i
−
1
{s_i} - 1
at points
x
i
{x_i}
for
i
⩾
0
i \geqslant 0
. Using these results, we prove the existence of continued fractions of the form
\[
c
0
+
c
1
⋅
(
x
−
y
0
)
+
…
+
c
k
⋅
(
x
−
y
0
)
…
(
x
−
y
k
−
1
)
+
c
k
+
1
⋅
(
x
−
y
0
)
…
(
x
−
y
k
)
1
+
c
k
+
2
⋅
(
x
−
y
k
+
1
)
1
+
c
k
+
3
⋅
(
x
−
y
k
+
2
)
1
+
…
,
{c_0} + {c_1} \cdot (x - {y_0}) + \ldots + {c_k} \cdot (x - {y_0}) \ldots (x - {y_{k - 1}}) + \frac {{{c_{k + 1}} \cdot (x - {y_0}) \ldots (x - {y_k})}}{1} + \frac {{{c_{k + 2}} \cdot (x - {y_{k + 1}})}}{1} + \frac {{{c_{k + 3}} \cdot (x - {y_{k + 2}})}}{1} + \ldots ,
\]
with the
y
k
{y_k}
suitably selected from among the
x
i
{x_i}
, whose convergents form the elements
r
k
,
0
,
r
k
+
1
,
0
,
r
k
+
1
,
1
,
r
k
+
2
,
1
,
…
{r_{k,0}},{r_{k + 1,0}},{r_{k + 1,1}},{r_{k + 2,1}}, \ldots
of the table. The properties of these continued fractions make it possible to derive an algorithm for constructing their coefficients
c
i
{c_i}
for
i
⩾
0
i \geqslant 0
. This algorithm is a generalization of the qd-algorithm.