For the limit periodic
J
J
-fraction
K
(
−
a
n
/
(
λ
+
b
n
)
)
K( - {a_n}/(\lambda + {b_n}))
,
a
n
{a_n}
,
b
n
∈
C
{b_n} \in \mathbb {C}
,
n
∈
N
n \in \mathbb {N}
, which is normalized such that it converges and represents a meromorphic function
f
(
λ
)
f(\lambda )
on
C
∗
:=
C
∖
[
−
1
,
1
]
{\mathbb {C}^{\ast } }: = \mathbb {C}\backslash [ - 1,1]
, the numerators
A
n
{A_n}
and denominators
B
n
{B_n}
of its
n
n
th approximant are explicitly determined for all
n
∈
N
n \in \mathbb {N}
. Under natural conditions on the speed of convergence of
a
n
{a_n}
,
b
n
{b_n}
,
n
→
∞
n \to \infty
, the asymptotic behaviour of the orthogonal polynomials
B
n
{B_n}
,
A
n
+
1
{A_{n + 1}}
(of first and second kind) is investigated on
C
∗
{\mathbb {C}^{\ast } }
and
[
−
1
,
1
]
[ - 1,1]
. An explicit representation for
f
(
λ
)
f(\lambda )
yields continuous extension of
f
f
from
C
∗
{\mathbb {C}^{\ast } }
onto upper and lower boundary of the cut
(
−
1
,
1
)
( - 1,1)
. Using this and a determinant relation, which asymptotically connects both sequences
A
n
{A_n}
,
B
n
{B_n}
, one obtains nontrivial explicit formulas for the absolutely continuous part (weight function) of the distribution functions for the orthogonal polynomial sequences
B
n
{B_n}
,
A
n
+
1
{A_{n + 1}}
,
n
∈
N
n \in \mathbb {N}
. This leads to short proofs of results which generalize and supplement results obtained by P. G. Nevai [7]. Under a stronger condition the explicit representation for
f
(
λ
)
f(\lambda )
yields meromorphic extension of
f
f
from
C
∗
{\mathbb {C}^{\ast } }
across
(
−
1
,
1
)
( - 1,1)
onto a region of a second copy of
C
\mathbb {C}
which there is bounded by an ellipse, whose focal points
±
1
\pm 1
are first order algebraic branch points for
f
f
. Then, by substitution, analogous results on continuous and meromorphic extension are obtained for limit periodic continued fractions
K
(
−
a
n
(
z
)
/
(
λ
(
z
)
+
b
n
(
z
)
)
)
K( - {a_n}(z)/(\lambda (z) + {b_n}(z)))
, where
a
n
(
z
)
{a_n}(z)
,
b
n
(
z
)
{b_n}(z)
,
λ
(
z
)
\lambda (z)
are holomorphic on a region in
C
\mathbb {C}
. Finally, for
T
T
-fractions
T
(
z
)
=
K
(
−
c
n
z
/
(
1
+
d
n
z
)
)
T(z) = K( - {c_n}z/(1 + {d_n}z))
with
c
n
→
c
{c_n} \to c
,
d
n
→
d
{d_n} \to d
,
n
→
∞
n \to \infty
, the exact convergence regions are determined for all
c
c
,
d
∈
C
d \in \mathbb {C}
. Again, explicit representations for
T
(
z
)
T(z)
yield continuous and meromorphic extension results. For all
c
c
,
d
∈
C
d \in \mathbb {C}
the regions (on Riemann surfaces) onto which
T
(
z
)
T(z)
can be extended meromorphically, are described explicitly.