Meromorphic extension of analytic continued fractions across their divergence line with applications to orthogonal polynomials
Authors: Runckel Hans-J. 
Published: June 2012

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-parenthesis minus a Subscript n Baseline left-parenthesis z right-parenthesis slash left-parenthesis lamda left-parenthesis z right-parenthesis plus b Subscript n Baseline left-parenthesis z right-parenthesis right-parenthesis right-parenthesis">

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.