Journal of Computational and Applied Mathematics 153 (2003) 79–88
www.elsevier.com/locate/cam

Szegő polynomials: some relations to L-orthogonal
and orthogonal polynomials�

C.F. Bracciali∗, A.P. da Silva, A. Sri Ranga
Departamento de Ciências de Computac�ão e Estat��stica, Instituto de Biociências, Letras e Ciências Exatas,

Universidade Estadual Paulista (UNESP), Rua Crist�ovão Colombo 2265,
15054-000 São Jos�e do Rio Preto, SP, Brazil

Received 5 November 2001; received in revised form 10 May 2002

Abstract

We consider the real Szegő polynomials and obtain some relations to certain self inversive orthogonal
L-polynomials de5ned on the unit circle and corresponding symmetric orthogonal polynomials on real intervals.
We also consider the polynomials obtained when the coe6cients in the recurrence relations satis5ed by the
self inversive orthogonal L-polynomials are rotated.
c© 2002 Elsevier Science B.V. All rights reserved.

MSC: 33C45; 40A15; 42C05

Keywords: Szegő polynomials; Chain sequences; Three term recurrence relations

1. Introduction

Let d�(z) be a positive measure on the unit circle C. This means �(ei�) is a real, bounded and
non-decreasing function for 06 �6 2�. We consider the Szegő polynomials {Sn} associated with
the measure d�(z) de5ned by

∫
C Sn(z)Sm(z) d�(z)=0, n �= m. These polynomials were introduced by

Szegő (see for example [8]). For a good source for some basic information on these polynomials
we refer to [10].
Since @z=1=z on the unit circle, the polynomials Sn can also be de5ned by

∫
C z−n+sSn(z)z d�(z)=0,

06 s6 n − 1. Hence, the Szegő polynomials also satisfy the L-orthogonality property on the unit

� This research was supported by grants from CNPq and FAPESP of Brazil.
∗ Corresponding author.
E-mail address: cleonice@dcce.ibilce.unesp.br (C.F. Bracciali).

0377-0427/03/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.
PII: S0377-0427(02)00605-2

mailto:cleonice@dcce.ibilce.unesp.br


80 C.F. Bracciali et al. / Journal of Computational and Applied Mathematics 153 (2003) 79–88

circle in relation to z d�(z). Polynomials satisfying the L-orthogonality property on the positive real
axis were introduced in [5]. For a study of these polynomials on the unit circle see for example [4].
The Szegő polynomials (given here in their monic form) are known to satisfy the system of

recurrence relations

Sn+1(z) = zSn(z) + an+1S∗
n (z);

(1− |an+1|2)zSn(z) = Sn+1(z)− an+1S∗
n+1(z) (1.1)

for n¿ 0. Here S∗
n (z) = zn @Sn(1=z) are the reciprocal polynomials. The numbers an = Sn(0), n¿ 1,

which are less than one in modulus, are known as the reHection coe6cients of the Szegő polynomials.
In this manuscript, we consider the Szegő polynomials with real reHection coe6cients and take

a look at the polynomials {Sn(z) + S∗
n (z)}, {Sn(z)− S∗

n (z)} and their relations to certain symmetric
orthogonal polynomials on the interval [− 1; 1]. These relations, found in [3], were very nicely ex-
plored in [12]. Zhedanov uses the information contained in the relations associated with Sn(z)+S∗

n (z)
(or Sn(z)−S∗

n (z)) to derive information about Sn from the corresponding orthogonal polynomials and
vice versa. However, diJerent to Zhedanov, we look at how one can use simultaneously the relations
associated with both Sn(z) + S∗

n (z) and Sn(z)− S∗
n (z) to do the same. We also give information on

the polynomials obtained when rotating the coe6cients of the three term recurrence relation satis5ed
by the polynomials Sn(z) + S∗

n (z).

2. The para-orthogonal polynomials

In [4], Jones et al. considered the polynomials Sn(z)+!nS∗
n (z), where |!n|=1. They called these

para-orthogonal polynomials and showed that their zeros are all distinct and lie on the unit circle.
Their proof is based on the self-inversive properties of these polynomials and the conditions∫

C
z−n+s[Sn(z) + !nS∗

n (z)] d�(z) = 0; 16 s6 n− 1: (2.1)

Here, restricting our selves to only real Szegő polynomials, we consider the two special cases of
para-orthogonal polynomials

S(1)n (z) =
Sn(z) + S∗

n (z)
1 + Sn(0)

and S(2)n (z) =
Sn(z)− S∗

n (z)
1− Sn(0)

; n¿ 1:

The denominators are chosen in order to make the polynomials monic.
The Szegő polynomials are real if and only if the measure d�(z) satis5es the symmetry d�(1=z)=

−d�(z). The following results up to the recurrence relations in Theorem 2.1 were 5rst given in [3].
From (1.1), the polynomials S(i)n , i=1; 2, can be shown to satisfy the simple three-term recurrence

relations

S(1)n+1(z) = (z + 1)S
(1)
n (z)− (1 + an−1)(1− an)zS

(1)
n−1(z);

S(2)n+1(z) = (z + 1)S
(2)
n (z)− (1− an−1)(1 + an)zS

(2)
n−1(z)

n¿ 1

with a0 = 1, S
(1)
0 (z) = 1, S

(2)
0 (z) = 1, S

(1)
1 (z) = z + 1 and S(2)1 (z) = z − 1.



C.F. Bracciali et al. / Journal of Computational and Applied Mathematics 153 (2003) 79–88 81

Observe from the recurrence relation that S(2)n (1) = 0 for n¿ 1. Thus, by letting R(1)n (z) = S(1)n (z),
n¿ 0 and R(2)n (z)=(z−1)−1S(2)n+1(z), n¿ 0, we obtain 2Sn(z)=(1+an)R

(1)
n (z)+(1−an)(z−1)R(2)n−1(z);

n¿ 1; or equivalently,

2zSn−1(z) = R(1)n (z) + (z − 1)R(2)n−1(z); n¿ 1: (2.2)

Theorem 2.1. The monic polynomials R(i)n , i = 1; 2, satisfy R(i)0 = 1, R
(i)
1 (z) = z + 1 and

R(i)n+1(z) = (z + 1)R
(i)
n (z)− 4�(i)n+1zR(i)n−1(z); n¿ 1;

with 4�(1)n+1 = (1 + an−1)(1− an)¿ 0 and 4�(2)n+1 = (1− an)(1 + an+1)¿ 0, n¿ 1.
Moreover, these polynomials satisfy the L-orthogonality relations∫

C
z−n+sR(1)n (z)

z
z − 1 d�(z) = 0; 06 s6 n− 1 (2.3)

and ∫
C
z−n+sR(2)n (z)(z − 1) d�(z) = 0; 06 s6 n− 1: (2.4)

Proof. The recurrence relations follows from above. The reason for choosing the multiplier 4 in the
recurrence relation will become apparent after Theorem 3.1. The recurrence relations also con5rm
the self inversive property znR(i)n (1=z) = R(i)n (z).
Now we give the proof of (2.4). Since

R(2)n (z) =
Sn+1(z)− S∗

n+1(z)
(z − 1)(1− Sn+1(0))

;

from (2.1)∫
C
z−(n+1)+sR(2)n (z)(z − 1) d�(z) = 0; 16 s6 n:

Clearly, this is equivalent to (2.4). Now to prove (2.3), again from (2.1)∫
C
z−n+sR(1)n (z)

z − 1
z − 1 d�(z) = 0; 16 s6 n− 1: (2.5)

This is equivalent to∫
C
z−n[Pn;s(z)]R(1)n (z)

z
z − 1 d�(z) = 0; 16 s6 n− 1; (2.6)

where Pn;s(z) = (z− 1)zs−1, s=1; 2; : : : ; n− 1. We now show that if we take Pn;n(z) = z+ zn−1 then
the relation (2.6) also holds for s= n. From (2.5),∫

C
z−n+sR(1)n (z)

z
z − 1 d�(z) =

∫
C
z−n+sR(1)n (z)

1
z − 1 d�(z); 16 s6 n− 1:



82 C.F. Bracciali et al. / Journal of Computational and Applied Mathematics 153 (2003) 79–88

Substituting z by 1=z on the right-hand side and using the symmetry of the measure and the self
inversive property of R(1)n , we obtain∫

C
z−n[zs + zn−s]R(1)n (z)

z
z − 1 d�(z) = 0; 16 s6 n− 1:

Hence, Pn;n(z) is obtained when letting s= 1 in the expression zs + zn−s.
One can verify that the set of polynomials Pn;s, s=1; 2; : : : ; n, of degree 6 n− 1, forms a linearly

independent set. In particular, for each of the monomials zs, 06 s6 n− 1, we get a unique linear
combination of the type zs = d(s)1 Pn;1(z) + d(s)2 Pn;2(z) + · · ·+ d(s)n−1Pn;n−1(z) + 1

2 Pn;n(z). Consequently
from (2.6) the result (2.3) of the theorem follows.

3. Relation to orthogonal polynomials on the real line

The results in the previous section lead to consider what one can say about a sequence of poly-
nomials {Rn} satisfying the recurrence relation

Rn+1(z) = (z + 1)Rn(z)− 4�n+1zRn−1(z); n¿ 1 (3.1)

with R0(z) = 1, R1(z) = z + 1 and �n+1¿ 0. We have the following theorem:

Theorem 3.1. Let {Rn} be the sequence of monic polynomials generated by the recurrence relation
(3.1). Then the zeros of Rn are distinct (except for a possible double zero at z = 1) and lie on
C ∪ (0;∞). In particular, if {�n+1} is a chain sequence then all the zeros are distinct and lie on
the open unit circle {z : z = ei�; 0¡�¡ 2�}. In this case, there exists a positive measure d�(z)
on the unit circle such that∫

C
z−n+sRn(z)

z
z − 1 d�(z) = 0; 06 s6 n− 1: (3.2)

Proof. From the recurrence relation (3.1), znRn(1=z) = Rn(z). Let x = x(z) = 1
2(z

1=2 + z−1=2). Here,
given z = rei� then z1=2 is understood as r1=2ei�=2. Hence, the polynomials Pn(x) = (4z)−n=2Rn(z),
n¿ 0, satisfy P0(x) = 1, P1(x) = x and

Pn+1(x) = xPn(x)− �n+1Pn−1(x); n¿ 1: (3.3)

From this it is well known that the zeros of Pn are real, distinct and lie symmetrically about the
origin. If we write, P2n(x) =

∏n
k=1 (x

2 − x22n;k), and P2n+1(x) = x
∏n

k=1 (x
2 − x22n+1; k), then from

Rn(z) = (4z)n=2Pn(x(z)) we see that

R2n(z) =
n∏

k=1

((z − z2n;k)(z − 1=z2n;k));

R2n+1(z) = (z + 1)
n∏

k=1

((z − z2n+1; k)(z − 1=z2n+1; k));

where zn;k = (2x2n;k − 1) + 2
√

x2n;k(x
2
n;k − 1). Hence, if x2n;k ¡ 1 then zn;k and 1=zn;k are a conjugate

pair of zeros of Rn on the unit circle and if x2n; i ¿ 1 then they are two positive zeros of Rn inverse



C.F. Bracciali et al. / Journal of Computational and Applied Mathematics 153 (2003) 79–88 83

to each other. If z = 1 is a zero of Rn then it is a zero of multiplicity 2. Hence, we conclude that
the zeros of Rn are either on the unit circle or on the positive real line.
Now if we assume {�n+1} to be a chain sequence. That is, there exists a second sequence {gn},

where 06 g0¡ 1 and 0¡gn ¡ 1, n¿ 1, such that �n+1 = (1 − gn−1)gn, n¿ 1. Then, it is well
known (see for example [2]) that all the zeros of Pn are inside (−1; 1). Hence, in this case, all the
zeros of Rn are on the open unit circle {z: z = ei�; 0¡�¡ 2�}.
When the zeros of Pn, n¿ 1, are within (−1; 1) then from Favard’s Theorem it follows that

these polynomials form a sequence of orthogonal polynomials in relation to a symmetric positive
measure d� with support inside [− 1; 1]. From the binomial expansion and the symmetry of Pn this
is equivalent to (see also [7])∫ 1

−1
{x + i

√
1− x2}−(n−1)+2s Pn(x)√

1− x2
d�(x) = 0; 06 s6 n− 1:

Letting x = 1
2(z

1=2 + z−1=2), that is letting z1=2 = x + i
√
1− x2, we obtain∫

−C
z−n+sRn(z)

z
z − 1 d�(x(z)) = 0; 06 s6 n− 1:

Since x(z) is a decreasing function of z = ei� as � varies from 0 to 2�, we obtain the positive
measure d�(z) =−d�(x(z)).

In the recurrence relations of R(1)n and R(2)n , the coe6cients {�(1)n+1} and {�(2)n+1} are chain sequences
with the respective parameter sequences {g(1)n = (1− an)=2} and {g(2)n = (1 + an+1)=2}. That is,

(1− g(1)n−1)g
(1)
n = �(1)n+1 [also g(1)n (1− g(1)n+1) = �(2)n+1];

(1− g(2)n−1)g
(2)
n = �(2)n+1 [also g(2)n−2(1− g(2)n−1) = �(1)n+1] (3.4)

for n¿ 1, with initial parameters g(1)0 = 0 and 0¡g(2)0 = 1− �(1)2 = (1+ a1)=2¡ 1. Here we have let
g(2)−1=1. The polynomials P(1)n (x)=(4z)−n=2R(1)n (z) and P(2)n (x)=(4z)−n=2R(2)n (z) satisfy the recurrence
relations

P(i)n+1(x) = xP(i)n (x)− �(i)n+1P
(i)
n−1(x); n¿ 1 (3.5)

and are orthogonal polynomials on [−1; 1] related to the measures d�(1)(x)=−d�(z) and d�(2)(x)=
−(1− x2) d�(z), respectively.
Earlier investigations (see [1]) have shown us that given two positive measures d�(1)(x) and

d�(2)(x) on the real line such that d�(2)(x) = (1 − q2x2) d�(1)(x), then there exists a sequence of
real numbers {‘n}, with ‘0 = 1, such that for the associated monic orthogonal polynomials P(1)n and
P(2)n the following hold:

P(1)n+1(x) = xP(1)n (x)− 1
4q2 (1− ‘n)(1 + ‘n−1)P

(1)
n−1(x);

P(2)n+1(x) = xP(2)n (x)− 1
4q2 (1− ‘n)(1 + ‘n+1)P

(2)
n−1(x);

P(1)n+1(x) = P(2)n+1(x)− 1
4q2 (1− ‘n)(1− ‘n+1)P

(2)
n−1(x)



84 C.F. Bracciali et al. / Journal of Computational and Applied Mathematics 153 (2003) 79–88

for n¿ 1. Here q2, diJerent from zero, can take any appropriate value including negative. Clearly,
if q = 0 then the two sets of polynomials are the same. The choice q = 1, so that the measures
have their support within [ − 1; 1], gives results (3.5) with an = ‘n, n¿ 0. Hence we can state the
following theorem.

Theorem 3.2. Let d�(1) and d�(2) be two positive measures on [ − 1; 1] such that d�(2)(x) =
(1 − x2) d�(1)(x). Let the respective monic orthogonal polynomials P(1)n and P(2)n associated with
these measures satisfy P(i)n+1(x) = xP(i)n (x)− �(i)n+1P

(i)
n−1(x), n¿ 1. Let

2zSn−1(z) = R(1)n (z) + (z − 1)R(2)n−1(z); n¿ 1;

where R(1)n (z)= (4z)n=2P
(1)
n (x(z)) and R(2)n (z)= (4z)n=2P

(2)
n (x(z)). Then Sn are the monic Szegő poly-

nomials associated with d�(z) = −d�(1)(x(z)). Furthermore, the reBection coeCcients an = Sn(0)
can be generated by

an = 1− 4�(1)n+1=(1 + an−1) and an+1 =−1 + 4�(2)n+1=(1− an); n¿ 1

with a0 = 1. Given explicitly (with �(i)0 , i = 1; 2, as the respective moments of order zero),

a2n−1 = 2
�(2)2n−1�

(2)
2n−3 · · · �(2)3 �(2)0

�(1)2n−1�
(1)
2n−3 · · · �(1)3 �(1)0

− 1 and a2n = 2
�(2)2n �

(2)
2n−2 · · · �(2)2

�(1)2n �
(1)
2n−2 · · · �(1)2

− 1; n¿ 1:

4. Examples of Szegő polynomials

We consider some examples to show how the above results can be used to obtain information
about real Szegő polynomials.
1. Gegenbauer–Szegő polynomials: We consider d�(1)(x) = (1 − x2)�−1=2 dx and d�(2)(x) =

(1− x2)�+1=2 dx in [− 1; 1], where �¿− 1=2. Then P(1)n = P(�)n and P(2)n = P(�+1)n are the respective
monic Gegenbauer polynomials. Hence,

�(1)n+1 =
n(n+ 2�− 1)

4(n+ �)(n+ �− 1) and �(2)n+1 =
n(n+ 2�+ 1)

4(n+ �+ 1)(n+ �)
; n¿ 1:

The polynomials S(�)n de5ned by

2zS(�)n (z) = R(1; �)n+1 (z) + (z − 1)R(2; �)n (z); n¿ 0;

where R(1; �)n (z) = (4z)n=2P(�)n (x(z)) and R(2; �)n (z) = (4z)n=2P(�+1)n (x(z)), are the well known monic
Gegenbauer–Szegő polynomials (see for example [10]) associated with the measure

d�(z) =
[
(z − 1)2
−4z

]� dz
2iz

=
1
2
|sin(�=2)|2� d�

with z = ei�. The reHection coe6cients are a(�)n = �=(n+ �), n¿ 1.
When � = 0, then a(0)n = 0 and we have the results associated with the Chebyshev polynomials.

In this case, we have R(1;0)n (z) = zn + 1 and R(2;0)n (z) = (zn+1 − 1)=(z − 1). Thus, we obtain S(0)n (z)=



C.F. Bracciali et al. / Journal of Computational and Applied Mathematics 153 (2003) 79–88 85

(1=2z)[R(1;0)n+1 (z) + (z − 1)R(2;0)n (z)] = zn, which are the monic Szegő polynomials associated with the
Lebesgue measure d�(z) =−d�(1)(x(z)) = (2iz)−1dz.
2. Koornwinder–Szegő polynomials: For �¿ − 3=2, M¿ 0 and 0¡ 6 1, let d�(1)(x), de5ned

on [− 1; 1], be such that
∫ 1

−1
f(x) d�(1)(x) =

1
2M + 1

{
M [f(−1) + f(1)] + c−1

∫  

− 
f(x)

( 2 − x2)�+1=2

1− x2
dx

}
:

Here

c =
∫  

− 

( 2 − x2)�+1=2

1− x2
dx

is such that the total mass of the measure d�(1) is 1 (a probability measure). Hence, the orthogonal
polynomials P(2)n associated with the measure d�(2)(x) = [(2M + 1)c]−1( 2 − x2)�+1=2 dx are the
Gegenbauer polynomials of parameter (�+1) scaled downed to the interval [− ;  ]. Hence, P(2)n+1(x)=
xP(2)n (x)− �(2)n+1P

(2)
n−1(x), n¿ 1, where

�(2)n+1 =
 2

4
n(n+ 2�+ 1)

(n+ �+ 1)(n+ �)
:

When  = 1, in this case � should be greater than − 1
2 , then P(1)n are the symmetric Koornwinder

polynomials [6].
Let us denote by S(�;M; )

n the Szegő polynomials associated with the measure d�(�, M;  ; z) =
−d�(1)(x(z)), which can be given by

∫ 2�

0
f(ei�) d�(�;M;  ; ei�) =

1
2M + 1

{
2Mf(1) +

c(f; �;  )
c(1; �;  )

}
;

where c(f; �;  )=
∫ 2�−�( )
�( ) f(ei�)[ 2− cos2(�=2)]�+1=2[sin(�=2)]−1 d�, with �(x)=2 arccos(x). For the

reHection coe6cients S(�;M; )
n (0) we then have {gn(�;M;  ) = [1 + S(�;M; )

n+1 (0)]=2}∞n=0 is a parameter
sequence of the chain sequence {�(2)n+1}. That is

[1− gn−1(�;M;  )]gn(�;M;  ) =
 2

4
n(n+ 2�+ 1)

(n+ �+ 1)(n+ �)
; n¿ 1:

Since g0(�;M;  ) = 1− �(1)2 , the initial parameter takes the value

g0(�;M;  ) =
1

(2M + 1)c(1; �;  )

∫ 2�

0
[ sin (�=2)]2�+2 d�:

Clearly, when M → ∞ then {gn(�;M;  )} moves towards the minimal parameter sequence and if
M = 0 then one can also show that gn(�;M;  ) represents the maximal parameter sequence.
3. Szegő polynomials associated with twin periodic recurrence relations: It was shown in [1] that

given the recurrence relation (3.3) with �2 = p�0, �2n−1 = �1 and �2n = �0, n¿ 2, where p, �0, �1
are all positive, then the polynomials Pn are the monic orthogonal polynomials in relation to the



86 C.F. Bracciali et al. / Journal of Computational and Applied Mathematics 153 (2003) 79–88

probability measure d� given by

�(x) =
$(�1 − �0)

A(p)
U (x) +

1
2�A(p)

∫ x

−∞

√
b2 − t2

√
t2 − a2

|t|(1 + q(p)t2)
I(t) dt

+
$((p− 1)2�0 − �1)
2(p− 1)A(p) [U (x + *(p)) + U (x − *(p))];

where b=
√
�0 +

√
�1, a= |√�0−√

�1|, A(p)= (p− 1)�0 + �1, q(p)=−1=[*(p)]2 = (1−p)=pA(p),
$(x) = xU (x), I(x) = U (x + b)− U (x + a) + U (x − a)− U (x − b) and the function U (x) is equal
to 0 for x¡ 0 and is equal to 1 for x¿ 0.
Clearly, the two jumps at ±*(p) have eJect only if p¿ 1+

√
�1=�0 or, when possible, 0¡p¡ 1−√

�1=�0. If p¿ 1 +
√

�1=�0 then [*(p)]2¿ b2. Hence, taking b6 1 and choosing the value of p
such that [*(p)]2 = 1 we obtain the following results. Let �0¿ 0, �1¿ 0 and let

b=
√
�0 +

√
�16 1; a= |√�0 −√

�1| and B(�0; �1) =
√
1− b2

√
1− a2:

Then, with

p= p(�0; �1) =
1 + (�0 − �1) + B(�0; �1)

2�0
;

the monic polynomials P(1)n generated by the recurrence relation P(1)n+1(x) = xP(1)n (x) − �(1)n+1P
(1)
n−1(x),

n¿ 1, where �(1)2 = p(�0; �1)�0, �
(1)
2n−1 = �1, �

(1)
2n = �0, n¿ 2, P(1)0 (x) = 1 and P(1)1 (x) = x, are the

orthogonal polynomials in relation to the measure d�(1) given by∫ 1

−1
f(x)d�(1)(x) =

B(�0; �1)
2A(�0; �1)

f(−1) + 1
2�A(�0; �1)

∫ −a

−b
f(x)

√
b2 − x2

√
x2 − a2

|x|(1− x2)
dx

+
$(�1 − �0)
A(�0; �1)

f(0) +
1

2�A(�0; �1)

∫ b

a
f(x)

√
b2 − x2

√
x2 − a2

|x|(1− x2)
dx

+
B(�0; �1)
2A(�0; �1)

f(1):

Here A(�0; �1)=A(p)=[1− (�0−�1)+B(�0; �1)]=2. It is easily veri5ed that, since p=p(�0; �1), the
sequence {an} obtained from the relation an = 1 − 4�(1)n+1=(1 + an−1), n¿ 1, where a0 = 1, satis5es
a2n−1 =−(�0 − �1)− B(�0; �1) and a2n = (�0 − �1)− B(�0; �1), for n¿ 1.
Consider the probability measure d�(z) =−d�(1)(x(z)) on C which can be given by∫ 2�

0
f(ei�)d�(ei�) =

B(�0; �1)
A(�0; �1)

f(1)+
1

2�A(�0; �1)

∫ �(a)

�(b)
f(ei�)

√
b2−cos2(�=2)

√
cos2(�=2)−a2

|sin(�)| d�

+
$(�1 − �0)
A(�0; �1)

f(−1)

+
1

2�A(�0; �1)

∫ 2�−�(b)

2�−�(a)
f(ei�)

√
b2 − cos2(�=2)

√
cos2(�=2)− a2

|sin(�)| d�:



C.F. Bracciali et al. / Journal of Computational and Applied Mathematics 153 (2003) 79–88 87

Here �(x) = 2 arccos(x). Then the Szegő polynomials S(�0 ; �1)n associated with d�(z) satisfy

S(�0 ; �1)2n−1 (0) =−B(�0; �1)− (�0 − �1) and S(�0 ; �1)2n (0) =−B(�0; �1) + (�0 − �1); n¿ 1:

If �1¿�0 there is a jump in the measure at the point z=−1. Note that √�0+
√
�1 can not exceed

1 and if
√
�0 +

√
�1¡ 1 then there is also a jump at z=1. This jump vanishes when

√
�0 +

√
�1 =1.

5. Rotating the coe%cients

Let {Rn} be the sequence of monic polynomials generated by the recurrence relation (3.1). Hence
by Theorem 3.1, there exists a positive measure d� on the unit circle such that (3.2) holds. We
now give some information regarding the sequence of monic polynomials {Rn( ; z)} given by the
recurrence relation

Rn+1( ; z) = (z + 1)Rn( ; z)− 4�n+1e2i zRn−1( ; z); n¿ 1:

For 5xed  such that 0¡ ¡�, let ,( ) represent the curve (path)

,( ) ≡ {z = z(t) = u( ; t) + iv( ; t): t2( )¿ t¿ t1( )};
where

u( ; t) = t
(t − 1)2 cot2( )− (t + 1)2
(t − 1)2 cot2( ) + (t + 1)2 and v( ; t) =

2t(t2 − 1) cot( )
(t − 1)2 cot2( ) + (t + 1)2 :

Here, t1( )¡t2( ), such that t1( )t2( ) = 1, are the two positive solutions of

(t − 1)2[cot2( ) + 1]
(t − 1)2 cot2( ) + (t + 1)2 =

4t
(t + 1)2

:

Note that ,(�=2) is the real interval from −3− 2√2 to −3+ 2√2. We also let ,(0) to be the unit
circle.

Theorem 5.1. The zeros of Rn( ; z) are distinct and lie on the curve ,( ). Furthermore, if −d�(w(z))
= d�(z) and d�( ; z) =−d�(w(z)e−i ), where 2w(z) = z1=2 + z−1=2, then∫

,( )
z−n+sRn( ; z)

z
z − 1 d�( ; z) = 0; 06 s6 n− 1:

Proof. We start with the polynomials Pn( ; w)=(4z)−n=2Rn( ; z), which satisfy Pn+1( ; w)=wPn( ; w)−
�n+1e2i Pn−1( ; w), n¿ 1. Clearly, they satisfy the orthogonality property∫

L( )
wsPn( ; w)d�(we−i ) = 0; 06 s6 n− 1;

where L( ) is the path represented by the straight line from −ei to ei . Equivalently,∫
L( )

{w + i
√
1− w2}−(n−1)+2s Pn( ; w)√

1− w2
d�(we−i ) = 0; 06 s6 n− 1:

This gives the required result as the line represented by L( ) is the image of the curve represented
by ,( ) under the transformation 2w(z) = z1=2 + z−1=2.



88 C.F. Bracciali et al. / Journal of Computational and Applied Mathematics 153 (2003) 79–88

These results, when  = �=2, lead to information given in [11].
As an example we consider the monic polynomials R(�)n ( ; z) given by

R(�)n+1( ; z) = (z + 1)R
(�)
n ( ; z)− 4�(�)n+1e

2i zR(�)n−1( ; z); n¿ 1;

where

�(�)n+1 =
n(n+ 2�− 1)

4(n+ �)(n+ �− 1) :

Note that the monic Gegenbauer polynomials are P(�)n (w) = (4z)−n=2R(�)n (0; z). We obtain from the
above theorem∫

,( )
z−n+sR(�)n ( ; z)z

−�[(b( ) − z)(z − 1=b( ))]�−1=2 dz = 0; 06 s6 n− 1;

where b( ) = (2e2i − 1) + 2ei √e2i − 1. When �= 1 and  = �=2 we obtain the polynomials which
are the denominator polynomials of the classical positive T-fraction (see [9])

z
z + 1 +

z
z + 1 +

z
z + 1 +

z
z + 1 +

· · · :

References

[1] A.C. Berti, A. Sri Ranga, Companion orthogonal polynomials: some applications, Appl. Numer. Math. 39 (2001)
127–149.

[2] T.S. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and its Applications Series, Gordon and
Breach, New York, 1978.

[3] P. Delsarte, Y. Genin, The split Levinson algorithm, IEEE Trans. Acoust. Speech Signal Process. 34 (1986)
470–478.

[4] W.B. Jones, O. NjRastad, W.J. Thron, Moment theory, orthogonal polynomials, quadrature, and continued fractions
associated with the unit circle, Bull. London Math. Soc. 21 (1989) 113–152.

[5] W.B. Jones, W.J. Thron, H. Waadeland, A strong Stieltjes moment problem, Trans. Amer. Math. Soc. 261 (1980)
503–528.

[6] T.H. Koornwinder, Orthogonal polynomials with weight function (1− x)�(1 + x)2 +M3(x+ 1) + N3(x− 1), Canad.
Math. Bull. 27 (1984) 205–214.

[7] A. Sri Ranga, Symmetric orthogonal polynomials and the associated orthogonal L-polynomials, Proc. Amer. Math.
Soc. 123 (1995) 3135–3141.

[8] G. Szegő, Orthogonal Polynomials, 4th Edition, in: American Mathematical Society Colloquium Publications, Vol.
23, American Mathematical Society, Providence, RI, 1975.

[9] W.J. Thron, Some properties of continued fractions 1 + d0 + K(z=(1 + dnz)), Bull. Amer. Math. Soc. 54 (1948)
206–218.

[10] W. Van Assche, Orthogonal polynomials in the complex plane and on the real line, in: M.E.H. Ismail, et al., (Eds.),
Field Institute Communications 14: Special Functions, q-Series and Related Topics, American Mathematical Society,
Providence, RI, 1997, pp. 211–245.

[11] L. Vinet, A. Zhedanov, Szegő polynomials on the real axis, Integral Transforms Spec. Functions 8 (1999) 149–164.
[12] A. Zhedanov, On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric

orthogonal polynomials on an interval, J. Approx. Theory 94 (1998) 73–106.


	Szego polynomials: some relations to L-orthogonaland orthogonal polynomials
	Introduction
	The para-orthogonal polynomials
	Relation to orthogonal polynomials on the real line
	Examples of Szego polynomials
	Rotating the coefficients
	References