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A class of orthogonal functions given by a three term recurrence formula∗

C.F. Braccialia, J.H. McCabeb, T.E. Pérezc and A. Sri Rangaa

aDepartamento de Matemática Aplicada, IBILCE,

UNESP - Universidade Estadual Paulista

15054-000, São José do Rio Preto, SP, Brazil

bDepartment of Applied Mathematics, School of Mathematics,

University of St.Andrews, Scotland

cDepartamento de Matemática Aplicada,

Universidad de Granada, 18071 Granada, Spain

September 28, 2018

Abstract

The main goal in this manuscript is to present a class of functions satisfying a certain
orthogonality property for which there also exists a three term recurrence formula. This class
of functions, which can be considered as an extension to the class of symmetric orthogonal
polynomials on [−1, 1], has a complete connection to the orthogonal polynomials on the unit
circle. Quadrature rules and other properties based on the zeros of these functions are also
considered.

1 Introduction

Let Ωm be the linear space of “real” functions on [−1, 1] defined as follows.

Ω0 ≡ P0 and Ωm for m ≥ 1 is such that if F ∈ Ωm then F(x) = B(0)(x)+
√
1− x2B(1)(x),

where B(0)(x) ∈ Pm and B(1)(x) ∈ Pm−1 satisfy

B(0)(−x) = (−1)mB(0)(x) and B(1)(−x) = (−1)m−1B(1)(x).

Here, Pm represents the linear space of real polynomials of degree at most m.
This means, if F ∈ Ω2n then B(0) is an even polynomial of degree at most 2n and B(1) is an

odd polynomial of degree at most 2n−1. Likewise, if F ∈ Ω2n+1 then B
(0) is an odd polynomial

of degree at most 2n+ 1 and B(1) is an even polynomial of degree at most 2n.
Note that the dimension of Ωm is m+ 1. As an example of a basis for Ω2n we have

{1, x
√

1− x2, x2, . . . , x2n−1
√

1− x2, x2n}

and as an example of a basis for Ω2n+1 we can state

{
√

1− x2, x, x2
√

1− x2, . . . , x2n
√

1− x2, x2n+1}.
∗This work was initiated during the exchange program CAPES(Brazil)/DGU(Spain) of 2008-2012. For this

research the first and the fourth authors have also received support from CNPq and FAPESP of Brazil. The third
author’s research was also supported by grants from Micinn of Spain and Junta de Andalućıa.

http://arxiv.org/abs/1305.3239v1


Our interest in the studies of such functions are many. Apart from the interesting properties
such as three term recurrence formula, orthogonality and quadrature formulas that can be
associated with these functions as shown in this manuscript, solutions of the following differential
equations, with integer m, are also of these type of functions (see [4]).

(1− x2)F ′′(x)−
[
(2λ+ 1)x− 2η

√
1− x2

]
F ′(x) +m

[
m+ 2λ+

2mηx√
1− x2

]
F(x) = 0.

When η = 0, the solutions of the above differential equations are the ultraspherical polynomials
(see, for example, [2, 12]).

We will also describe in section 6 of this manuscript the connection that the functions
considered here have with orthogonal polynomials on the unit circle (OPUC).

For F(x) = B(0)(x) +
√
1− x2B(1)(x) ∈ Ωm, by setting

B(0)(x) =

⌊m/2⌋∑

j=0

b
(0)
2j x

m−2j and B(1)(x) =

⌊(m−1)/2⌋∑

j=0

b
(1)
2j x

m−1−2j ,

we say that the function F is of exact degree m if (b
(0)
0 )2+(b

(1)
0 )2 > 0. The nonnegative number

(b
(0)
0 )2+(b

(1)
0 )2 may be called the lead factor of F . The coefficients b

(0)
0 and b

(1)
0 may be referred

to as the first and second leading coefficients of F , respectively.
The specific aim of this manuscript is to consider some properties, in particular, the orthog-

onal properties, of the sequence of functions {Wm(x)}, where Wm(x) ∈ Ωm, given by

W0(x) = γ0, W1(x) = (γ1x− β1
√
1− x2)γ0,

Wm+1(x) =
[
γm+1x− βm+1

√
1− x2

]
Wm(x)− αm+1Wm−1(x), m ≥ 1.

(1.1)

Here, {αm}∞m=2, {βm}∞m=1 and {γm}∞m=0 are sequences of real numbers.

2 Functions in Ωm and self-inversive polynomials

It is known that a polynomialQm of degree at mostm is self-inversive of degreem if zmQm(1/z) =
cmQm(z), where |cm| = 1. As one of the earliest references to self-inversive polynomials we cite
Bonsall and Marden [1]. The most interesting self-inversive polynomials are those polynomials
with all their zeros on the unit circle. Characterizing self-inversive polynomials with zeros on
the unit circle has been of considerable interest (see, for example, [8, 10, 11]).

In this manuscript we adopt the definition that Qm is a self-inversive polynomial of degree
m if Qm is a polynomial of degree at most m and satisfy

Q∗
m(z) = zmQm(1/z) = Qm(z).

Note that we have assumed cm = 1 from the original definition of self-inversive polynomials. We
remark that what is considered in the present manuscript as self-inversive polynomials are, as
in [11], are known as conjugate reciprocal polynomials.

Functions belonging to Ωm are connected to self-inversive polynomials of degree m. That
is, given Fm ∈ Ωm then associated with it there exists a unique Qm which is a self-inversive
polynomials of degree m. Precisely, e−imθ/2Qm(eiθ) = Fm(x), where x = cos(θ/2).

The following lemma, which will be of considerable use in this manuscript, gives a more
precise statement regarding this connection.

Lemma 2.1 Let x = cos(θ/2). Then the polynomial Qm is self-inversive of degree m if and
only if

e−imθ/2Qm(eiθ) = Fm(x) = B(0)
m (x) +

√
1− x2B(1)

m (x),

2



where B
(0)
m and B

(1)
m are real polynomials of degree at most m and m − 1, respectively, and

satisfying the symmetry

B(0)
m (−x) = (−1)mB(0)

m (x) and B(1)
m (−x) = (−1)m−1B(1)

m (x).

Thus,

|Qm(eiθ)|2 =
[
B(0)

m (x) +
√

1− x2B(1)
m (x)

]2
and Qm(1) = B(0)

m (1).

Moreover, Qm is a self-inversive polynomial with real coefficients if and only if B
(1)
m is identically

zero.

Proof. This Lemma has also been stated and proved in [5]. However, for completeness and for
a better understanding of the use of this lemma, we give here a sketch of its proof.

Given any polynomial Qm of degree at most m, not necessarily self-inversive, by setting

Qm(z) =
∑m

j=0(c
(m)
j + i d

(m)
j )zj we can write

z−m+1/2Q2m−1(z) =

m−1∑

j=0

(c
(2m−1)
m−1−j + c

(2m−1)
m+j )

zj+1/2 + z−j−1/2

2

−
m−1∑

j=0

i(d
(2m−1)
m−1−j − d

(2m−1)
m+j )

zj+1/2 − z−j−1/2

2

+

m−1∑

j=0

i(d
(2m−1)
m−1−j + d

(2m−1)
m+j )

zj+1/2 + z−j−1/2

2

−
m−1∑

j=0

(c
(2m−1)
m−1−j − c

(2m−1)
m+j )

zj+1/2 − z−j−1/2

2

and

z−mQ2m(z) = c(2m)
m +

m∑

j=1

[
(c

(2m)
m−j + c

(2m)
m+j )

zj + z−j

2
− i(d

(2m)
m−j − d

(2m)
m+j )

zj − z−j

2

]

+ i d(2m)
m +

m∑

j=1

[
i(d

(2m)
m−j + d

(2m)
m+j )

zj + z−j

2
− (c

(2m)
m−j − c

(2m)
m+j )

zj − z−j

2

]
.

Then for z = eiθ, with x = (z1/2 + z−1/2)/2 = cos(θ/2), we obtain

e−imθ/2Qm(eiθ) = Fm(x) + i F̃m(x) and |Qm(eiθ)|2 = [Fm(x)]2 + [F̃m(x)]2,

where the functions Fm and F̃m, defined for x ∈ [−1, 1], satisfy

F2m−1(x) =

m−1∑

j=0

(c
(2m−1)
m−1−j + c

(2m−1)
m+j )T2j+1(x) + (d

(2m−1)
m−1−j − d

(2m−1)
m+j )

√
1− x2 U2j(x),

F̃2m−1(x) =
m−1∑

j=0

(d
(2m−1)
m−1−j + d

(2m−1)
m+j )T2j+1(x)− (c

(2m−1)
m−1−j − c

(2m−1)
m+j )

√
1− x2 U2j(x),

F2m(x) = c(2m)
m +

m∑

j=1

(c
(2m)
m−j + c

(2m)
m+j )T2j(x) + (d

(2m)
m−j − d

(2m)
m+j )

√
1− x2 U2j−1(x),

F̃2m(x) = d(2m)
m +

m∑

j=1

(d
(2m)
m−j + d

(2m)
m+j )T2j(x)− (c

(2m)
m−j − c

(2m)
m+j )

√
1− x2 U2j−1(x).

3



Here,

Tj(x) = cos(jθ/2) =
1

2
(zj/2 + z−j/2) and Uj(x) =

sin((j + 1)θ/2)

sin(θ/2)
=

(z(j+1)/2 − z−(j+1)/2)

(z1/2 − z−1/2)
,

are respectively the Chebyshev polynomials of the first and second kind.
These relations enable one to establish a connection between any polynomial Qm(z) and two

functions Fm(x) and F̃m(x) both in Ωm.

However, if Qm(z) =
∑m

j=0(c
(m)
j + i d

(m)
j )zj is self-inversive then c

(m)
j = c

(m)
m−j and d

(m)
j =

−d(m)
m−j and hence F̃m(x) is identically zero.

An immediate consequence of this Lemma is the following.

Theorem 2.2 If F ∈ Ωm then the number of zeros of F in [−1, 1] can not exceed m.

Another interesting result regarding functions in Ωm is their interpolation property, which
is virtually the interpolation property of self-inversive polynomials on the unit circle |z| =
1. Results regarding interpolation by polynomials, including the idea behind the proof of the
theorem below, are well known and can be found in any numerical analysis texts. For a recent
reference to such a text we cite [9].

Theorem 2.3 Given the m+ 1 pairs of real numbers (xj , yj), j = 1, 2, . . . ,m+ 1, where −1 <
x1 < x2 < . . . < xm+1 < 1, then there exists a unique F ∈ Ωm such that

F(xj) = yj, j = 1, 2, . . . ,m+ 1.

Moreover, this interpolation function can be given by

F(x) =

m+1∑

k=1

Lk(x) yk, (2.1)

where

Lk(x) = z−m/2 z
m/2
k

m+1∏

l = 1
l 6= k

z − zl
zk − zl

, k = 1, 2, . . . ,m+ 1,

with z = eiθ, θ = 2arccos(x), zk = eiθk and θk = 2arccos(xk).

Proof. Uniqueness follows from Theorem 2.2. That is if F and F̃ are two different functions in
Ωm such that

F(xj) = yj and F̃(xj) = yj, j = 1, 2, . . . ,m+ 1,

then G(x), where G(x) = F(x) − F̃(x) ∈ Ωm and Gm(x) 6= 0, has m + 1 zeros in (−1, 1)
contradicting Theorem 2.2.

To show the existence we construct the required function as follows. It is easily seen that
the scaled Lagrange polynomials

z
m/2
k

m+1∏

l = 1
l 6= k

z − zl
zk − zl

, k = 1, 2, . . . ,m+ 1,

defined on the set {z1, z2, . . . , zm} are self-inversive. Consequently, Lk(x) are in Ωm and that
they satisfy

Lk(xj) = δjk, j = 1, 2, . . . ,m+ 1.

Hence, the formula (2.1) immediately leads to the required interpolation function.

4



3 Some basic properties

Theorem 3.1 Let the sequence functions {Fm}, where F0(x) = b
(0)
0,0,

Fm(x) =

⌊m/2⌋∑

j=0

b
(0)
m,2j x

m−2j +
√

1− x2
⌊(m−1)/2⌋∑

j=0

b
(1)
m,2j x

m−1−2j , m ≥ 1,

be such that b
(0)
m+1,0b

(0)
m,0 + b

(1)
m+1,0b

(1)
m,0 6= 0, m ≥ 0. Here, b

(1)
0,0 = 0. Then the following hold.

1. A basis for Ω2m is

{F2m(x),
√

1− x2F2m−1(x),F2m−2(x), . . . ,F2(x),
√

1− x2F1(x),F0(x)};

2. A basis for Ω2m+1 is

{F2m+1(x),
√

1− x2F2m(x),F2m−1(x), . . . ,F3(x),
√

1− x2F2(x),F1(x),
√

1− x2F0(x)}.

Proof. Since b
(0)
m+1,0b

(0)
m,0 + b

(1)
m+1,0b

(1)
m,0 6= 0 for m ≥ 0 implies (b

(0)
m,0)

2 + (b
(1)
m,0)

2 > 0 for m ≥ 0,
the function Fm is of exact degree m for m ≥ 0.

Now to prove the theorem, by observing that the dimension of Ωm is m+ 1, all we have to
do is to verify that the above sets are linearly independent. We prove this for the even indices
and the proof for the odd indices is similar.

Clearly F0(x) = b
(0)
0,0 6= 0 is a basis for Ω0. Now we verify that

{F2(x),
√

1− x2F1(x),F0(x)}

is a basis for Ω2.
Let c0, c1, c2 be such that c0F2(x) + c1

√
1− x2F1(x) + c2 F0(x) = 0. Since the dimension

of Ω2 is 3, we need to verify that this is possible only if c0 = c1 = c2 = 0. By considering the
coefficients of x2 and x

√
1− x2, we have

c0 b
(0)
2,0 − c1 b

(1)
1,0 = 0,

c0 b
(1)
2,0 + c1 b

(0)
1,0 = 0.

The determinant of this system is b
(0)
2,0b

(0)
1,0 + b

(1)
2,0b

(1)
1,0, which is different from zero. We must

therefore have c0 = c1 = 0. This reduces our verification to finding c2 such that c2 F0(x) = 0.

Clearly c2 = 0, which follows from F0(x) = b
(0)
0,0 6= 0.

Now assuming

{F2m(x),
√

1− x2F2m−1(x),F2m−2(x), . . . ,F2(x),
√

1− x2F1(x),F0(x)} (3.1)

is a basis for Ω2m, we show that

{F2m+2(x),
√

1− x2F2m+1(x),F2m(x), . . . ,F2(x),
√

1− x2F1(x),F0(x)},

is a basis for Ω2m+2. Let c0, c1, c2, . . . , c2m+1, c2m+2 be such that

c0 F2m+2(x) + c1
√
1− x2F2m+1(x) + . . .+ c2m+1

√
1− x2F1(x) + c2m+2 F0(x) = 0.

By considering the coefficients of x2m+2 and x2m+1
√
1− x2, we have

c0 b
(0)
2m+2,0 − c1 b

(1)
2m+1,0 = 0,

c0 b
(1)
2m+2,0 + c1 b

(0)
2m+1,0 = 0.

5



The determinant of this system is b
(0)
2m+2,0b

(0)
2m+1,0 + b

(1)
2m+2,0b

(1)
2m+1,0. Since this determinant

is different from zero, we must have c0 = c1 = 0. This reduces our verification to finding
c2, c3, . . . , c2m+2, such that

c2 W2m(x) + c3
√

1− x2W2m−1(x) + . . .+ c2m+1

√
1− x2W1(x) + c2m+2 W0(x) = 0.

Clearly c2 = c3 = . . . = c2m+1 = c2m+2 = 0, which follow from the assumption in (3.1). Thus,
the results of the theorem for even indices follow by induction.

Now with the assumption γ20 > 0, γ2m + β2m > 0, m ≥ 1, we consider the functions Wm given
by the recurrence formula (1.1). It is easily seen that Wm takes the form

Wm(x) = A(0)
m (x) +

√
1− x2A(1)

m (x), (3.2)

where A
(0)
m and A

(1)
m are, respectively, polynomials of degree at most m and m− 1, such that

A(0)
m (−x) = (−1)mA(0)

m (x) and A(1)
m (−x) = (−1)m−1A(1)

m (x).

Setting

A(0)
m (x) =

⌊m/2⌋∑

j=0

a
(0)
m, 2j x

m−2j and A(1)
m (x) =

⌊(m−1)/2⌋∑

j=0

a
(1)
m, 2j x

m−1−2j , (3.3)

we have the following.

Theorem 3.2 Let γ20 > 0, γ2m + β2m > 0, m ≥ 1 in (1.1). Then for the leading coefficients a
(0)
m,0

and a
(1)
m,0 and lead factor λm = (a

(0)
m,0)

2 + (a
(1)
m,0)

2 of Wm the following hold.


 a

(0)
m,0

a
(1)
m,0


 =

[
γm βm

−βm γm

] 
 a

(0)
m−1,0

a
(1)
m−1,0


 , m ≥ 1,

with a
(0)
0,0 = 1 and a

(1)
0,0 = 0. Consequently, there hold

λ0 = γ20 , λm = (γ2m + β2m)λm−1, m ≥ 1,

λm,1 = a
(0)
m+1,0 a

(0)
m,0 + a

(1)
m+1,0 a

(1)
m,0 = γm+1λm, m ≥ 1,

a
(0)
m+1,0 a

(1)
m,0 − a

(1)
m+1,0 a

(0)
m,0 = βm+1λm, m ≥ 1,

a
(0)
m+1,0 a

(0)
m−1,0 + a

(1)
m+1,0 a

(1)
m−1,0 = (γmγm+1 − βmβm+1)λm−1, m ≥ 1

and

a
(0)
m+1,0 a

(1)
m−1,0 − a

(1)
m+1,0 a

(0)
m−1,0 = (γm+1βm + γmβm+1)λm−1, m ≥ 1.

Proof. From (1.1), (3.2) and (3.3), by equating the coefficients of xm+1 and xm
√
1− x2,

a
(0)
m+1,0 = γm+1a

(0)
m,0 + βm+1 a

(1)
m,0,

a
(1)
m+1,0 = −βm+1 a

(0)
m,0 + γm+1a

(1)
m,0.

(3.4)

From these the matrix formula in the theorem follows. The equalities and recurrence can be
obtained as follows.

6



From the matrix formula,

(a
(0)
m,0)

2 + (a
(1)
m,0)

2 =

[
a
(0)
m,0 a

(1)
m,0

]
 a

(0)
m,0

a
(1)
m,0




=

[
a
(0)
m−1,0 a

(1)
m−1,0

][
γm −βm
βm γm

][
γm βm

−βm γm

]
 a

(0)
m−1,0

a
(1)
m−1,0


 .

Thus,

(a
(0)
m,0)

2 + (a
(1)
m,0)

2 =

[
a
(0)
m−1,0 a

(1)
m−1,0

][
γ2m + β2m 0

0 γ2m + β2m

] 
 a

(0)
m−1,0

a
(1)
m−1,0




= (γ2m + β2m)
[
(a

(0)
m−1,0)

2 + (a
(1)
m−1,0)

2
]
= (γ2m + β2m)λm−1,

which gives the recurrence formula for λm. Similarly,

a
(0)
m+1,0 a

(0)
m,0 + a

(1)
m+1,0 a

(1)
m,0 =

[
a
(0)
m,0 a

(1)
m,0

]
 a

(0)
m+1,0

a
(1)
m+1,0




=

[
a
(0)
m,0 a

(1)
m,0

][
γm+1 βm+1

−βm+1 γm+1

]
 a

(0)
m,0

a
(1)
m,0




= γm+1

[
(a

(0)
m,0)

2 + (a
(1)
m,0)

2
]
= γm+1λm.

From (3.4),

a
(0)
m+1,0 a

(1)
m,0 − a

(1)
m+1,0 a

(0)
m,0

= [γm+1a
(0)
m,0 + βm+1a

(1)
m,0]a

(1)
m,0 − [−βm+1a

(0)
m,0 + γm+1a

(1)
m,0]a

(0)
m,0

= βm+1

[
(a

(0)
m,0)

2 + (a
(1)
m,0)

2
]
= βm+1λm.

Again from (3.4),

a
(0)
m+1,0 a

(0)
m−1,0 + a

(1)
m+1,0 a

(1)
m−1,0

= [γm+1a
(0)
m,0 + βm+1a

(1)
m,0]a

(0)
m−1,0 + [−βm+1a

(0)
m,0 + γm+1a

(1)
m,0]a

(1)
m−1,0

= γm+1

[
a
(0)
m,0 a

(0)
m−1,0 + a

(1)
m,0 a

(1)
m−1,0

]
− βm+1

[
a
(0)
m,0 a

(1)
m−1,0 − a

(1)
m,0 a

(0)
m−1,0

]

= (γmγm+1 − βmβm+1)λm−1.

Similarly, the value associated with a
(0)
m+1,0 a

(1)
m−1,0 − a

(1)
m+1,0 a

(0)
m−1,0 is also obtained.

Clearly, with the assumptions γ20 > 0 and β2m + γ2m > 0, m ≥ 1, there holds

λm = (a
(0)
m,0)

2 + (a
(1)
m,0)

2 > 0, m ≥ 0,

which means the leading coefficients of A
(0)
m and A

(1)
m can not be zero simultaneously and Wm ∈

Ωm is of exact degree m.
With a more restrictive condition than γ20 > 0 and β2m + γ2m > 0, m ≥ 1, we can state the

following theorem.

7



Theorem 3.3 Let γm 6= 0, m ≥ 0 and let {Wm} be the sequence of functions given by the
recurrence formula (1.1). Then for any m ≥ 0,

1. A basis for Ω2m is

{W2m(x),
√

1− x2W2m−1(x),W2m−2(x), . . . ,W2(x),
√

1− x2W1(x),W0(x)};

2. A basis for Ω2m+1 is

{W2m+1(x),
√

1− x2W2m(x),W2m−1(x), . . . ,W3(x),
√

1− x2W2(x),W1(x),
√

1− x2W0(x)}.

Proof. From Theorem 3.2 we observe that, with γm 6= 0, m ≥ 0, the leading coefficients of Wm,

m ≥ 0 are such that λm,1 = a
(0)
m+1,0 a

(0)
m,0 + a

(1)
m+1,0 a

(1)
m,0 6= 0, m ≥ 0. Hence, the present theorem

follows from Theorem 3.1.

Finally, by denoting the self-inversive polynomial associated with Wm(x) by Km(z), we have
the following.

Theorem 3.4 Let Km(z) =
∑m

j=0 k
(m)
j zj be such that

e−imθ/2Km(eiθ) = Wm(x),

where x = cos(θ/2). Then

k
(m)
0 = k

(m)
m = 2−m[a

(0)
m,0 + i a

(1)
m,0] and |k(m)

0 |2 = |k(m)
m |2 = 2−2mλm.

4 Orthogonal properties associated with Wm

Let ψ be a positive measure on [−1, 1]. We consider the sequence of functions {Wm}, where
Wm ∈ Ωm is of exact degree m, is such that

∫ 1

−1
W2n(x)W2m(x)

√
1− x2 dψ(x) = ρ2m δn,m,

∫ 1

−1
W2n+1(x)W2m+1(x)

√
1− x2 dψ(x) = ρ2m+1 δn,m,

∫ 1

−1
W2n+1(x)W2m(x) dψ(x) = 0,

(4.1)

for n,m = 0, 1, 2, . . . .

We use the notation W0(x) = a
(0)
0,0,

Wm(x) =

⌊m/2⌋∑

j=0

a
(0)
m, 2j x

m−2j +
√

1− x2
⌊(m−1)/2⌋∑

j=0

a
(1)
m, 2j x

m−1−2j , m ≥ 1,

and
λm = (a

(0)
m,0)

2 + (a
(1)
m,0)

2 and λm,1 = a
(0)
m+1,0a

(0)
m,0 + a

(1)
m+1,0a

(1)
m,0, m ≥ 0,

with a
(1)
0,0 = 0.

Observe that, in the case a
(1)
m,2j = 0, j = 0, 1, . . . , ⌊(m − 1)/2⌋, m ≥ 1, then {Wm} are

symmetric polynomials and (4.1) reduces to the orthogonality of symmetric polynomials with
respect to

√
1− x2 dψ(x) in [−1, 1]. For references to some of the classical texts on orthogonal

polynomials on the real line we cite [2, 7, 12].

8



Theorem 4.1 The sequence of functions {Wm} that satisfies the orthogonality property (4.1),
where Wm ∈ Ωm and is of exact degree m, exists and satisfies the three term recurrence formula

W0(x) = γ0, W1(x) =
[
γ1x− β1

√
1− x2

]
γ0,

Wm+1(x) =
[
γm+1x− βm+1

√
1− x2

]
Wm(x)− αm+1Wm−1(x), m ≥ 1,

(4.2)

where γm 6= 0, m ≥ 0, β1 = γ1 ρ
−1
0

∫ 1
−1 xW2

0 (x) dψ(x),

βm+1 = γm+1
1

ρm

∫ 1

−1
xW2

m(x) dψ(x), m ≥ 1 and

αm+1 =
1

ρm−1

∫ 1

−1

[
γm+1x− βm+1

√
1− x2

]
Wm−1(x)Wm(x)

√
1− x2 dψ(x), m ≥ 1.

(4.3)

Here, ρm =
∫ 1
−1 W2

m(x)
√
1− x2 dψ(x), m ≥ 0.

Proof. It follows from Theorem 3.2 that, with γm 6= 0, m ≥ 0, the function Wm obtained from
the three term recurrence formula (4.2) is of exact degree m. Hence, to prove the existence part
of the above theorem, we show that the sequence of functions {Wm} generated by the above
three term recurrence formula satisfies (4.1).

Since β1 = γ1 ρ
−1
0

∫ 1
−1 xW2

0 (x) dψ(x), with W1(x) =
[
γ1x − β1

√
1− x2

]
γ0 there follows∫ 1

−1 W0(x)W1(x) dψ(x) = 0.
Since β2 and α2 are as in (4.3), with

W2(x) =
[
γ2x− β2

√
1− x2

]
W1(x)− α2W0(x),

we have

∫ 1

−1
W1(x)W2(x) dψ(x) = 0 and

∫ 1

−1
W0(x)W2(x)

√
1− x2 dψ(x) = 0.

Now suppose that for N ≥ 2 the sequence of functions {Wm}Nm=0 obtained from the three
term recurrence formula (4.2) satisfies (4.1).

Since xWN−2k(x) ∈ ΩN+1−2k, by Theorem 3.3 there exists c0, c1, . . . , cN−2k such that

xWN−2k(x) = c0WN+1−2k(x) + c1
√

1− x2WN−2k(x) + c2WN−1−2k(x) + . . . .

Hence, we have ∫ 1

−1
xWN−2k(x)WN (x) dψ(x) = 0, k = 1, 2, . . . ⌊N/2⌋.

Likewise, since x
√
1− x2WN−1−2k(x) ∈ ΩN+1−2k, we have

∫ 1

−1
x
√

1− x2WN−1−2k(x)WN (x) dψ(x) = 0, k = 1, 2, . . . ⌊(N − 1)/2⌋ (4.4)

and, since (1− x2)WN−1−2k(x) ∈ ΩN+1−2k, we also have

∫ 1

−1
(1− x2)WN−1−2k(x)WN (x) dψ(x) = 0, k = 1, 2, . . . ⌊(N − 1)/2⌋. (4.5)

Note that in (4.4) and (4.5) the value of N is assumed to be ≥ 3.

9



Hence, from

WN+1(x) =
[
γN+1x− βN+1

√
1− x2

]
WN (x)− αN+1WN−1(x),

it follows that

∫ 1
−1 WN−2k(x)WN+1(x) dψ(x) = 0, k = 1, 2, . . . , ⌊N/2⌋,
∫ 1
−1 WN−1−2k(x)WN+1(x)

√
1− x2dψ(x) = 0, k = 1, 2, . . . , ⌊(N − 1)/2⌋.

Moreover, since αN+1 and βN+1 are as in (4.3), we also have

∫ 1
−1 WN (x)WN+1(x) dψ(x) = 0 and

∫ 1
−1 WN−1(x)WN+1(x)

√
1− x2dψ(x) = 0.

Thus, by induction we conclude that the sequence of functions given by the three term recurrence
formula satisfies (4.1).

On the other hand, to show that any sequence of functions {Wm}, where Wm ∈ Ωm is of
exact degree m, for which (4.1) holds, must also satisfy the three term recurrence formula (4.2),
we proceed as follows.

Clearly we can writeW0(x) = γ0 6= 0 andW1(x) = a
(0)
1,0x+a

(1)
1,0

√
1− x2 = [γ1x−β1

√
1− x2]γ0.

Then for
∫ 1
−1 W0(x)W1(x) dψ(x) = 0 to hold such that W1(x) is of exact degree 1, one must

have γ1 6= 0 and β1 = γ1 ρ
−1
0

∫ 1
−1 xW2

0 (x) dψ(x).
With the next element W2 of the given orthogonality sequence, let γ2 and β2 be such that

W2(x)−
[
γ2x− β2

√
1− x2

]
W1(x) ∈ Ω0. With respect to the leading coefficients of W2 and W1,

the elements γ2 and β2 must satisfy


 a

(0)
2,0

a
(1)
2,0


 =


 a

(0)
1,0 a

(1)
1,0

a
(1)
1,0 −a(0)1,0






γ2

β2


 .

The determinant of this system is −[(a
(0)
1,0)

2 + (a
(1)
1,0)

2]. Since W1 is of exact degree 1 this deter-
minant is different from zero. Hence, the values of γ2 and β2 are uniquely found. Writing

W2(x) =
[
γ2x− β2

√
1− x2

]
W1(x)− α2W0(x),

we find, with the orthogonality and with the additional observation that W2 is of exact degree
2, that γ2 6= 0,

β2 = γ2
1

ρ1

∫ 1

−1
xW2

1 (x) dψ(x) and

α2 =
1

ρ0

∫ 1

−1

[
γ2x− β2

√
1− x2

]
W0(x)W1(x)

√
1− x2 dψ(x).

Now for N ≥ 2 assume that the orthogonal functions Wm, m = 0, 1, . . . , N satisfy the three
term recurrence formula (4.2), with γm 6= 0, m = 0, 1, . . . , N . Let γN+1 and βN+1 be such that
WN+1(x)−

[
γN+1x− βN+1

√
1− x2

]
WN (x) ∈ ΩN−1. With respect to the leading coefficients of

WN+1 and WN , the elements γN+1 and βN+1 must satisfy


 a

(0)
N+1,0

a
(1)
N+1,0


 =


 a

(0)
N,0 a

(1)
N,0

a
(1)
N,0 −a(0)N,0




[
γN+1

βN+1

]
.

10



The determinant of this system is −[(a
(0)
N,0)

2 +(a
(1)
N,0)

2], which is different form zero because WN

is of exact degree N . Hence, the values for γN+1 and βN+1 are uniquely found.
Now using Theorem 3.3, there exist c0, c1, . . . , cN−1 such that

WN+1(x) =
[
γN+1x− βN+1

√
1− x2

]
WN (x)

+ c0WN−1(x) + c1
√
1− x2WN−2(x) + c2WN−3(x) + . . . .

Applications of the orthogonality properties (4.1) of the sequence {Wm}N+1
m=0, together with the

observation that WN+1 is of exact degree N + 1, lead to c1 = c2 = . . . = cN−1 = 0, γN+1 6= 0,

βN+1 = γN+1
1

ρN

∫ 1

−1
xW2

N (x) dψ(x) and

αN+1 = −c0 =
1

ρN−1

∫ 1

−1

[
γN+1x− βN+1

√
1− x2

]
WN−1(x)WN (x)

√
1− x2 dψ(x).

This concludes the proof of the Theorem.

Following as in Theorem 3.2 we have γm+1 = λm,1/λm. Thus, the orthogonal functions Wm,
m ≥ 1, are such that

λm,1 = a
(0)
m+1,0 a

(0)
m,0 + a

(1)
m+1,0 a

(1)
m,0 6= 0, m ≥ 0.

Corollary 4.1.1 The sequence of functions {Wm}, where Wm ∈ Ωm and is of exact degree m,
satisfies the orthogonality property (4.1) if and only if, for m ≥ 1,

∫ 1

−1
F(x)Wm(x) dψ(x) = 0 whenever F ∈ Ωm−1. (4.6)

Proof. First we assume that (4.6) holds. Observe that if F ∈ Ωm+1−2k for k = 1, 2, . . . , ⌊(m +
1)/2⌋ then F ∈ Ωm−1. Hence, Wm+1−2k(x) ∈ Ωm+1−2k and Wm−2k(x)

√
1− x2 ∈ Ωm+1−2k leads

immediately to (4.1).
On the other hand, if (4.1) holds then from Theorem 4.1 and from Theorem 3.3 we can write

Fm+1−2k(x) = c0Wm+1−2k(x) + c1
√

1− x2Wm−2k(x) + c2Wm−1−2k(x) + . . . .

for k = 1, 2, . . . , ⌊(m+ 1)/2⌋. Hence, (4.6) is immediate.

It is quite straight forward that another way to present the above corollary is the following.

Corollary 4.1.2 The sequence of functions {Wm}, where Wm ∈ Ωm and is of exact degree m,
satisfies the orthogonality property (4.1) if and only if, for m ≥ 1,

∫ 1

−1
B(0)(x)Wm(x) dψ(x) = 0 and

∫ 1

−1
B(1)(x)Wm(x)

√
1− x2 dψ(x) = 0,

where B(0) ∈ Pm−1 and B(1) ∈ Pm−2 satisfy

B(0)(−x) = (−1)m−1B(0)(x) and B(1)(−x) = (−1)m−2B(1)(x).

The following corollary provides one other way to express the orthogonality (4.1) of the
sequence {Wn}.

11



Corollary 4.1.3 The sequence of functions {Wm}, where Wm ∈ Ωm and is of exact degree m,
satisfies the orthogonality property (4.1) if and only if, for m ≥ 1,

∫ 1

−1
(x+ i

√
1− x2 )−m+1+2s Wm(x) dψ(x) = 0, s = 0, 1, . . . ,m− 1. (4.7)

Proof. We just give the proof for m = 2n and the proof for m = 2n+ 1 is similar.
Since (x+ i

√
1− x2 )(x− i

√
1− x2 ) = 1 the orthogonality (4.7) for m = 2n can be written

as

∫ 1

−1
(x± i

√
1− x2 )2l+1 W2n(x) dψ(x) = 0, l = 0, 1, . . . , n− 1.

Observe that

(x± i
√
1− x2 )2l+1

=
l∑

k=0

(
2l + 1
2k + 1

)
x2k+1(x2 − 1)l−k ± i

√
1− x2

l∑

k=0

(
2l + 1
2k

)
x2k(x2 − 1)l−k.

Since the polynomials represented by the above sums are, respectively, odd and even polynomials
of exact degrees 2l + 1 and 2l, the required result follows from Corollary 4.1.2.

The following theorem, in addition to showing some further orthogonality properties of the
functions {Wm} given by Theorem 4.1, also gives another expression for the coefficients αn given
in Theorem 4.1.

Theorem 4.2 ∫ 1

−1
x
√

1− x2 W1(x)W2(x) dψ(x) =
γ2γ3 − β2β3
(γ22 + β22)γ3

ρ2,

∫ 1

−1
(1− x2)W1(x)W2(x) dψ(x) = −γ3β2 + γ2β3

(γ22 + β22)γ3
ρ2

and for m ≥ 2,

∫ 1

−1
x
√

1− x2 Wm−1−2k(x)Wm(x) dψ(x) =





γmγm+1 − βmβm+1

(γ2m + β2m)γm+1
ρm, k = 0,

0, 1 ≤ k ≤ ⌊(m− 1)/2⌋,

∫ 1

−1
(1− x2)Wm−1−2k(x)Wm(x) dψ(x) =





−γm+1βm + γmβm+1

(γ2m + β2m)γm+1
ρm, k = 0,

0, 1 ≤ k ≤ ⌊(m− 1)/2⌋,

Consequently,

αm+1 =
γ2m+1 + β2m+1

γ2m + β2m

γm
γm+1

ρm
ρm−1

, m ≥ 1.

Proof. Since x
√
1− x2 Wm−1−2k(x) and (1 − x2)Wm−1−2k(x) are in Ωm+1−2k, the results

corresponding to 1 ≤ k ≤ ⌊(m− 1)/2⌋ follows from Corollary 4.1.1.
We now prove the result associated with

∫ 1
−1 x

√
1− x2 Wm−1(x)Wm(x) dψ(x).

Since x
√
1− x2 Wm−1(x) ∈ Ωm+1, there exist c0, c1, . . . , cm+1 such that

x
√

1− x2 Wm−1(x) = c0Wm+1(x) + c1
√

1− x2Wm(x) + c2Wm−1(x) + . . .

12



and that
∫ 1
−1 x

√
1− x2 Wm−1(x)Wm(x) dψ(x) = c1ρm. Moreover, comparing the leading coeffi-

cients on both sides
−a(1)m−1,0 = c0 a

(0)
m+1,0 − c1 a

(1)
m,0,

a
(0)
m−1,0 = c0 a

(1)
m+1,0 + c1 a

(0)
m,0.

This gives c1 =
[
a
(0)
m+1,0a

(0)
m−1,0+a

(1)
m+1,0a

(1)
m−1,0

]
/
[
a
(0)
m+1,0a

(0)
m,0+a

(1)
m+1,0a

(1)
m,0

]
. Thus, from Theorem

3.2 we have

c1 =
γmγm+1 − βmβm+1

(γ2m + β2m)γm+1

and the required result for
∫ 1
−1 x

√
1− x2 Wm−1(x)Wm(x) dψ(x).

The result associated with
∫ 1
−1(1− x2)Wm−1(x)Wm(x) dψ(x) is obtained similarly, and the

latter result of the theorem then follows from (4.3).

Now we can state the following theorem with respect to the zeros of Wm(x).

Theorem 4.3 Let {Wm(x)} be the sequence of functions defined as in Theorem 4.1. Then for
m ≥ 1, the function Wm(x) has exactly m distinct zeros in (−1, 1).

Proof. From (4.1) since at least one of the integrals
∫ 1

−1
W0(x)Wm(x)dψ(x) and

∫ 1

−1
W0(x)Wm(x)

√
1− x2dψ(x)

is zero, we can say that Wm(x) changes sign at least once in (−1, 1). According to Theorem 2.2
the number of sign changes of Wm(x) also can not exceed m.

Suppose that Wm(x) changes sign k (1 ≤ k ≤ m) times in (−1, 1), namely at the points
y1, y2, . . . , yk. Let θj = 2arccos(yj), j = 1, 2, . . . , k and consider the self-inversive polynomial (of
degree k) defined by

qk(z) = e−i(kπ+θ1+θ2+...+θk)/2(z − eiθ1)(z − eiθ2) · · · (z − eiθk).

Now, with x = cos(θ/2) = (z1/2 + z−1/2)/2, if we consider the function Fk(x) given by

Fk(x) = e−ikθ/2qk(e
iθ),

then by Lemma 2.1 Fk(x) ∈ Ωk and further Fk(x) has exactly the k zeros y1, y2, . . . , yk, which
are the points of sign changes in Wm(x). Hence, the function Fk(x)Wm(x) does not change sign
in (−1, 1) which then leads to the conclusion

∫ 1

−1
Fk(x)Wm(x)dψ(x) 6= 0 and

∫ 1

−1
Fk(x)Wm(x)

√
1− x2dψ(x) 6= 0.

On the other hand, if k < m then from Corollary 4.1.1 at least one of the integrals
∫ 1

−1
Fk(x)Wm(x)dψ(x) and

∫ 1

−1
Fk(x)Wm(x)

√
1− x2dψ(x)

must be equal to zero, contradicting the earlier conclusion. Hence, the only possibility is k = m
and that Wm has m sign changes in (−1, 1). That is, Wm has exactly m zeros in (−1, 1).

Now we look at the sequence of functions {Ŵm}, obtained from the sequence of orthogonal
functions {Wm} by the scaling

Ŵ0(x) =
1

γ0
W0(x), Ŵm(x) =

1

γ0 · · · γm
Wm(x), m ≥ 0.

By considering the properties of {Ŵm} we can state the following.

13



Theorem 4.4 Let the sequence of functions {Ŵm} be given by

Ŵ0(x) = 1, Ŵ1(x) = x− β̂1
√
1− x2,

Ŵm+1(x) =
[
x− β̂m+1

√
1− x2

]
Ŵm(x)− α̂m+1Ŵm−1(x), m ≥ 1,

where β̂1 =
1

ρ̂0

∫ 1

−1
x Ŵ2

0 (x) dψ(x) and for m ≥ 1,

β̂m+1 =
1

ρ̂m

∫ 1

−1
x Ŵ2

m(x) dψ(x) and

α̂m+1 =
1

ρ̂m−1

∫ 1

−1

[
x− β̂m+1

√
1− x2

]
Ŵm−1(x) Ŵm(x)

√
1− x2 dψ(x) =

1 + β̂2m+1

1 + β̂2m

ρ̂m
ρ̂m−1

.

Here, ρ̂m =
∫ 1
−1 Ŵ2

m(x)
√
1− x2 dψ(x), m ≥ 0. Then {Ŵm}, where Ŵm ∈ Ωm and is of exact

degree m, satisfies the orthogonality (4.1).

Observe that α̂m+1 > 0, m ≥ 1. In fact, see Theorem 6.2, that more can be said about these
coefficients if the measure ψ is such that

∫ 1
−1(1− x2)−1/2dψ(x) exists.

5 Quadrature rules associated with Wm

In order to be able to obtain the quadrature rules based on the zeros of Wm we first present the
following theorem.

Theorem 5.1 Let m ≥ 1 and let {Wm} be as defined before.
Given any function E ∈ Ω4m−1 there exists a function F ∈ Ω2m−1 and a function G ∈ Ω2m−1

such that
E(x) = F(x)W2m(x) + G(x).

Likewise, given any E ∈ Ω4m there exists a function F ∈ Ω2m−1 and a function G ∈ Ω2m such
that

E(x) = F(x)W2m+1(x) + G(x).

Proof. We give a proof of the latter formula. Let P , Q, R and K2m+1 be the respective self-
inversive polynomials associated with the functions E , F , G and W2m+1. Then one needs to
prove that given the self inversive-polynomial P of degree at most 4m there exists a self-inversive
polynomial Q of degree at most 2m− 1 and a self-inversive polynomial R of degree at most 2m
such that

P (z) = Q(z)K2m+1(z) + zmR(z). (5.1)

With the self-inversive property, we can write

P (z) =

4m∑

j=0

pj z
j =

2m−1∑

j=0

pj z
j + p2m z

2m +

4m∑

j=2m+1

p4m−j z
j ,

Q(z) =

2m−1∑

j=0

qj z
j =

m−1∑

j=0

qj z
j +

2m−1∑

j=m

q2m−1−j z
j ,

R(z) =

2m∑

j=0

rj z
j =

m−1∑

j=0

rj z
j + rm z

m +

2m∑

j=m+1

r2m−j z
j

14



and

K2m+1(z) =

2m+1∑

j=0

k
(2m+1)
j zj =

m∑

j=0

k
(2m+1)
j zj +

2m+1∑

j=m+1

k
(2m+1)
2m+1−j z

j .

Note that p2m and rm are both real.
Comparing the coefficients of the potentials 1, z, z2, . . . , z2m on both sides of (5.1) we have

q0 = p0/k
(2m+1)
0

qj =
[
pj −

j−1∑

l=0

ql k
(2m+1)
j−l

]
/k

(2m+1)
0 , j = 1, 2, . . . ,m− 1,

rj = pm+j −
m+j∑

l=0

ql k
(2m+1)
m+j−l , j = 0, 1, . . . ,m− 1,

and

rm = p2m −
2m−1∑

l=0

ql k
(2m+1)
2m−l = p2m −

m−1∑

l=0

[ql k
(2m+1)
2m−l + ql k

(2m+1)
2m−l ].

From Theorem 3.4 since k
(2m+1)
0 6= 0 the above formulas are well defined. Thus, with the further

observation that equalities in the higher potentials lead to the same results, the existence of Q
and R in (5.1) is verified.

The proof of the first formula of the theorem is similar.

We now look at the interpolatory type quadrature rule at the zeros of Wm. For the notion
of polynomial interpolatory quadrature rules we cite [6].

First we denote the zeros of Wm by x
(m)
k , k = 1, 2, . . . ,m and let z

(m)
k = ei2 arccos(x

(m)
k

),
k = 1, 2, . . . ,m. Hence, if G ∈ Ωm−1 then by Theorem 2.3 we can write

G(x) =
m∑

k=1

Lm,k(x)G(x(m)
k ),

where Lm,k ∈ Ωm−1 are functions associated with Wm, give by

Lm,k(x) = z−(m−1)/2 (z
(m)
k )(m−1)/2

m∏

l = 1
l 6= k

z − z
(m)
l

z
(m)
k − z

(m)
l

, k = 1, 2, . . . ,m,

with x = cos(θ/2) and z = eiθ. Consequently, if

λ̂
(m)
k =

∫ 1

−1
Lm,k(x) dψ(x) and λ̃

(m)
k =

∫ 1

−1
Lm,k(x)

√
1− x2 dψ(x), k = 1, 2, . . . ,m,

then, for any G ∈ Ωm−1, we have the following interpolatory quadrature rules

∫ 1

−1
G(x) dψ(x) =

m∑

k=1

λ̂
(m)
k G(x(m)

k ) and

∫ 1

−1
G(x)

√
1− x2 dψ(x) =

m∑

k=1

λ̃
(m)
k G(x(m)

k ). (5.2)

Clearly, these quadrature rules hold for any distinct set of points x
(m)
k , k = 1, 2, . . . ,m. However,

since x
(m)
k are the zeros of the functions Wm we can say more.

15



Theorem 5.2 Let x
(m)
k , k = 1, 2, . . . ,m be the zeros of Wm and let

λ
(m)
k =

∫ 1

−1
[Lm,k(x)]

2
√

1− x2 dψ(x), k = 1, 2, . . . ,m.

If E ∈ Ω4m−1, m ≥ 1, then

∫ 1

−1
E(x) dψ(x) =

2m∑

k=1

1√
1− (x

(2m)
k )2

λ
(2m)
k E(x(2m)

k )

and if E ∈ Ω4m, m ≥ 0, then

∫ 1

−1
E(x)

√
1− x2 dψ(x) =

2m+1∑

k=1

λ
(2m+1)
k E(x(2m+1)

k ).

Proof. To obtain the quadrature rule associated with E ∈ Ω4m−1, we have from Theorem 5.1
that there exist F ∈ Ω2m−1 and G ∈ Ω2m−1 such that E(x) = F(x)W2m(x)+G(x). Hence, from
the orthogonality given by Corollary 4.1.1 that

∫ 1

−1
E(x) dψ(x) =

∫ 1

−1
G(x) dψ(x).

Therefore, from E(x(2m)
k ) = G(x(2m)

k ), k = 1, 2, . . . , 2m and from (5.2)

∫ 1

−1
E(x) dψ(x) =

2m∑

k=1

λ̂
(2m)
k E(x(2m)

k ),

which holds for E ∈ Ω4m−1. With the choice E(x) =
√
1− x2 [L2m,j(x)]

2 ∈ Ω2m−1 we then
obtain that

λ̂
(2m)
j =

1√
1− (x

(2m)
j )2

λ
(2m)
j , j = 1, 2, . . . , 2m.

Now to obtain the quadrature rule associated with E ∈ Ω4m, it follows from Theorem 5.1
that there exist F ∈ Ω2m−1 and G ∈ Ω2m such that E(x) = F(x)W2m(x) + G(x). This leads to
the interpolatory quadrature rule

∫ 1

−1
E(x)

√
1− x2 dψ(x) =

2m+1∑

k=1

λ̃
(2m+1)
k E(x(2m+1)

k ),

which holds for E ∈ Ω4m. With the choice E(x) = [L2m+1,j(x)]
2 ∈ Ω2m we then obtain that

λ̃
(2m+1)
j = λ

(2m+1)
j , j = 1, 2, . . . , 2m+ 1.

This completes the proof of the theorem.

16



6 Connection with orthogonal polynomials on the unit circle

From now on let ψ be a positive measure on [−1, 1] such that
∫ 1
−1(1− x2)−1/2dψ(x) exists. Let

µ be a positive measure on the unit circle that satisfy

− sin(θ/2) dµ(eiθ) = dψ(x), (6.1)

where x = (z1/2 + z−1/2)/2 = cos(θ/2).
Observe that the measure µ is not unique. If µ̃ is a positive measure on the unit circle such

that ∫

C
f(z) dµ̃(z) =

∫

C
f(z) dµ(z) + δ f(1),

where δ is some nonzero constant, then (6.1) also holds for µ̃.
Now consider the sequence of self-inversive polynomials {K̂m(z)} given by

e−imθ/2K̂m(eiθ) = 2mŴm(x), m ≥ 0,

where {Ŵm} are the normalized orthogonal functions given in Theorem 4.4. Hence, one easily
obtains from Corollary 4.1.3 and Theorem 4.4 the following.

Theorem 6.1 The elements of sequence of polynomials {K̂m(z)} satisfy

∫

C
z−m+sK̂m(z) (1 − z)dµ(z) = 0, s = 0, 1, . . . ,m− 1, m ≥ 1.

Moreover,

K̂0(z) = 1, K̂1(z) = (1 + iβ̂1)z + (1− iβ̂1),

K̂m+1(z) =
[
(1 + iβ̂m+1)z + (1− iβ̂m+1)]K̂m+1(z)− 4α̂m+1z K̂m+1(z), m ≥ 1,

(6.2)

where β̂m, α̂m+1, m ≥ 1 are as in Theorem 4.4.

The remaining results in this section, stated mainly without any proofs, follows from recent
results obtained in [3].

The polynomials K̂m, m ≥ 0, are constant multiples of the CD kernels

Km(z, 1) =
s∗m+1(1) s

∗
m+1(z)− sm+1(1) sm+1(z)

1− z
=

m∑

j=0

sj(1) sj(z), m ≥ 0,

where sm, m ≥ 0, are the orthonormal polynomials with respect to the positive measure µ.
The following result can be stated for the coefficients {α̂m+1}∞m=1.

Theorem 6.2 Let the positive measure ψ on [−1, 1] be such that the integral
∫ 1
−1(1−x2)−1/2dψ(x)

exists. Then the sequence of positive numbers {α̂m+1}∞m=1 that appear in the three term recur-
rence formula given in Theorem 4.4 (and also in the three term recurrence formula given in
Theorem 6.1) is a positive chain sequence. Moreover, this positive chain sequence is such that
its maximal parameter sequence does not coincide with its minimal parameter sequence.

One consequence of {α̂m+1}∞m=1 being a positive chain sequence is that this together with
(6.2) enables one to prove also the interlacing of the zeros of Wm and Wm+1 (see [4]).

The results of the above Theorem may be true even without the assumed condition for the
measure ψ. However, we have not been able to verify this.

17



Even though the polynomials K̂m are uniquely defined in terms of the measure ψ, we have
already observed that the measure µ that satisfy (6.1) is not unique (varying according to the
size of the jump at z = 1). Hence, with such distinct measures there exist distinct sets of OPUC.
However, with 0 ≤ t < 1, if one defines the probability measure µ(t) such that

− sin(θ/2) dµ(t)(eiθ) = c(t) dψ(x),

µ(t) has a jump t at z = 1 (i.e. µ(t) has a pure point of size t at z = 1),
(6.3)

where x = (z1/2+ z−1/2)/2 = cos(θ/2) and c(t) is the normalizing constant so that
∫
C dµ

(t)(z) =

1, then we can say more about the associated monic OPUC S
(t)
n and hence, also the orthonormal

polynomials s
(t)
n .

Let {Mm}∞m=1 be the maximal parameter sequence of the positive chain sequence {α̂m}∞m=2.
Using the value of t and the sequence {Mm}∞n=1, we now consider the new positive chain sequence
{α̃m}∞m=1 given by

α̃1 = (1− t)M1, α̃m+1 = α̂m+1 = (1−Mm)Mm+1, m ≥ 1.

It is easily verified (see [2]) that the maximal parameter sequence of the positive chain sequence

{α̃m}∞m=1 is precisely {Mm}∞m=0, withM0 = t. Let {m(t)
m }∞m=0 be the minimal parameter sequence

of {α̃m}∞m=1. That is,

m
(t)
0 = 0, m

(t)
1 = α̃1 = (1− t)M1, m

(t)
m+1 = α̃m+1/(1 −m

(t)
m ), m ≥ 1.

Then we can state the following.

Theorem 6.3 Let S
(t)
m , m ≥ 0, be the monic OPUC with respect to the measure µ(t) given by

(6.3). Then the associated Verblunsky coefficients a
(t)
m−1 = −S(t)

m (0), m ≥ 1, satisfy

a
(t)
m−1 =

1

τm−1

1− 2m
(t)
m − iβ̂m

1− iβ̂m
m ≥ 1,

where τ0 = 1 and τm =
1− iβ̂m

1 + iβ̂m
τm−1, m ≥ 1.

7 Examples

Given any measure ψ on [−1, 1], one can easily obtain by numerical computation the coefficients
in the three term recurrence formulas in Theorem 4.1, hence, also information about the re-
quired orthogonal functions Wm. However, for two reasons we like to consider the normalized
orthogonal functions Ŵm given by Theorem 4.4. One of these reasons is that when the measure
is symmetric then these functions turn out to be monic polynomials. The other reason is because
of Theorem 6.2.

Example 1. Let dψ(x) = (1−x)dx. We obtain by numerical computation the following values
for the first few α̂m’s and β̂m’s in the three term recurrence formula in Theorem 4.4.

m 1 2 3 4 5 6

β̂m −0.4244132 −0.3029978 −0.2398161 −0.2003582 −0.1730831 −0.1529639

α̂m 0.2229581 0.2408213 0.2455306 0.2473987 0.2483152

18



- 1.0 - 0.5 0.5 1.0

- 0.6

- 0.4

- 0.2

0.2

0.4

0.6

W 4 H x L

W 3 H x L

- 1.0 - 0.5 0.5 1.0

- 0.6

- 0.4

- 0.2

0.2

0.4

0.6

W 5 H x L

W 4 H x L

Figure 1: Plots of the functions Ŵ3, Ŵ4 and Ŵ5 when dψ(x) = (1− x)dx.

In the two graphs in Figure 1 we give, respectively, plots of the functions Ŵ3 and Ŵ4 and plots
of the functions Ŵ4 and Ŵ5, separated in this way to be able see clearly the interlacing of the
zeros as pointed out after Theorem 6.2. Glancing at the plot of Ŵ4 it appears though this
function has a zero at the origin. To be precise, the zero is very near to the origin and the value
of this zero is roughly equal to −0.0055075.

The second example we give here is interesting from the point of view of knowing many
things explicitly.

Example 2. Let dψ(x) = [e− arccos(x)]2η [1 − x2]λ−1dx, where η, λ ∈ R and λ > 1/2. Here, we
assume arccos(x) between 0 and π. From results given in [5] we have

Ŵm(x) = 2−m (2λ)m
(λ)m

e−imθ/2
2F1(−m, b; b+ b̄; 1− eiθ),

where x = cos(θ/2), b = λ + iη and the hypergeometric polynomial 2F1(−m, b; b + b̄; 1 − z) is

self-inversive. The orthogonality of {Ŵm} can be explicitly written as

∫ 1

−1
Ŵ2n(x) Ŵ2m(x) [e− arccos(x)]2η [1− x2]λ−1/2dx = ρ̂2m δn,m,

∫ 1

−1
Ŵ2n+1(x) Ŵ2m+1(x) [e

− arccos(x)]2η [1− x2]λ−1/2dx = ρ̂2m+1 δn,m,

∫ 1

−1
Ŵ2n+1(x) Ŵ2m(x) [e− arccos(x)]2η [1− x2]λ−1dx = 0,

for n,m = 0, 1, 2, . . . , where

ρ̂m =
πm! (λ+m) Γ(2λ+m)

22λ+2m−1eηπ |Γ(b+m+ 1)|2
1

[(λ)m]2
[(
Re[(b)m]

)2
+

(
Im[(b)m]

)2
].

Here, Γ represents the gamma function and that (b)0 = 1 and (b)m = b(b+1) · · · (b+m− 1) for
m ≥ 1 are the Pochhammer symbols.

Moreover, in the three term recurrence formula (see Theorem 4.4) for {Ŵm},

β̂m =
η

m+ λ− 1
and α̂m+1 =

1

4

m(m+ 2λ− 1)

(m+ λ− 1)(m+ λ)
, m ≥ 1.

Observe that when η = 0, the functions Ŵm reduce to the monic ultraspherical polynomials

C
(λ−1/2)
m .

19



References

[1] F.F. Bonsall and M. Marden, Zeros of self-inversive polynomials, Proc. Amer. Math. Soc.,
3 (1952), 471-475.

[2] T.S. Chihara, “An Introduction to Orthogonal Polynomials”, Mathematics and its Appli-
cations Series, Gordon and Breach, 1978.

[3] M.S. Costa, H.M. Felix and A. Sri Ranga, Orthogonal polynomials on the unit circle and
Chain Sequences, J. Approx. Theory, to appear.

[4] D.K. Dimitrov and A. Sri Ranga, Zeros of a family of hypergeometric para-orthogonal
polynomials on the unit circle, submitted.

[5] D.K. Dimitrov, M.E.H. Ismail and A. Sri Ranga, A class of hypergeometric polynomials
with zeros on the unit circle: Extremal and orthogonal properties and quadrature formulas,
Applied Numerical Mathematics, 65 (2013), 41-52.

[6] W. Gautschi, “Orthogonal Polynomials: Computation and Approximation”, Numerical
Mathematics and Scientific Computation, Oxford Univ. Press, 2003.

[7] M.E.H. Ismail, “Classical and Quantum Orthogonal Polynomials in One Variable”, Cam-
bridge Univ. Press, Cambridge, 2005.

[8] W.B. Jones, O. Nj̊astad and W.J. Thron, Moment theory, orthogonal polynomials, quadra-
ture, and continued fractions associated with the unit circle, Bull. Lond. Math. Soc., 21
(1989), 113-152.

[9] G.M. Phillips, “Interpolation and Approximation by Polynomials”, CMS Books in Mathe-
matics, Springer, 2003.

[10] A. Schinzel, Self-inversive polynomials with all zeros on the unit circle, Ramanujan J., 9
(2005), 19-23.

[11] C.D. Sinclair and J.D. Vaaler, Self-inversive polynomials with all zeros on the unit circle,
in Number Theory and Polynomials (J. McKee and C. Smyth, eds.), London Mathematical
Society Lecture Notes Series, vol. 352, Cambridge Univ. Press, 2008.

[12] G. Szegő, “Orthogonal Polynomials”, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23,
(Amer. Math. Soc., Providence, RI, 1975).

20


	1 Introduction 
	2 Functions in m and self-inversive polynomials
	3 Some basic properties 
	4 Orthogonal properties associated with Wm 
	5 Quadrature rules associated with Wm 
	6 Connection with orthogonal polynomials on the unit circle 
	7 Examples