Full Terms & Conditions of access and use can be found at
https://www.tandfonline.com/action/journalInformation?journalCode=gitr20

Integral Transforms and Special Functions

ISSN: 1065-2469 (Print) 1476-8291 (Online) Journal homepage: https://www.tandfonline.com/loi/gitr20

Orthogonal polynomials on the unit circle
satisfying a second-order differential equation
with varying polynomial coefficients

J. Borrego-Morell & A. Sri Ranga

To cite this article: J. Borrego-Morell & A. Sri Ranga (2017) Orthogonal polynomials on the unit
circle satisfying a second-order differential equation with varying polynomial coefficients, Integral
Transforms and Special Functions, 28:1, 39-55, DOI: 10.1080/10652469.2016.1249866

To link to this article:  https://doi.org/10.1080/10652469.2016.1249866

Published online: 12 Nov 2016.

Submit your article to this journal 

Article views: 83

View Crossmark data

https://www.tandfonline.com/action/journalInformation?journalCode=gitr20
https://www.tandfonline.com/loi/gitr20
https://www.tandfonline.com/action/showCitFormats?doi=10.1080/10652469.2016.1249866
https://doi.org/10.1080/10652469.2016.1249866
https://www.tandfonline.com/action/authorSubmission?journalCode=gitr20&show=instructions
https://www.tandfonline.com/action/authorSubmission?journalCode=gitr20&show=instructions
http://crossmark.crossref.org/dialog/?doi=10.1080/10652469.2016.1249866&domain=pdf&date_stamp=2016-11-12
http://crossmark.crossref.org/dialog/?doi=10.1080/10652469.2016.1249866&domain=pdf&date_stamp=2016-11-12


INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS, 2017
VOL. 28, NO. 1, 39–55
http://dx.doi.org/10.1080/10652469.2016.1249866

RESEARCH ARTICLE

Orthogonal polynomials on the unit circle satisfying a
second-order differential equation with varying polynomial
coefficients

J. Borrego-Morella and A. Sri Rangab

aDepartamento de Matemática, Universidade Federal do Rio de Janeiro, Polo de Xerém, Rio de janeiro, Brazil;
bDepartamento de Matemática Aplicada, Universidade Estadual Paulista, Campus São José do Rio Preto, São
Paulo, Brazil

ABSTRACT
Consider the linear second-order differential equation

An(z)y
′′ + Bn(z)y

′ + Cny = 0, (1.1)

where An(z) = a2,nz2 + a1,nz + a0,n with a2,n �= 0, a21,n − 4a2,na0,n �=
0, ∀n ∈ N or a2,n = 0, a1,n �= 0, ∀n ∈ N, Bn(z) = b1,n + b0,nz are
polynomials with complex coefficients and Cn ∈ C. Under some
assumptions over a certain class of lowering and raising operators,
we show that for a sequence of polynomials (φn)

∞
n=0 orthogonal on

the unit circle to satisfy the differential equation (1.1), the polynomial
φn must be of a specific form involving and extension of the Gauss
and confluent hypergeometric series.

ARTICLE HISTORY
Received 13 May 2016
Accepted 11 October 2016

KEYWORDS
Orthogonal polynomials on
the unit circle; differential
equations; special functions;
complex analysis

AMS SUBJECT
CLASSIFICATION
42C05; 34M03; 33C05;
33C15; 30Axx

1. Introduction

The Bochner classification theorem [1] (also given by Routh in 1885, [2]) characterizes,
under a complex linear change of the variable z, the sequences (yn)∞n=0 of orthogonal poly-
nomials with respect to a positive Borel measure having finite moments of all orders that
simultaneously solve a second-order differential of the form

A(z)y′′ + B(z)y′ + Cny = 0,

whereA,B are polynomials of degree 2 and 1, respectively,Cn ∈ C. Such sequences of poly-
nomials turn out to be the classical families of orthogonal polynomials Laguerre, Jacobi and
Hermite.

Askey [3] introduced the two-parameter system {Rn, Sn}n≥0 of polynomials given by

Rn(z;α,β) = 2F1(−n,α + β + 1;β − α + 1 − n; z),

Sn(z;α,β) = Rn(z;α,−β), (1.2)

and pointed out that this system is biorthogonal with respect to the complex-
valued weight of beta type ω(θ) = (1 − eiθ )α+β(1 − e−iθ )α−β = (2 − 2 cos θ)α(−eiθ )β ,

CONTACT J. Borrego-Morell jborrego@xerem.ufrj.br, jbmorell@gmail.com

© 2016 Informa UK Limited, trading as Taylor & Francis Group

http://www.tandfonline.com
mailto:jborrego@xerem.ufrj.br, jbmorell@gmail.com


40 J. BORREGO-MORELL AND A. SRI RANGA

θ ∈ [−π ,π],�(α) > − 1
2 . That is

1
2π

∫ π

−π

Rn(eiθ ;α,β)Sm(e−iθ ;α,β)ω(θ) dθ

= �(2α + 1)
�(α + β + 1)�(α − β + 1)

n!
(2α + 1)n

δn,m,

where � denotes the Euler Gamma function. The bi-orthogonality was stated in [3] in a
slightly different form and a formal proof was given later in [4, p.16–17]. Other proofs of
the bi-orthogonality have been given by several authors, please see [5] for some historical
considerations.

The author in [6] using a different approach also proved that when α ∈ R,α > − 1
2 and

iβ ∈ R, the sequence (Rn(z;α,β))∞n=0 is orthogonal with respect to the weight ω, which is
now positive and can be given by ω(θ) = 22α e(π−θ)	(β) sin2α(θ/2).

We also mention another known system of orthogonal polynomials on the unit circle of
hypergeometric type which arise in a class of random unitary matrix ensembles, cf. [7], the
circular Jacobi polynomials, defined as Cn(z; a) = 2F1(−n, a + 1;−a + 1 − n; z). These
polynomials are a particular case of the Rn, by taking α = 2a and β = 0, we obtain Cn.

From known results on hypergeometric functions, the element Rn in the orthogonal
system (Rn(z;α,β))∞n=0 satisfies the differential equation

z(1 − z)y′′ + (β − α + 1 − n − (−n + 2 + α + β)z)y′ + n(α + β + 1)y = 0.

Hence, it is natural to question if there exists other classes of orthogonal polynomials on
the unit circle satisfying a linear second-order differential equation similar to the Jacobi,
Hermite and Laguerre systems of orthogonal polynomials. The above differential equation
satisfied by the sequence (Rn)∞n=0 suggest that we should consider a differential equation
with varying coefficients in the index n and the associated sequence of orthogonal polyno-
mials as solution. In the present manuscript, under some assumptions on a certain class
of lowering and raising operators, we give a necessary condition for the existence of a
sequence of orthogonal polynomials solving the differential equations.

We state the results and notation in the subsection below and in Section 2 we prove the
results.

1.1. Statement of the results

Let μ be a probability measure supported on an infinite subset of the unit circle. We say
that (φn)

∞
n=0 is the sequence of orthonormal polynomials with respect to μ if

∫
|z|=1

φm(z)φn(z) dμ(z) = δm,n,

where φn(z) = κnzn + lnzn−1 + lower order terms and κn > 0. Let �n(z) = φn(z)/κn be
the monic polynomials.

A general background to orthogonal polynomials systems on the unit circle can be
found in the monographs [8–12]. More recent surveys in [13–17].



INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 41

If f is a polynomial of degree n then the reverse polynomial f ∗ is znf (1/z̄), that is

f ∗(z) =
n∑

k=0

ākzn−k if f (z) =
n∑

k=0

akzk and an �= 0.

The sequence (φn)
∞
n=0 of orthonormal polynomials on the unit circle (OPUC for short)

satisfy the recurrence relations [8, (11.4.6), (11.4.7)],

κnzφn(z) = κn+1φn+1(z) − φn+1(0)φ∗
n+1(z),

κnφn+1(z) = κn+1zφn(z) + φn+1(0)φ∗
n(z).

(1.3)

In terms of the monic polynomials, the above recursion also can be expressed as [14,
(1.5.10), (1.5.40)]

�n+1(z) = z�n(z) − ᾱn�
∗
n(z), (1.4)

z�n(z) = ρ−2
n �n+1(z) + ᾱn�

∗
n+1(z), (1.5)

where αn = −�n+1(0) and ρn = κn/κn+1. The coefficients (αn) are called the recursion
coefficients or the Geronimus coefficients. Simon [14] makes a strong case and calls these
coefficients as Verblunsky coefficients.

Consider the hypergeometric equation

z(1 − z)y′′ + (c − (a + b + 1)z)y′ − aby = 0, (1.6)

we denote by 2F1(a, b; c; z) the Gauss hypergeometric series or hypergeometric function of
the variable z with parameters a,b,c; cf. [18, p.56], defined as

2F1(a, b; c; z) =
∞∑
k=0

(a)k(b)k
(c)k

zk

k!
, |z| < 1,

where c �= 0,−1,−2, . . ., which is a solution to (1.6), holomorphic at z=0 and can be
extended appropriately by analytic continuation, see please [18, Section 2.1.4]. Here (a)k =
a(a + 1) · · · (a + k − 1), k ∈ N; (a)0 = 1 denotes the Pochhammer symbol. All the solu-
tions to (1.6) can be expressed in terms of the hypergeometric series (also expressible in
terms of the Riemann P function, if we consider the point of view of Riemann for the
hypergeometric equation).

We will follow [18, p.57] in supplementing the definition of the hypergeometric series
for the case c = −m,m ∈ N ∪ {0}. If a=−n or b=−n, where n ∈ N ∪ {0} and if c=−m
wherem = n, n + 1, n + 2, . . ., then

2F1(−n, b;−m; z) =
n∑

k=0

(−n)k(b)k
(−m)k

zk

k!
, (1.7)

2F1(a,−n;−m; z) =
n∑

k=0

(a)k(−n)k
(−m)k

zk

k!
. (1.8)



42 J. BORREGO-MORELL AND A. SRI RANGA

If c = −n, n ∈ N ∪ {0} and −a,−b /∈ {n, n − 1, . . . , 0}, we define

2F1(a, b;−n; z) =
∞∑
k=0

(a + n + 1)k(b + n + 1)k
(n + 2)k

zk

k!

= zn+1
2F1(a + n + 1, b + n + 1; n + 2; z), (1.9)

which is also holomorphic at z=0 and a solution to (1.6).
We unify our notation by using the symbol 2F1(a, b; c; z) to referring to 2F1(a, b; c; z)

or (1.7)–(1.9) when the parameters a,b,c assume the values specified. Notice that 2F1 is a
polynomial whenever −a or −b is a non-negative integer.

In a similar way, we denote by 1F1 the confluent hypergeometric function or Kummer’s
series

1F1(a; c; z) =
∞∑
k=0

(a)k
(c)k

zk

k!
,

where c �= 0,−1,−2, . . ., which is holomorphic at z=0 and is a solution of the confluent
hypergeometric equation, cf. [18, p.248]

zy′′ + (c − z)y′ − ay = 0.

If a=−n, where n ∈ N ∪ {0} and if c=−m wherem = n, n + 1, n + 2, . . ., then

1F1(−n;−m; z) =
n∑

k=0

(−n)k
(−m)k

zk

k!
. (1.10)

We define

1F1(a;−n; z) =
∞∑
k=0

(a + n + 1)k
(n + 2)k

zk

k!
= zn+1

1F1(a + n + 1; n + 2; z), (1.11)

when c = −n, n ∈ N ∪ {0} and −a /∈ {n, n − 1, . . . , 0}.
We will use the symbol 1F1(a; c; z) to refer to 1F1(a; c; z) or (1.10) and (1.11) when the

parameters a,c assume the values specified.
Let us consider the differential equation

An(z)y′′ + Bn(z)y′ + Cny = 0, (1.12)

whereAn(z) = a2,nz2 + a1,nz + a0,nwith a2,n �= 0, a21,n − 4a2,na0,n �= 0, ∀n ∈ N or a2,n =
0, a1,n �= 0, ∀n ∈ N, Bn(z) = b1,n + b0,nz and (ai,n)n∈N, i = 0, 1, 2; (bi,n)n∈N, i = 0, 1;
(Cn)n∈N are sequences of complex numbers. Under a linear complex change in the variable
z, the differential equation (1.12) can be transformed to

θ(z)y′′ + (cn − bnz)y′ + λny = 0, (1.13)

where (bn)n∈N, (cn)n∈N and (λn)n∈N are sequences of complex numbers and

θ(z) =
{
z(1 − z), if a2,n �= 0, a21,n − 4a2,na0,n �= 0, ∀n ∈ N,
z, if a2,n = 0, a1,n �= 0, ∀n ∈ N.

We are interested in those differential equations of the form (1.13) for which there exists
a unique sequence (φn)

∞
n=0 of OPUC with φn solving (1.13).



INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 43

It is clear that for every n ∈ N fixed, the existence and uniqueness of a polynomial solu-
tion of degree n to the differential equation (1.13) will depend on the values assumed by
the parameters bn, cn and λn. This study is done in the following proposition, whose proof
will be given in Section 2.

Proposition 1.1: Let n ∈ N be fixed. There exists a unique polynomial solution of degree n
for the differential equation (1.13) if and only if

λn =
{
n(n − 1 + bn), θ(z) = z(1 − z),
nbn, θ(z) = z,

(1.14)

and

bn �=
{

−2n + 2,−2n + 3, · · ·,−n,−n + 1, θ(z) = z(1 − z),
0, θ(z) = z.

(1.15)

This solution is

πn(z) = const.

{
2F1(−n, bn + n − 1; cn; z), if θ(z) = z(1 − z),

1F1(−n; cn; bnz), if θ(z) = z.

By virtue of the above proposition, we study up to a complex linear change in the variable
z, those sequences of OPUC satisfying a second-order differential equation of the form

y′′ + Pn(z)y′ + Qn(z)y = 0, (1.16)

where

Pn(z) = cn − bnz
θ(z)

, Qn(z) = n(n − 1 + bn)
θ(z)

if θ(z) = z(1 − z),

Pn(z) = cn − z
θ(z)

, Qn(z) = n
θ(z)

if θ(z) = z,

and bn /∈ {−2n + 2,−2n + 3, . . . ,−n,−n + 1} for θ(z) = z(1 − z).

1.1.1. Raising and lowering operators for orthogonal polynomials on the unit circle.
Let w be a weight function supported on a subset of the unit circle and assume that w is
normalized by ∫

|ξ |=1
w(ξ)

dξ
iξ

= 1,

and define the external field v, cf. [19]

w(z) = e−v(z).

Ismail andWitte [20] derived raising and lowering operators L1,n and L2,n whose expres-
sion is given by Ismail andWitte [20, (2.21)]. Under the assumptions thatw is differentiable
in a neighbourhood of the unit circle and the Verblunsky coefficients do not vanish, these



44 J. BORREGO-MORELL AND A. SRI RANGA

lowering and raising operators define a pair of second-order differential equations such
that

φ′′
n + P1,nφ

′
n + Q1,nφn = 0,

φ′′
n + P2,nφ

′
n + Q2,nφn = 0,

with coefficients given by Ismail and Witte [20, (2.22),(3.8)] and depending on functional
coefficientsAn and Bn

An(z) = n
κn−1

κn
+ i

κn−1

φn(0)
z
∫

|ξ |=1

v′(z) − v′(ξ)

z − ξ
φn(ξ)φ∗

n(ξ)w(ξ) dξ ,

φ′
n(z) = An(z)φn−1(z) − Bn(z)φn(z).

The authors proved that, cf. [20, Theorem 3.1 & Remark 3], if v is a meromorphic
function on the unit disk then

P1,n = P2,n, (1.17)

therefore,Q1,n = Q2,n.
From Ismail and Witte [20, Example 1], for the orthonormal circular Jacobi polynomi-

als φn

φn(z) = (a)n√
n!(2a + 1)n

2F1(−n, a + 1;−a + 1 − n; z),

relation (1.17) gives

φ′′
n +

(
1 − n − a

z
− 2a + 1

1 − z

)
φ′
n + n(1 + a)

z(1 − z)
φn = 0,

that is, by taking the weight associated with the circular Jacobi polynomials, the rela-
tion (1.17) defines a linear differential equation of the form (1.16).

It is reasonable to ask if we drop the differentiability condition over the weight and allow
An and Bn varying in a wider class of functions, does the relation (1.17) give other classes
of OPUC satisfying a linear differential equation of the form (1.13)? In Theorem 1.6, we
give a necessary condition over (bn)n∈N and (cn)n∈N for this to happen. In particular, we
obtain the system of OPUC defined by Rn which is orthogonal with respect to ω. Note that
this weight is not in general a meromorphic function on the unit disk.

Before we enunciate the results in this article, we introduce some auxiliary notation used
in the sequel.

We denote by �θ the region given by

�θ =
{
D∗(0, ε0) ∪ D∗(1, ε1), θ(z) = z(1 − z),
D∗(0, ε0), θ(z) = z,

here D∗(0, ε0) and D∗(1, ε1) are punctured open disks of respective radius ε0 and ε1
sufficiently small and with their respective centres at the points 0 and 1.



INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 45

Let (φn)
∞
n=0 be a sequence of orthogonal polynomials with respect to a Borel measure

supported on the unit circle satisfying (1.16). We denote by {mn}n∈N the set of indices
corresponding to non-null Verblunsky coefficients. Define A0 = B0 = 0, for each n ∈ N

fixed, denote by An(z) and Bn(z) for z ∈ �θ a solution of

φ′
n(z) = An(z)φn−1(z) − Bn(z)φn(z), (1.18)

in particular the functionsAn and Bn are a special case.
Before we enunciate the main result, we need some auxiliary propositions whose proof

will be given in Section 2.

Proposition 1.2: Let (φn)
∞
n=0 be the sequence of orthogonal polynomials with respect to a

Borel measure supported on the unit circle and suppose that (φn)
∞
n=0 satisfies (1.16), then

φn(z) = zn, ∀n ∈ N if and only if there exists n0 ∈ N such that φn0(z) = zn0 .

Proposition 1.3: Let (φn)
∞
n=0 be a sequence of OPUC satisfying (1.16)with φ1(z) �= z, then

{mn}n∈N is infinite and consecutive Verblunsky coefficients cannot vanish simultaneously.

Proposition 1.4: Let (φn)
∞
n=0 be a sequence of OPUC such that φ1(z) �= z and An,Bn

satisfying (1.18). Then for n ≥ 1

φ′′
mn + P1,mnφ

′
mn + Q1,mnφmn = 0, (1.19)

φ′′
mn + P2,mnφ

′
mn + Q2,mnφmn = 0, (1.20)

where

P1,mn = Bmn + Bmn−1 − A′
mn

Amn

+ mn−1

z
− mn

z
+ 1

z
− κmn−1

κmn−1−1

Amn−1

z

− zmn−mn−1−1 κmn

κmn−1−1

φmn−1(0)
φmn(0)

Amn−1 ,

Q1,mn = Amn

(
Bmn

Amn

)′
+ Bmn−1Bmn − κmn−1

κmn−1−1

Amn−1Bmn

z

− κmn

κmn−1−1

φmn−1(0)
φmn(0)

z−mn−1+mn−1Amn−1Bmn

+ κmn−1

κmn−1−1

φmn−1(0)
φmn(0)

z−mn−1+mn−2Amn−1Amn + mn−1 − mn + 1
z

Bmn ,

and

P2,mn = Bmn+1 + Bmn − A′
mn

Amn

− κmn

κmn−1

Amn

z
− zmn+1−mn−1 κmn+1

κmn−1

φmn(0)
φmn+1(0)

Amn + 1
z



46 J. BORREGO-MORELL AND A. SRI RANGA

Q2,mn = Amn

(
Bmn

Amn

)′
+ BmnBmn+1 − κmn

κmn−1

AmnBmn+1

z

− κmn+1

κmn−1

φmn(0)
φmn+1(0)

z−mn+mn+1−1AmnBmn+1

+ φmn(0)
φmn+1(0)

κmn+1−1

κmn−1
z−mn+mn+1−2AmnAmn+1

+ Bmn

z
− κmn+1

κmn−1

φmn(0)
φmn+1(0)

(mn+1 − mn)z−mn+mn+1−2Amn .

Proposition 1.5: Let (φn)
∞
n=0 be a sequence of OPUC satisfying (1.16), φ1(z) �= z. Then

there exist functions Amn and Bmn analytic in �θ satisfying (1.18) such that for n ≥ 1

P1,mn = P2,mn , (1.21)

Q1,mn = Q2,mn , (1.22)

where P1,mn ,Q1,mn and P2,mn ,Q2,mn are as in Proposition 1.4.

By analogy with (1.17), let Pmn be the common value in (1.21). In the present article,
we prove the following theorem.

Theorem 1.6: Let (φn)
∞
n=0 be a sequence of orthonormal polynomials with respect to a

positive Borel measure on the unit circle satisfying (1.16). Then the following statements
hold:

(a) If c1 = 0 the whole sequence reduces to (zn)∞n=0.
(b) If Pmn = Pmn , ∀n ∈ N, then there exist pn, qn, rn ∈ Z, p1, q1 = 0, c1 �= 0, b1 + pn /∈

{−n + 1,−n + 2, . . . , 0} and γn an appropriate non-null complex constant such that

φn(z) = γn

{
2F1(−n, b1 + pn; c1 + qn; z), if θ(z) = z(1 − z),

1F1(−n; c1 + rn; z), if θ(z) = z,
(1.23)

The above theorem gives as a particular case, the sequence (Rn)
∞
n=0 which is orthog-

onal with respect to ω when ω is positive by choosing pn = 0, qn = 1 − n and b1 =
α + β + 1, c1 = β − α; α ∈ R, α > − 1

2 and iβ ∈ R. It seems plausible to conjecture that
for θ(z) = z(1 − z) the family (Rn)

∞
n=0 is the unique sequence of orthogonal polynomials

on the unit circle satisfying (1.16). Furthermore, when θ(z) = z also there exists a unique
sequence and is given by φn(z) = zn.

As a restatement of the above theorem, we obtain

Corollary 1.7: With the same hypothesis of Theorem 1.6,

(a) If c1 = 0 the whole sequence reduces to (zn)∞n=0.
(b) If Pmn = Pmn , ∀n ∈ N, then bn − b1, cn − c1 ∈ Z.



INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 47

2. Proof of the results

The sketch of the proof for Theorem1.6 is as follows.We start by obtaining a pair of second-
order differential equations in terms of two functional coefficients An,Bn satisfied for an
arbitrary sequence of orthogonal polynomials for a measure supported on the unit cir-
cle. This will be done in Proposition 1.4. Later, in Proposition 1.5, we unify the differential
equations obtained in Proposition 1.4. Under the assumption that the sequence of orthogo-
nal polynomials is of hypergeometric or confluent hypergeometric type and the differential
equation, they satisfy is equal to the differential equations obtained in Proposition 1.4,
we obtain a system of equations with An and Bn as functional variables. By analysing the
singular points of An, we obtain the possible sequences of orthogonal polynomials.

Proof of Proposition 1.1: For any n ∈ N, let yn(z) = ∑n
k=0 akz

k be a monic polynomial.
Then yn satisfies (1.13) if and only if

(k + 1)(cn + k)ak+1 + (λn − ηk)ak = 0, 0 ≤ k ≤ n,

an+1 = 0,
(2.1)

where

ηk =
{
k(k − 1 + bn), θ(z) = z(1 − z),
kbn, θ(z) = z.

From (2.1), a0, . . . , an−1 is uniquely determined if and only if λn − ηk �= 0, ∀0 ≤ k ≤
n − 1 and λn = ηn.

For k=n, we have

λn =
{
n(n − 1 + bn), θ(z) = z(1 − z),
nbn, θ(z) = z.

(2.2)

For the case θ(z) = z, it is straightforward that relation (2.2) and bn �= 0 is the necessary
and sufficient conditions for (1.13) to have a unique polynomial solution.

Consider now the case θ(z) = z(1 − z). The condition λn − ηk �= 0, ∀0 ≤ k ≤ n − 1
is equivalent to saying that bn /∈ {−2n + 2,−2n + 3, . . . ,−n,−n + 1} and this completes
the proof of the existence and uniqueness of a polynomial solution of degree n for (1.13).

Assume now that λn is given as in (1.14). It follows from the theory for the hypergeomet-
ric (confluent hypergeometric) equation, see [18, p.56, 248] that a holomorphic solution
in a neighbourhood of z=0 for (1.13) is given by

πn(z) = const.

{
2F1(−n, bn + n − 1; cn; z), if θ(z) = z(1 − z),

1F1(−n; cn; bnz), if θ(z) = z,

the condition bn /∈ {−2n + 2,−2n + 3, . . . ,−n,−n + 1} for θ(z) = z(1 − z) implies that
bn + n − 1 /∈ {0,−1, . . . ,−n + 1}, therefore, from the existence and uniqueness condition
it follows that πn is the polynomial solution of degree n. �

Proof of Proposition 1.2: Assume that there exists n0 ∈ N such that �n0(z) = zn0 . From
the recurrence formula for the monic polynomials (1.4), it is straightforward that�n(z) =



48 J. BORREGO-MORELL AND A. SRI RANGA

zn, n ≤ n0. A straightforward argument by induction shows that �n(z) = zn, n ≥ n0.
Indeed, for n = n0 + 1, from the recurrence formula for the monic polynomials (1.4)

�n0+1(z) = zn0+1 + �n0+1(0), |�n0+1(0)| < 1. (2.3)

From Proposition 1.1

�n0+1(z) = const.

{
2F1(−n0 − 1, bn0+1 + n0; cn0+1; z), if θ(z) = z(1 − z),

1F1(−n0 − 1; cn0+1; z), if θ(z) = z,
(2.4)

therefore, from (1.9) and (1.11) we have that �n0+1(0) = 0, hence φn(z) = zn, ∀n ∈ N.
The converse implication is trivial. �

Proof: Let us assume that {mn}n∈N is finite. Denote μ = max{mn}n∈N, by hypothesis
φ1(z) �= z, hence μ ≥ 1. Since φμ satisfies (1.16), from Proposition 1.1, it follows that

φμ(z) = γμ

{
2F1(−μ, bμ + μ − 1; cμ; z), θ(z) = z(1 − z),

1F1(−μ; cμ; z), θ(z) = z,
(2.5)

where γμ ∈ C \ {0} is an appropriate constant and cμ /∈ {−μ + 1, . . . , 0}.
From the condition φn(0) = 0 for n > μ, we have that κμ = κn, therefore from (1.3)

and (2.5)

φn(z) = zn−μφμ(z). (2.6)

If cn ∈ N ∪ {0}, cn < n, then

2F1(−n, bn + n − 1;−cn; z) = zcn+1
2F1(−n + cn + 1, bn + cn + n; cn + 2; z),

1F1(−n;−cn; z) = zcn+1
1F1(−n + cn + 1; cn + 2; z).

(2.7)

Since φn satisfies (1.16) and φn(0) = 0, for n > μ, Proposition 1.1 gives

φn(z) = constn

{
2F1(−n, bn + n − 1;−cn; z), θ(z) = z(1 − z),

1F1(−n;−cn; z), θ(z) = z,
(2.8)

where cn ∈ N ∪ {0}, cn < n, therefore from (2.7) and (2.8)

φn(z) = constnzcn+1

{
2F1(−n + cn + 1, bn + cn + n; cn + 2; z), θ(z) = z(1 − z),

1F1(−n + cn + 1; cn + 2; z), θ(z) = z.
(2.9)

Since the order of the zero at z=0 in the relations (2.6) and (2.9) must coincide, we
obtain cn = n − μ − 1. From (2.6), limz→0(φn(z)/zn−μ) = γμ and from (2.9) with cn =
n − μ − 1 it follows that constn = γμ.

We prove now that (2.6) gives a contradiction, unless n = μ + 1. As a consequence, we
have that consecutive Verblunsky coefficients cannot vanish simultaneously.



INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 49

Indeed, by comparing the coefficients of the expressions (2.6) and (2.9) that define the
polynomial φn, we find that for 1 ≤ k ≤ n and n > μ

(bn + 2n − μ − 1)k
(n − μ + 1)k

= (bμ + μ − 1)k
(cμ)k

, θ(z) = z(1 − z),

(n − μ + 1)k = (cμ)k, θ(z) = z.
(2.10)

Consider the case θ(z) = z(1 − z). Relation (2.10) for k=1 and k=2 gives

bn + 2n − μ − 1
n − μ + 1

= bμ + μ − 1
cμ

, (2.11)

(bn + 2n − μ − 1)(bn + 2n − μ)

(n − μ + 1)(n − μ + 2)
= (bμ + μ − 1)(bμ + μ)

cμ(cμ + 1)
. (2.12)

From (2.11) and (2.12)

bn + 2n − μ

n − μ + 2
= bμ + μ

cμ + 1
. (2.13)

From relations (2.11) and (2.13)

(cμ − n + μ − 1)(−bμ + cμ − μ + 1)
cμ(cμ + 1)

= 0, (2.14)

it follows that

bμ = cμ − μ + 1, (2.15)

cμ = n − μ + 1. (2.16)

The alternative (2.15) is not possible. Indeed, by substituting the value bμ obtained in
(2.15) into (2.5) we find that

φμ(z) = γμ2F1(−μ, cμ; cμ; z),

which is impossible, since |�μ(0)| = 1.
Since cμ is fixed, relation (2.16) is only possible for n = μ + 1, it follows that (2.6) only

holds for n = μ + 1 and in this case cμ = 2, bμ+1 = bμ − 2.
Consider now the case θ(z) = z. Relation (2.10) gives cμ = n − μ + 1, n > μ, that is,

relation (2.16) is only possible for n = μ + 1. We conclude that (2.6) only holds for n =
μ + 1 with cμ = 2 and this completes the proof of the lemma. �

Proof: From Proposition 1.3, the set {mn}n∈N is infinite, therefore, from the recurrence
relations (1.3), for n ≥ 2

κmn−1−1zφmn−1−1(z) = κmn−1φmn−1(z) − φmn−1(0)φ
∗
mn−1

(z), (2.17)

κmn−1φmn(z) = zκmnφmn−1(z) + φmn(0)φ
∗
mn−1(z). (2.18)



50 J. BORREGO-MORELL AND A. SRI RANGA

Since {mn}n∈N is the set of indices for which φmn(0) �= 0, then κmn−1 = κmn−1, hence

φmn−1(z) = zmn−mn−1−1φmn−1(z), (2.19)

φ∗
mn−1(z) = φ∗

mn−1(z). (2.20)

Relations (2.17)–(2.20) give

z(κmn−1−1φmn(0)φmn−1−1(z) − φmn−1(0)κmnz
mn−mn−1−1φmn−1(z))

= κmn−1φmn(0)φmn−1(z) − κmn−1φmn−1(0)φmn(z). (2.21)

Let An,Bn be such that (1.18) holds. Define the operators

Tmn,1 = d
dz

+ Bmn(z),

Tmn,2 = − d
dz

− Bmn−1(z) + Amn−1(z)
κmn

κmn−1−1

φmn−1(0)
φmn(0)

zmn−mn−1−1 + Amn−1(z)
z

κmn−1

κmn−1−1
.

From (2.21) and (1.18), the operators Tmn,1 and Tmn,2 are anhilation and creation
operators in the sense that they satisfy

Tmn,1[φmn(z)] = Amn(z)z
mn−mn−1−1φmn−1(z), (2.22)

Tmn,2[φmn−1(z)] = Amn−1(z)
z

κmn−1

κmn−1−1

φmn−1(0)
φmn(0)

φmn(z), (2.23)

Relations (2.22) and (2.23) give that φn satisfy the differential equations

Tmn,2

[
1

Amn(z)zmn−mn−1−1Tmn,1[φmn(z)]
]

= Amn−1(z)
z

κmn−1

κmn−1−1

φmn−1(0)
φmn(0)

φmn(z), (2.24)

Tmn+1,1

[
z

Amn(z)
Tmn+1,2[φmn(z)]

]

= Amn+1(z)z
mn+1−mn−1 κmn+1−1

κmn−1

φmn(0)
φmn+1(0)

φmn(z). (2.25)

By expanding (2.24) and (2.25) we obtain the coefficients of the differential equation for
n ≥ 2. From the definition ofA0 = B0 = 0 we find that the lemma also holds for n=1 and
this completes the proof. �

Lemma 2.1: The Riccati equation

y′ + P1y − y2 − Q1 = 0, (2.26)

has an analytic solution in �θ .



INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 51

Proof: The change of variable

y = v′

v
, (2.27)

transforms the differential equation (2.26) in the second-order differential equation

v′′ − P1v′ + Q1v = 0. (2.28)

Let us consider the case θ(z) = z(1 − z) and express (2.28) as the hypergeometric
differential equation

z(1 − z)v′′ + (−c1 − (η1 + η2 + 1)z)v′ − η1η2v = 0, (2.29)

where η1, η2 are the solutions of the equation

η2 + (b1 + 1)η − b1 = 0.

From the theory of hypergeometric functions [18, Section 2.1.1 (3)], we find that the
function

u(z) =
{
2F1(η1, η2;−c1; z), c1 /∈ N ∪ {0},
zc1+1

2F1(η1 + c1 + 1, η2 + c1 + 1; c1 + 2; z), c1 ∈ N ∪ {0}, (2.30)

is an holomorphic solution to (2.29) at z=0, therefore, from (2.27) we have that there exists
an analytic solution for the Equation (2.26) of the form

y(z) = u′(z)
u(z)

, z ∈ D∗(0, ε0), (2.31)

where ε0 is sufficiently small.
For the point z=1 we consider [18, Section 2.9 (5)], that is

u(z) =
{
2F1(η1, η2; c1 − b1; 1 − z), b1 − c1 /∈ N ∪ {0},
(1 − z)b1−c1+1

2F1(−η2 − c1,−η1 − c1; b1 − c1 + 2; 1 − z), b1 − c1 ∈ N,

which is holomorphic in a neighbourhood of z=1. Applying now (2.27) we obtain the
lemma for θ(z) = z(1 − z).

The analysis for the case θ(z) = z is similar. �

Lemma 2.2: There exist analytic functions A1,B1 in�θ such that the following system holds

P1(z) = −A′
1(z)

A1(z)
+ B1(z), (2.32)

Q1(z) = B′
1(z) − B1(z)

A′
1(z)

A1(z)
, (2.33)

κ1 = A1(z) − B1(z)φ1(z). (2.34)



52 J. BORREGO-MORELL AND A. SRI RANGA

Proof: From (1.16)

κ1P1(z) + Q1(z)φ1(z) = 0.

From this last relation, it follows that the substitution of (2.34) into (2.32) or (2.33) gives
the Riccati equation

B′
1 + B1P1 − B21 − Q1 = 0. (2.35)

From Lemma 2.1, there exists an analytic solution for (2.35) in �θ . The statement for
the function A1 follows immediately from (2.34). �

Proof of Proposition 1.5: For n=1 we obtain the system defined by (2.32)–(2.34) and
from Lemma 2.2, there exists analytic functions in �θ , A1 and B1 such that this system
holds.

For n ≥ 1, let us write

P1,mn(z) = �mn(z) − A′
mn(z)

Amn(z)
, (2.36)

where

�mn(z) = Bmn(z) + Bmn−1(z) − κmn−1

κmn−1−1

Amn−1(z)
z

− zmn−mn−1−1 κmn

κmn−1−1

φmn−1(0)
φmn(0)

Amn−1(z) + mn−1

z
− mn

z
+ 1

z
, (2.37)

and �1 = B1.
Note that if (1.18) and (1.21) hold then (1.22) also holds. Indeed, since φmn satis-

fies (1.19) and (1.20)

φ′′
mn + P1,mnφ

′
mn + Q1,mnφmn = 0,

φ′′
mn + P1,mnφ

′
mn + Q2,mnφmn = 0,

the difference between these two relations gives Q1,mn = Q2,mn , hence, Equation (1.22) is
redundant.

From the expressions for P1,mn and P2,mn given in Proposition 1.4 we find that the
relation P1,mn = P2,mn in (1.21) holds if and only if �mn satisfies the difference equation

�mn+1(z) = �mn(z) + mn

z
− mn+1

z
,

�1(z) = B1(z).
(2.38)

By hypothesis φ1(z) �= z. Hence, from Proposition 1.2, m1 = 1, it follows that the
solution to the difference Equation (2.38) for n ≥ 1 is

�mn(z) = B1(z) + 1 − mn

z
. (2.39)

From (2.36) and (2.38) we have that (1.21) holds. Therefore, for n ≥ 1, if �mn is given
by (2.39), we define the analytic functions in �θ , Amn and Bmn as the unique solution of



INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 53

(1.18) and (2.37). We have that (1.18), (1.21) and (1.22) hold and this completes the proof
of the lemma. �

From the preceding lemmas we now are able to give a proof of Theorem 1.6.

Proof of Theorem 1.6: Statement (a) follows immediately from Proposition 1.2.
To prove (b) note that we have two cases, the set of indices {mn}n∈N for which φmn(0) �=

0 is finite or infinite. Results in Propositions 1.2 and 1.3 show that the first case implies that
the whole sequence reduces to φn(z) = zn.

Let us analyse the second case.
FromProposition 1.5 there existAmn and Bmn such that P1,mn = P2,mn . Fromhypothesis

Pmn = P1,mn = P2,mn . Therefore, from (1.21), (2.36) and (2.39)

Pmn(z) = 1 − mn

z
− A′

mn(z)
Amn(z)

+ B1(z). (2.40)

We have that φ1(z) �= z and from (2.40) of Proposition 1.5

P1(z) = −A′
1(z)

A1(z)
+ B1(z),

Pmn(z) = 1 − mn

z
− A′

mn(z)
Amn(z)

+ B1(z), n > 1.

Therefore, A′
mn/Amn = A′

1/A1 + πn, where

πn(z) =

⎧⎪⎨
⎪⎩
c1 − cmn + 1 − mn − (b1 − bmn + 1 − mn)z

θ(z)
, θ(z) = z(1 − z),

c1 − cmn + 1 − mn

θ(z)
, θ(z) = z.

That is

Amn = A1τn, (2.41)

where

τn(z) = c

{
zumn (1 − z)vmn , θ(z) = z(1 − z),
zumn , θ(z) = z,

here c ∈ C \ {0} is an appropriate constant and

umn = c1 − cmn + 1 − mn (2.42)

vmn = b1 − bmn − (c1 − cmn). (2.43)

From Proposition 1.5 and from the assumption of the theorem, there exists B1 analytic
in �θ , hence, from Proposition 1.5, the function Amn is analytic in �θ . Therefore, from
(2.41), (2.42) and (2.43), we have umn , vmn ∈ Z, ∀n ≥ 2.



54 J. BORREGO-MORELL AND A. SRI RANGA

Now, from (2.42) and (2.43)

φmn(z) = γmn2F1(−mn, b1 + pmn ; c1 + qmn ; z),

where pmn , qmn ∈ Z; p1 = 0, q1 = 0, c1 �= 0, b1 + pmn /∈ {−mn + 1,−mn + 2, . . . , 0} and
γmn an appropriate complex constant.

For the indices kn such that φkn(0) = 0

φkn(z) = γkn2F1(−kn, b1 + pkn ; c1 + qkn ; z),

where pkn , qkn ∈ Z; c1 + qkn = 0, b1 + pkn /∈ {−kn + 1,−kn + 2, . . . , 0} and γkn an appro-
priate complex constant, and this completes the proof of theorem. �

Corollary 1.7 follows immediately from relation (1.23).

Acknowledgments

We thank the anonymous reviewer for the careful reading of our article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

The first author acknowledges the financial support of FAPESP of Brazil, under (grant no.
2012/21042–0). The second author acknowledges the support of CNPq (grant no. 305073/2014–1)
and FAPESP (grant no. 2009/13832–9).

References

[1] Bochner S. Über sturm–liouvillesche polynomsysteme. Math Z. 1929;89:730–736.
[2] Routh EJ. On some properties of certain solutions of a differential equation of the second order.

Proc London Math Soc. 1885;S1–16(1):245–262.
[3] Askey R. Discussion of Szegö’s paper “Beiträge zur Theorie der Toeplitzschen Formen”.

In: Askey R, editor. Gabor Szegö. Collected works. Vol. I. Boston, MA: Birkhäuser; 1982;
p. 303–305.

[4] Askey R. Some open problems about special functions and computations. Rend SemMat Univ
Politec Torino, Special Volume. 1985:1–22.

[5] Temme NM. Uniform asymptotic expansion for a class of polynomials biorthogonal on the
unit circle. Constr Approx. 1986;2:369–376.

[6] Sri Ranga A. Szegö polynomials from hypergeometric functions. Proc Amer Math Soc.
2010;138(12):4259–4270.

[7] Witte NS, Forrester PJ. Gap probabilities in the finite and scaled Cauchy random matrix
ensembles. Nonlinearity. 2000;13:1965–1986.

[8] Szegő G. Orthogonal polynomials. 4th ed. Providence, RI: American Mathematical Society;
1975. (American Mathematical Society Colloquium Publication; Vol 23).

[9] Geronimus LY. Polynomials orthogonal on a circle and their applications. American Mathe-
matical Society Translation; Vol. 3, p. 1–78. Providence, RI: American Mathematical Society;
1962.

[10] Geronimus LY. Orthogonal polynomials. New York: Consultants Bureau; 1971.
[11] Geronimus LY. Orthogonal polynomials. American Mathematical Society Translation;

Vol. 108, p. 3–130. Providence, RI: American Mathematical Society; 1977.



INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 55

[12] Freud G. Orthogonal polynomials. Oxford: Pergamon Press; 1971.
[13] Van AsscheW. Orthogonal polynomials in the complex plane and on the real line. Fields Instit

Commun. 1997;14:211–245.
[14] Simon B. Orthogonal polynomials on the unit circle, Part 1: Classical theory. Providence, RI:

American Mathematical Society; 2005 (AMS Colloquium Series).
[15] Simon B. Orthogonal polynomials on the unit circle. Part 2: Spectral theory. Providence, RI:

American Mathematical Society; 2005 (AMS Colloquium Series).
[16] Ismail ME. Classical and quantum orthogonal polynomials in one variable. Cambridge, UK:

CambridgeUniversity Press; 2005. (Encyclopedia ofMathematics and Its Applications; Vol 98).
[17] Nevai P. Géza Freud, orthogonal polynomials, and Christoffel functions. A case study. J Approx

Theory. 1986;48:3–167.
[18] BatemanH, Erdélyi A. Higher transcendental functions, Vol. I. New York: McGraw-Hill; 1953.
[19] Ismail MEH. An electrostatics model for zeros of general orthogonal polynomials. Pacific J

Math. 2000;193:355–369.
[20] Ismail MEH, Witte NS. Discriminants and functional equations for polynomials orthogonal

on the unit circle. J Approx Theory. 2001;110:200–228.


	1. Introduction
	1.1. Statement of the results
	1.1.1. Raising and lowering operators for orthogonal polynomials on the unit circle.


	2. Proof of the results
	Acknowledgments
	Disclosure statement
	Funding
	References