PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 139, Number 11, November 2011, Pages 3929–3936
S 0002-9939(2011)10806-2
Article electronically published on March 10, 2011

MONOTONICITY AND ASYMPTOTICS OF ZEROS OF

LAGUERRE-SOBOLEV-TYPE ORTHOGONAL POLYNOMIALS

OF HIGHER ORDER DERIVATIVES

FRANCISCO MARCELLÁN AND FERNANDO R. RAFAELI

(Communicated by Walter Van Assche)

Abstract. In this paper we analyze the location of the zeros of polynomials
orthogonal with respect to the inner product

(0.1) 〈p, q〉 =
∫ ∞

0
p(x)q(x)xαe−xdx+Np(j)(0)q(j)(0),

where α > −1, N ≥ 0, and j ∈ N. In particular, we focus our attention on
their interlacing properties with respect to the zeros of Laguerre polynomials
as well as on the monotonicity of each individual zero in terms of the mass
N. Finally, we give necessary and sufficient conditions in terms of N in order
for the least zero of any Laguerre-Sobolev-type orthogonal polynomial to be
negative.

1. Introduction and statement of results

Given a nontrivial probability measure μ supported in an infinite subset of the
real line and a real number c, sequences of polynomials orthogonal with respect to
the Sobolev inner product,

(1.1) 〈p, q〉 =
∫
R

p(x)q(x)dμ(x) +Np(j)(c)q(j)(c),

have been introduced in [11]. There, the authors obtain a higher order recurrence
relation that such polynomials satisfy as well as second order linear differential
equations of holonomic type assuming that the measure μ is semi-classical. The
zero distribution of such polynomials when the support of μ is located in the positive
real semi-axis and c = 0 is analyzed in [12]. There the author proves that the zeros
of the polynomials of degree n are real, simple, and at most one of them is outside
(0,+∞). The location of these zeros is given in terms of the location of the zeros
of the polynomials orthogonal with respect to the measures dμ and x2dμ.

Next, we will focus our attention on a canonical case of the above inner product
related to the Laguerre weight. Indeed, let us consider the sequence of polynomials

Received by the editors November 25, 2009 and, in revised form, September 7, 2010.
2010 Mathematics Subject Classification. Primary 42C05; Secondary 33C47.
Key words and phrases. Laguerre orthogonal polynomials, Laguerre-Sobolev-type orthogonal

polynomials, zeros, interlacing, monotonicity, asymptotics.

c©2011 American Mathematical Society
Reverts to public domain 28 years from publication

3929

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3930 FRANCISCO MARCELLÁN AND FERNANDO R. RAFAELI

{L(α,N)
n (x)}∞n=0 orthogonal with respect to the Sobolev-type inner product

(1.2) 〈p, q〉 =
∫ ∞

0

p(x)q(x)xαe−xdx+Np(j)(0)q(j)(0),

where α > −1, N ≥ 0, and j ∈ N as well as the Laguerre polynomials {L(α)
n (x)}∞n=0,

normalized as in Szegő’s book [13], i.e., L
(α)
n (0) =

(
n+α
n

)
.

The case j = 1 has attracted the interest of many researchers from many points
of view. First, the analysis of the asymptotic properties of such orthogonal poly-
nomials is done (see [1] and the survey paper [10]). Second, the study of analytic
properties of their zeros such as interlacing with the zeros of Laguerre polynomi-
als and monotonicity of each zero in terms of N, among others, are completely
discussed in [4]. Furthermore, the limit behavior of such zeros is also deduced in
[10]. Third, the study of spectral properties of these orthogonal polynomials in the
sense that they are eigenfunctions of an infinite order linear differential operator
which reduces to order 2α + 8 if α is a nonnegative integer number and N > 0 is
presented in [9]. Finally, a second order linear differential equation of holonomic
type that such polynomials satisfy is deduced in [7]. As an immediate consequence,
an electrostatic interpretation of these zeros is given.

In recent years, some attention was paid to the asymptotic properties of these
orthogonal polynomials when j > 1. In particular, in [6] the authors obtain a
Mehler-Heine formula for such polynomials as well as their outer relative asymp-
totics in terms of the standard Laguerre orthogonal polynomials. On the other
hand, in [2] it is proved that these polynomials are eigenfunctions of an infinite
order linear differential operator which reduces to order 2α + 4 + 4j if α is a non-
negative integer number and N > 0. But, the analysis of their zeros in terms of N
remains an open problem. This is the aim of our contribution.

The structure of the manuscript is as follows. In Section 2, a connection formula
for the sequences of standard Laguerre orthogonal polynomials and the Laguerre-
Sobolev type orthogonal polynomials is given. Notice that this connection formula
is quite different from the connection formula given in Theorem 3.1 of [12]. It will
be very useful in the study of some analytic properties of the zeros of Laguerre-
Sobolev-type orthogonal polynomials. Indeed, in Section 3 we deduce interlacing
properties between the zeros of such sequences as well as locating the least zero of

L
(α,N)
n (x) with respect to the interval (0,+∞) in terms of the massN . Furthermore,

we prove that each zero is a decreasing function of N and we analyze its limit when
N tends to infinity as well as the speed of convergence. These results are new in
the literature as far as we know. Some numerical examples allow us to check our
results.

2. Connection formula

In the following, we adopt the convention
(
i
0

)
:= 1 and

(
i

−j

)
:= 0, i, j > 0.

Theorem 1. Let {L(α,N)
n (x)}∞n=0 be the Laguerre-Sobolev-type orthogonal polyno-

mials corresponding to the inner product (1.2). Then, for every n > j and each
α > −1,

(2.3) L(α,N)
n (x) =

j+1∑
k=0

An,kL
(α+k)
n−k (x),

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LAGUERRE-SOBOLEV-TYPE ORTHOGONAL POLYNOMIALS 3931

where

(2.4)

An,0 = 1 +N
Γ(j + 1)

Γ(α+ j + 1)

j+1∑
k=1

(−1)k+1

(
n+ α

n− j − k

)(
n− k

j + 1− k

)
,

An,k = (−1)kN
Γ(j + 1)

Γ(α+ j + 1)

(
n+ α

n− j

)(
n− k

j + 1− k

)
, k = 1, . . . , j + 1.

Proof. Let {L(α,N)
n (x)}∞n=0 be the sequence of polynomials defined by (2.3) and

(2.4). We shall prove the orthogonality of these polynomials with respect to the
inner product (1.2). Denote by πn the set of polynomials of degree at most n. Let
pl+j+1(x) = xj+1ql(x), l = 0, 1, . . . , n−(j+2), where ql(x) is a polynomial of degree
exactly l. Thus, the polynomials 1, x, . . . , xj and pl+j+1(x), l = 0, 1, . . . , n− (j+2),
constitute a basis in πn−1. Then, for n ≥ j + 2,

〈pl+j+1, L
(α,N)
n 〉 =

∫ ∞

0

xj+1ql(x)L
(α,N)
n (x)xαe−xdx

=

j+1∑
k=0

An,k

∫ ∞

0

xj+1−kql(x)L
(α+k)
n−k (x)xα+ke−xdx

= 0,

because of the orthogonality property of the classical Laguerre polynomials. Now,
we define the integral Ik,i by (see [8, (3.3)])

Ik,i =

∫ ∞

0

L
(α+i)
n−i (x)xα+ke−xdx =

(
n− k − 1

n− i

)
Γ(α+ k + 1).

Thus,

〈1, L(α,N)
n 〉 =

∫ ∞

0

L(α,N)
n (x)xαe−xdx

= An,0.0 +An,1.I0,1 +An,2.I0,2 + · · ·+An,j .I0,j +An,j+1.I0,j+1,

〈x, L(α,N)
n 〉 =

∫ ∞

0

xL(α,N)
n (x)xαe−xdx

= An,0.0 +An,1.0 +An,2.I1,2 + · · ·+An,j .I1,j +An,j+1.I1,j+1,

...

〈xj−1, L(α,N)
n 〉 =

∫ ∞

0

xj−1L(α,N)
n (x)xαe−xdx

= An,0.0 +An,1.0 +An,2.0 + · · ·+An,j .Ij−1,j +An,j+1.Ij−1,j+1,

and, finally,

〈xj , L
(α,N)
n 〉 =

∫ ∞

0

xjL(α,N)
n (x)xαe−xdx+Nj!

[
L(α,N)
n (x)

](j) ∣∣∣∣
x=0

= An,0.0 +An,1.0 +An,2.0 + · · ·+An,j .0 +An,j+1.I0,j+1

+NΓ(j + 1)(−1)j
j+1∑
k=0

An,kL
(α+k+j)
n−k−j (0).

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3932 FRANCISCO MARCELLÁN AND FERNANDO R. RAFAELI

Using the formula of Ik,i, the identity (5.1.7) in Szegő’s book [13], and the expression
of An,k, 0 ≤ k ≤ j + 1, given in Theorem 1, our statement follows. �

Notice that a different expression for the connection formula appears in Theo-
rem 3.1 in [12]. Using the structure relation for Laguerre polynomials and after
cumbersome computations, (2.3) can be deduced. Our proof is more simple as well
as here the connection coefficients are explicitly given.

Lemma 1. For n > j and each α > −1, the inequalities

An,0 ≥ 1, (−1)kAn,k ≥ 0, k = 1, 2, . . . , j + 1

hold.

Proof. The positivity of the numbers (−1)kAn,k, 1 ≤ k ≤ j+1 is obvious. To prove
that An,0 − 1 ≥ 0, notice that(

n+ α

n− j − ν

)(
n− ν

j + 1− ν

)
−
(

n+ α

n− j − ν − 1

)(
n− ν − 1

j − ν

)
≥ 0

holds for n > j, α > −1, and ν = 1, 3, 5, . . . , j or j − 1 depending on the parity of
j. �

3. The zeros

From (2.3), L
(α,N)
n (x) can be written as

(3.5) L(α,N)
n (x) = (1 + Ãn,0N)L(α)

n (x) +

j+1∑
k=1

NÃn,kL
(α+k)
n−k (x),

where Ãn,0 = (An,0−1)/N and Ãn,k = An,k/N , k = 1, . . . , j+1. Now, we introduce
the polynomial

(3.6) Fn,α,l(x) = −
j+1∑
k=0

Ãn,kL
(α+k)
n−k (x).

Then, (3.5) reads

(3.7) L(α,N)
n (x) = L(α)

n (x)−NFn,α,l(x).

Let us denote by xN
n,k(α), xn,k(α), and ζn,k(α) the zeros of L

(α,N)
n (x), L

(α)
n (x),

and Fn,α,l(x), respectively, arranged in a decreasing order. It was proved in [6],

Theorem 3, that at most one of the zeros of L
(α,N)
n (x) is located outside (0,∞).

Notice that this is a particular case of Theorem 4.1 in [12].
In the present work, we will give explicitly the value N0 of the mass such that

for N > N0 this situation occurs; i.e., the least zero is negative. Moreover, in [6], it
was shown that xN

n,n(α) < xn,n(α) < · · · < xN
n,1(α) < xn,1(α) (see also Theorem 4.3

in [12]). In such a sense, here we complete this result proving that the zeros xN
n,k(α)

and xn,k(α) interlace with the zeros ζn,k(α).

Theorem 2. For every n > j and each α > −1, the inequalities

(3.8) ζn,n(α) < xN
n,n(α) < xn,n(α) < · · · < ζn,1(α) < xN

n,1(α) < xn,1(α)

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LAGUERRE-SOBOLEV-TYPE ORTHOGONAL POLYNOMIALS 3933

hold. Moreover, the smallest zero xN
n,n(α) satisfies

xN
n,n(α) > 0, for N < N0,

xN
n,n(α) = 0, for N = N0,

xN
n,n(α) < 0, for N > N0,

where

N0 = N0(n, α, j) =
L
(α)
n (0)

Fn,α,l(0)
=

(
n+ α

n

)

−
j+1∑
k=0

Ãn,k

(
n+ α

n− k

) .

Proof. Observe that the leading coefficients of L
(α,N)
n (x), L

(α)
n (x), and Fn,α,l(x) are

equal to

(−1)nAn,0

n!
,

(−1)n

n!
and

(−1)n−1Ãn,0

n!
,

respectively. Since (see [6, Thm. 4]) xN
n,n(α) < xn,n(α) < · · · < xN

n,1(α) < xn,1(α),
then, using the connection formula (3.7), we get

sign
(
L(α,N)
n (xn,k(α))

)
= −sign

(
Fn,α,l(xn,k(α))

)
= (−1)n−k+1

and

sign
(
L(α)
n (xN

n,k(α))
)
= sign

(
Fn,α,l(x

N
n,k(α))

)
= (−1)n−k,

for k = 1, . . . , n. Thus, the interlacing property (3.8) follows. In order to investi-
gate the location of xN

n,n(α) with respect to the origin, it suffices to observe that

L
(α,N)
n (0) = 0 if and only if N = N0. �
In order to show the behavior of the least zero xN

n,n(α) of the Laguerre-Sobolev-
type orthogonal polynomials, we will do some numerical computations using the
Mathematica software. We present, for n, j, and α fixed, a table that shows the
behavior of xN

n,n(α) with respect to N . In particular, for N > N0, the least zero is
negative.

For the case j = 1, N0 reduces to

N0 = N0(n, α, 1)=
Γ(n− 1)Γ(α+ 2)Γ(α+ 4)

Γ(n+ α+ 2)
.

Then, for n = 2, 3 and α = −1/2, 1, 5 we get

N xN
2,2(−1/2) N xN

2,2(1) N xN
2,2(5)

1/2 0.115964 3/2 0.271499 710 0.0419159
N0 =

√
π/2 0 N0 = 2 0 N0 = 720 0

1 -0.0313955 5/2 -0.230139 730 -0.0414199

N xN
3,3(−1/2) N xN

3,3(1) N xN
3,3(5)

1/4 0.00211646 1/5 0.407703 79 0.0251697
N0 =

√
π/7 0 N0 = 2/5 0 N0 = 80 0

1/2 -0.133233 3/5 -0.275762 81 -0.0248324

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3934 FRANCISCO MARCELLÁN AND FERNANDO R. RAFAELI

For the case j = 2, N0 reduces to

N0 = N0(n, α, 2) =
Γ(n− 2)Γ(α+3)Γ(α+6)

2[n(α+4)− (α+2)]Γ(n+α+2)
.

Thus, for n = 3, and α = −1/2, α = 1, α = 5 we obtain

N xN
3,3(−1/2) N xN

3,3(1)

1/4 0.0715899 1 0.415775
N0 = 3

√
π/16 0 N0 = 3/2 0

1/3 -0.00100291 2 -0.450761

N xN
3,3(5)

1259 0.00317496
N0 = 1260 0

1261 -0.00317424

For the case j = 3, N0 reduces to

N0 = N0(n, α, 3)

=
Γ(n− 3)Γ(α+ 4)Γ(α+ 8)

3[n2(α+ 5)(α+ 6)− 3n(α+ 4)(α+ 5) + 2(α+ 2)(α+ 3)]Γ(n+ α+ 2)
.

Thus, for n = 4, and α = −1/2, α = 1, α = 5 we obtain

N xN
4,4(−1/2) N xN

4,4(1)

1/10 0.102777 1 0.322548
N0 = 5

√
π/48 0 N0 = 4/3 0

2/10 -0.0346506 5/3 -0.41383

N xN
4,4(5)

2239 0.00200886
N0 = 2240 0

2241 -0.00200899

In the next result, we obtain the monotonicity of the zeros xN
n,k(α) with respect

to N as well as their convergence when N tends to infinity to the zeros ζn,k(α) with
a speed of convergence of order 1/N . For the proof of these statements, we need
the following lemma concerning the behavior of the zeros of linear combinations of
two polynomials with interlacing zeros.

Lemma 2. Let hn(x) = (−1)na(x − x1) · · · (x − xn) and gn(x) = (−1)n−1b(x −
ζ1) · · · (x− ζn) be polynomials with real and interlacing zeros,

ζn < xn < · · · < ζ1 < x1,

where a and b are real positive constants. Then, for any real constant c > 0, the
polynomial

f(x) = hn(x)− cgn(x)

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LAGUERRE-SOBOLEV-TYPE ORTHOGONAL POLYNOMIALS 3935

has n real zeros ηn < ηn−1 < · · · < η1 which interlace with both the zeros of hn(x)
and gn(x) in the following form:

ζn < ηn < xn < · · · < ζ1 < η1 < x1.

Moreover, each ηk is a decreasing function of c and, for each k = 1, . . . , n,

(3.9) lim
c→∞

ηk = ζk and lim
c→∞

c[ηk − ζk] =
hn(ζk)

g′n(ζk)
.

Recent results concerning zeros of linear combinations of orthogonal polynomials
have been used in [4] and [5] in order to analyze monotonicity and the asymptotics
for the zeros of some class of orthogonal polynomials. We omit here the proof of
the above lemma (see [3, Lemma 1] and [5, Lemma 3]).

Theorem 3. For every n > j and each α > −1, the zeros xN
n,k(α) are decreasing

functions of N . Moreover,

lim
N→∞

xN
n,k(α) = ζn,k(α)

and
lim

N→∞
N [xN

n,k(α)− ζn,k(α)] = gn,k,l(α), k = 1, . . . , n,

where

gn,k,l(α) =
L
(α)
n (ζn,k(α))

F ′
n,α,l(ζn,k(α))

.

Proof. This is an immediate consequence of (3.7), the inequalities (3.8), and Lem-
ma 2. �

Acknowledgements

We thank the referee for suggestions in order to improve the presentation of the
manuscript. In particular, we have emphasized the connections of our results with
some previous ones done in [12] (for section 2) and [3] (for section 3), where quite
different approaches are presented. The work of the first author has been supported
by Dirección General de Investigación, Ministerio de Ciencia e Innovación of Spain,
grant MTM2009-12740-C03-01. The work of the second author has been supported
by FAPESP under grant 07/02854-6. This manuscript was finished during the
second author’s stay at the Departmento de Matemáticas, Universidad Carlos III
de Madrid, in the framework of a joint project of Dirección General de Investigación,
Ministerio de Educación y Ciencia of Spain and the Brazilian Science Foundation
CAPES, Project CAPES/DGU 160/08.

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Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos

III de Madrid, 28911 Leganés, Spain

Instituto de Matemática, Estat́ıstica e Computação Cient́ıfica, Universidade

Estadual de Campinas, São Paulo, Brazil

Current address: Departamento de Matemática, Estat́ıstica e Computação/FCT, Universidade
Estadual Paulista-UNESP, 19060-900 Presidente Prudente, São Paulo, Brazil

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http://www.ams.org/mathscinet-getitem?mr=2581392
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http://www.ams.org/mathscinet-getitem?mr=1433121
http://www.ams.org/mathscinet-getitem?mr=1433121
http://www.ams.org/mathscinet-getitem?mr=2273888
http://www.ams.org/mathscinet-getitem?mr=2273888
http://www.ams.org/mathscinet-getitem?mr=1106092
http://www.ams.org/mathscinet-getitem?mr=1106092
http://www.ams.org/mathscinet-getitem?mr=1201003
http://www.ams.org/mathscinet-getitem?mr=1201003
http://www.ams.org/mathscinet-getitem?mr=0372517
http://www.ams.org/mathscinet-getitem?mr=0372517

	1. Introduction and statement of results
	2. Connection formula
	3. The zeros
	Acknowledgements
	References