Orthogonal polynomials on the real line corresponding to a perturbed chain sequence

0.18 MB·

by  Kiran Kumar Behera

ORTHOGONAL POLYNOMIALS ON THE REAL LINE
CORRESPONDING TO A PERTURBED CHAIN SEQUENCE
KIRAN KUMAR BEHERA AND A. SWAMINATHAN
Abstract. In recent years, chain sequences and their perturbations have played a sig-
nificantroleincharacterisingthe orthogonalpolynomialsbothonthe realline aswellas
onthe unit circle. Inthis note, a particulardisturbance of the chain sequence relatedto
orthogonal polynomials having their true interval of orthogonality as a subset of [0, )
7 ∞
is studied leading to an important consequence related to the kernel polynomials. Such
1
perturbations are shown to be related to transformations of symmetric measures. An
0
2 illustration using the generalized Laguerre polynomials is also provided.

n
a
J

7 1. Preliminaries
2
Let be a moment functional defined on the vector space of polynomials with real
] L
A coefficients. The quantities µ = [xk], k 0 are called the moments of order k and are
k
C used to construct the Hankel matLrix H =≥(µ )∞ . In case the principal submatrices
i+j i,j=0
. H are non-singular, is said to be quasi-definite and there always exists a sequence of
h n ∞L
t polynomials p (x) , where for n 0,
a { n }n=0 ≥
m p (x) = c xn +c xn−1 + +c x+c , c > 0,
n n,n n,n−1 n,1 n,0 n,n
···
[
satisfying the property,

1
v [p (t)p (t)] = δ , m,n 0.
n m m,n
0 L ≥
6 The sequence p (x) ∞ is said to be orthonormal with respect to .
9 { n }n=0 ∞L
Following the notation used in [16, Page 12], the sequence p (x) satisfies the three
7 { n }n=0
0 term recurrence relation (the so called Favard’s Theorem),
.
1
a p (x) = (x b )p (x) a p (x), n 0, (1.1)
0 n+1 n+1 n+1 n n n−1
− − ≥
7 where p (x) and p (x) are pre-defined. Further, b R and a > 0, n 1 and are
1 −1 0 n n
∈ ≥
: related to the coefficients of p (x) as [10],
v n
Xi a = cn−1,n−1 and b = cn,n−1 cn+1,n , n 1.
n n+1
c c − c ≥
r n,n n,n n+1,n+1
a
Moreover, though a is arbitrary, it is usually taken to be equal to [1] = µ > 0 for
0 0
L
convenience. It may be noted that the recurrence coefficients b and a satisfy
n+1 n
{ } { }
the above conditions if and only if is positive definite, that is, if [r(x)] > 0 for every
polynomial r(x) such that r(x) > 0Lfor all x R. L
∈
In many cases, the polynomials p (x) are normalized to make their leading coefficient
n
equal to one. These monic orthogonal polynomials denoted as (x) ∞ and defined by
{Pn }n=0
c (x) = p (x), n 0, satisfy the recurrence relation
n,n n n
P ≥
(x) = (x b ) (x) a2 (x), n 0, (1.2)
Pn+1 − n+1 Pn − nPn−1 ≥
2010 Mathematics Subject Classification. 42C05,33C45, 15B99 .
Key words and phrases. Momentfunctional, Jacobimatrices,Chainsequences,Orthogonalpolynomi-
als, Kernel polynomials.
1
2 Kiran KumarBehera and A.Swaminathan
with (x) = 0 and (x) = 1. In the sequel, these will be called an orthogonal
−1 0
P P
polynomials sequence (OPS). The system (1.2) is usually written in the more precise
form: xP¯ = JP¯, where P¯ = [ (x) (x) (x) ]T and
0 1 2
P P P ···
b 1 0 0
1
···
a2 b 1 0
 1 2 ··· 
J =  00 a022 ab32 b1 ···  (1.3)
 ... ... ...3 ...4 ·.·..· 
 
is called the monic Jacobi matrix associated with the orthogonal polynomials generated
by (1.2). The polynomials defined as [( (y) (t))/(y t)], n 0, are called
n n
L P − P − ≥
the associated polynomials associated with µ and constitute a second solution of the
recurrence relation
z = (x b )z a2z , n 1,
n+1 − n+1 n − n n−1 ≥
with the modified initial conditions z = 0 and z = 1. Following the notation used in [9],
0 1
(1)
these polynomials will be denoted as (x), n 0. By induction, it can be seen that
n
P ≥
(1)
(x), n 1 is a monic polynomial of degree n 1.
n
P ≥ −
It is interesting to note that if is positive definite, it can be used to define an inner
L
product on the space of polynomials. In such a case, there exists a positive measure
µ(t) supported on a subset E of the real line R with E¯ denoting the smallest closed
interval containing this support. The linear functional now has the representation [tk] =
L
tkdµ(t). Further, the Stieltjes transform of the measure
E
R
dµ(t)
(x) = ,
S ZE x t
−
yields the Jacobi continued fraction expansion
dµ(t) 1 a2 a2
= 1 2 ... (1.4)
Z x t x b - x b - x b -
E 1 2 3
− − − −
wherethe a2 and b arethesamecoefficientsappearingin(1.2). TheStieltjesfunction
{ n} { n}
(x) plays a very fundamental role in the theory of orthogonal polynomials on the real
S
line. It admits the power series expansion at infinity
µ µ µ
0 1 2
(x) = + + +
S x x2 x3 ···
and hence acts as the generating function for the moments µ′s associated with dµ(t).
k
Further, the nth convergent of the continued fraction in the right hand side of (1.4) is
(1) (1)
easily seen to be (x)/ (x) and hence as n , (x)/ (x) converges uniformly
n n n n
to 1 (x) on comPpact subPsets of C¯ E¯ [4]. Here→C¯∞is tPhe extenPded complex plane.
µ0S \
An equivalence transformation of the nth convergent of the continued fraction in (1.4)
yields [4]
1 d (x) d (x) d (x)
1 2 n−1
(x) = ...
Fn 1 - 1 - 1 - - 1
where
(1)(x) a2
(x) = (x b )Pn and d (x) = n .
n 1 n
F − (x) (x b )(x b )
n n n+1
P − −
Orthogonal polynomials on thereal line corresponding to a perturbed chain sequences 3
The quantities d (t) can also be obtained, [9, Page 110], from the recurrence relation (1.2)
n
as
(t) (t)
n n+1
d (t) = P 1 P , n 1. (1.5)
n
(t bn) n−1(t) (cid:20) − (t bn+1) n(t)(cid:21) ≥
− P − P
Such structures, called chain sequences have been studied in [21], followed by a systematic
treatment in [9] (see also [11, Section 7.2]), particularly in the context of orthogonal
polynomials on the real line.
Formally stated, a sequence d ∞ is called a positive chain sequence if it can be
{ n}n=1 ∞
expressed in terms of another sequence g as d = (1 g )g , n 1. Here,
g ∞ called aparameter sequence of the{chna}inn=0sequennce d ∞− nis−s1ucnh that≥0 g < 1
{ n}n=0 { n}n=1 ≤ 0
and 0 < g < 1, n 1. The parameter sequence is in general not unique, and in
n
≥
such cases it is interesting to study the bounds for each parameter g , n 0. The
n
minimal parameter sequence m ∞ of the positive chain sequence d is≥defined as
{ n}n=0 { n}
m = 0, 0 < m < 1, with d = (1 m )m , n 1. Thus, for instance, the
0 n n n−1 n
minimal parameter sequence of the chain−sequence d (t)≥∞ as defined in (1.5) is given
{ n }n=1
by m (t) = [1 (t)/(t b ) (t)], n 0. At the other end, there is the maximal
n n+1 n+1 n
parameter sequ−enPce M ∞−, whosPe parame≥ters are defined as,
{ n}n=0
M = sup g : g any other parameter sequence of d
k k n n
{ { } { }}
where sup denotes the supremum of the set. It is known that m g M , k 0, for
k k k
≤ ≤ ≥
any parameter sequence g of d . In case, M = m = 0, the parameter sequence
n n 0 0
{ } { }
is unique and d is called a single parameter positive chain sequence, abbreviated as
n
{ }
SPPCS in the sequel.
2. Symmetric Orthogonal Polynomials
A positive measure dφ(x) is called symmetric if it satisfies dφ( x) = dφ(x). The
− −
monic polynomials (x), n 0, which are orthogonal with respect to dφ(x) satisfy
n
S ≥
(x) = x (x) γφ (x), n 1,
Sn Sn−1 − nSn−2 ≥
with (x) = 0 and (x) = 1.
−1 0
S S
In a series of research articles [3,17–19], many properties of symmetric orthogonal
polynomialshavebeendiscussed inthecontextofcertainorthogonalLaurentpolynomials.
Particularly, if dφ (x) and dφ (x) are two different symmetric distributions related by the
1 2
Christoffel transformation as dφ (x) = (1+kx2)dφ (x), where k is positive real constant,
2 1
then
Theorem 2.1. [19, Theorem 1] Associated with dφ (x) and dφ (x) there exists a sequence
1 2
of positive numbers l , with l = 1 and l > 1 for n 1, such that
n 0 n
{ } ≥
4kγφ1 = (l 1)(l +1) and 4kγφ2 = (l 1)(l +1) (2.1)
n+1 n − n−1 n+1 n − n+1
This result was established using the transformation t(x) = ( αx2 +β + √αx)2 for
x ( , ) and the monic polynomials (t) ∞ , uniquely depfined by
∈ −∞ ∞ {Bn }n=0
∞
t−n+s (t)dψ(t) = 0, 0 s n 1,
n
Z B ≤ ≤ −
0
where dψ(t) is strong distribution on (0, ). Note that if x = d corresponds to t = b, the
∞
transformation maps ( d,d) to (β2/b,b) (0, ). For any strong distribution dψ(t) on
− ∞ ⊆ ∞
(0, ), the monic polynomials (t) satisfy
∞ {Bn }n=0
(t) = (t βψ ) (t) αψ t (t), n 1,
Bn+1 − n+1 Bn − n+1 Bn−1 ≥
4 Kiran KumarBehera and A.Swaminathan
with (t) = 1 and (t) = t βψ. Further, the unique coefficients βψ , n 0 and αψ ,
B0 B1 − 1 n+1 ≥ n+1
n 1 are all positive real numbers.
≥
Two kinds of strong distributions play important role in this analysis. The first is the
ScS(a,b) distribution and the second is the Sc¯S(a,b) distribution. A strong distribution
dψ(t) with its support inside (β2/b,b) is called a ScS(β2/b,b) distribution if [19]
dψ(t) dψ(β2/t)
= , t (β2/b,b).
√t β/√t ∈
A strong distribution dψ(t) with its support inside (β2/b,b) is called a Sc¯S(β2/b,b) dis-
tribution if dψ(t) = dψ(β2/b), t (β2/b,b) [19]. Then it is known that dψ (t) is a
2
− ∈
ScS(β2/b,b) distribution if and only if (t + β)dψ (t) is a Sc¯S(β2/b,b) distribution. Fur-
0
ther, if l2 1 = αψ0 /βψ0, then
n − n+1 n
αψ2 = β(l 1)(l +1), n 1.
n+1 n − n+1 ≥
Conversely, dψ (t) is a Sc¯S(β2/b,b) distribution if and only if t+βdψ (t) is a ScS(β2/b,b)
0 t 1
distribution and in this case
αψ1 = β(l 1)(l +1), n 1.
n+1 n − n−1 ≥
Then choosing
t t
dψ (t) = A dφ (x(t)), dψ (t) = A dφ (x(t)),
1 1 1 2 2 2
t+β t+β
with k = α/β and appropriate choices of A and A , the relations (2.1) are obtained.
1 2
In this note, we consider the particular case when the sequence l satisfy l = l and
n 1 2
{ } 6
l = l = l = l = l =
1 4 5 8 9
···
l = l = l = l = l =
2 3 6 7 10
···
This sequence can be obtained from appropriate conditions imposed on the coefficients
βψ and αψ , for instance, choosing the coefficients such that they satisfy αψ20 = αψ50
n+1 n+1 β1ψ0 β4ψ0
and so on. Consequently, the coefficients associated with the Christoffel transformation
mentioned above satisfy
γφ2,γφ2,γφ2,γφ2, = γφ1,γφ1,γφ1,γφ1,
{ 1 2 3 4 ···} { 2 1 4 3 ···}
Thus it can be seen that transformations of the symmetric measures are related to
transformations of strong distributions on (0, ) which possess special properties like
∞
symmetry. In this note, we study this particular case of the transformation from the point
of view of a perturbation of the chain sequences associated with polynomial sequences
orthogonal on (0, ) and some related consequences. We also provide an illustration
∞
using the Laguerre polynomials in the last section.
3. Perturbed chain sequences
The theory of chain sequences has been used to study many properties of a given or-
thogonalpolynomial system onthereal line, forinstance, itstrueinterval oforthogonality,
which is the smallest closed interval that contains all the zeros of all the polynomials of
such a system. Denoting the true interval of orthogonality of (x) satisfying (1.2) as
n
{P }
[ξ,η] and
a2
ω (t) = n , n 1,
n
(t b )(t b ) ≥
n n+1
− −
Orthogonal polynomials on thereal line corresponding to a perturbed chain sequences 5
the following are known to be equivalent, [11, Corollary 7.2.4],
(i) [ξ,η] is contained in (a,b),
(ii) b (a,b) for n 0 and both ω (a) and ω (b) are chain sequences.
n+1 n n
∈ ≥ { } { }
We note that here t ( , ) E¯. In particular, the true interval of orthogonality is
∈ −∞ ∞ \
a subset of [0, ] if and only if b > 0 for n 0 and there are numbers g such that
n+1 n
∞ ≥
0 g < 1, 0 < g < 1, n 1, satisfying,
0 n
≤ ≥
a2
(1 g )g = n = ω (0), n 1.
n−1 n n
− b b ≥
n n+1
As in [9], the sequence g ∞ is constructed using another sequence γ ∞ , where
{ n}n=0 { n}n=1
γ 0 and γ > 0, n 2. For details, the reader is referred to [9, Chapter 1, Theorems
1 n
≥ ≥
9.1, 9.2], where it is shown that
b = γ +γ , n 0 and a2 = γ γ , n 1.
n+1 2n+1 2n+2 ≥ n 2n 2n+1 ≥
and the parameters are given by g = γ /b , n 0. Hence when γ = 0, we obtain
n 2n+1 n+1 1
the minimal parameter sequence m ∞ . ≥
{ n}n=0
Associated with the chain sequence ω (0) , another sequence ω˜ (0) arises in a very
n n
{ } { }
natural way. Defining ω˜ (0) = (1 k )k = γ /b and
1 0 1 4 2
−
γ γ
2n−1 2n+2
ω˜ (0) = = (1 k )k , n 2,
n n−1 n
b b − ≥
n n+1
it can be seen that ω˜ ∞ becomes a chain sequence with the minimal parameter se-
quence k ∞ whe{renk}n==1 0 and k = 1 g , n 1. Hence, we give the following
{ n}n=0 0 n − n ≥
definition which was given earlier in [2] with reference to orthogonal polynomials on the
unit circle.
Definition 3.1. Suppose d ∞ is a chain sequence with m ∞ as its minimal pa-
rameter sequence. Let k{ ∞n}n=b1e another sequence given b{y kn}=n=00 and k = 1 m
{ n}n=0 ∞ ∞ 0 n − n
for n 1. Then the chain sequence a having k as its minimal parameter
sequen≥ce is called as complementary ch{ainn}ns=eq1uence of{ dn}n∞=0 .
{ n}n=1
If γ > 0, a non-minimal parameter sequence g ∞ is obtained for the chain sequence
1 { n}n=0
∞ ˆ ∞
ω (0) . In this case, the associated chain sequence ϑ (0) , is defined as
{ n }n=1 { n }n=1
γ γ
ˆ ′ ′ 2n−1 2n+2
ϑ (0) = (1 k )k = , n 1.
n − n−1 n b b ≥
n n+1
where k′ = 1 g for n 0.
n − n ≥
Definition 3.2. Suppose d ∞ is a chain sequence with g ∞ as its non-minimal
{′n}∞n=1 { n}n′=0
parameter sequence. Let k be another sequence given by k = 1 g for n 0.
Then the chain sequence {an}n∞=0 having k′ ∞ as its parametenr seque−ncen is calle≥d as
generalised complementary{cnh}ani=n1sequence{ofn}nd=0∞ .
{ n}n=1
It may be noted from the above two definitions that for a fixed chain sequence, while its
complementary chain sequence is unique, its generalised complementary chain sequence
neednotbeunique. Infact,achainsequencewillhaveasmanygeneralisedcomplementary
chain sequences as its non-minimal parameter sequences. Naturally, the complementary
chain sequence and all the generalised complementary chain sequences will coincide only
for a SPPCS.
∞ ˆ ∞
We would like to mention that the chain sequences ω˜ (0) and ϑ (0) have
{ n }n=1 { n }n=1
definite sources in the theory of orthogonal polynomials on the real line. To see this, first
6 Kiran KumarBehera and A.Swaminathan
note that the symmetric polynomials (x) satisfy the property ( x) = ( 1)n (x),
n n n
n 1, which implies the existence of{twSo OP}S (x) ∞ and S(x−) ∞ su−ch thSat
≥ {Pn }n=1 {Kn }n=1
(x) = (x2), (x) = x (x2).
2n n 2n+1 n
S P S K
It is interesting to note that (x) = (0;x) is the sequence of kernel polynomial
n n
{K } {K }
corresponding to (x) based at the origin and abbreviated as KOPS in the sequel.
n
P
(i)
Further, if the polynomials (x) , i = 1,2, satisfy the recurrence relation,
n
{R }
(i) (x) = (x b(i) ) (i)(x) (a2)(i) (i) (x), n 0, (3.1)
Rn+1 − n+1 Rn − n Rn−1 ≥
(i) (i)
with (x) = 0 and (x) = 1 then,
R−1 R0
(1)
(i) (x) (x), n 1, if and only if
n n
R ≡ P ≥
(1) (1)
b = γ , b = γ +γ , n 1
1 2 n+1 2n+1 2n+2 ≥
(a2)(1) = γ γ , n 1,
n 2n 2n+1 ≥
(2)
(ii) (x) (x), n 1, if and only if
n n
R ≡ K ≥
b(2) = γ +γ , n 0 and (a2)(2) = γ γ , n 1.
n+1 2n+2 2n+3 ≥ n 2n+1 2n+2 ≥
(1)
With these notations, the parameter sequences can be denoted as m = γ /b and
n 2n+1 n+1
g = γ /b(1) , n 0. Further, denoting a˜2 = γ γ , n 1, the following theorem
n 2n+1 n+1 ≥ n 2n−1 2n+2 ≥
shows that the polynomials ˜ (x) and ˆ (x) associated respectively, with the com-
n n
{P } {P }
plementary chain sequence ω˜ (0) and the generalised complementary chain sequence
n
{ }
ˆ
ϑ (0) , can be attributed to a particular perturbation of the recurrence coefficients of
n
{ }
the polynomials S (x) .
n
{ }
Remark 3.1. Such perturbations of the recurrence coefficients as well as of the Stieltjes
function have been studied deeply. The reader is referred to [7,10,22] for some details.
In most of the cases only a single modification or a finite composition of modifications is
considered. In this note however, all the recurrence coefficients are perturbed.
Theorem 3.1. Let the symmetric polynomials S˜ (x) ∞ satisfy
{ n }n=0
S˜ (x) = xS˜ (x) ν˜ S˜ (x), n 1, (3.2)
n n−1 n n−2
− ≥
with S˜ (x) = 0, S˜ (x) = 1 and where, for n 1,
−1 0
≥
γ , n=2j, j=1,2,
ν˜ = 2j−1 ··· (3.3)
n (cid:26) γ2j+2, n=2j+1, j=0,1, .
···
Then, with γ = 0, ˜ (x) ∞ , where S˜ (x) = ˜ (x2), satisfy,
1 6 {Pn }n=0 2n Pn
˜ (x) = (x b(1) )˜ (x) a˜2 ˜ (x), n 1, (3.4)
Pn+1 − n+1 Pn − nPn−1 ≥
with the initial conditions ˜ (x) = 1 and ˜ (x) = (x γ ).
0 1 1
P P −
Proof. First note that, the perturbation (3.3) implies that the sequence of coefficients
γ ,γ ,γ ,γ , is replaced by γ ,γ ,γ ,γ , . That is, γ ,γ are pair-wise
1 2 3 4 2 1 4 3 2k−1 2k
{ ···} { ···} { }
interchanged to γ ,γ , k 1. Then, for n = 2m,
2k 2k−1
{ } ≥
S˜ (x) = xS˜ (x) γ S˜ (x), m 1
2m 2m−1 2m−1 2m−2
− ≥
which implies,
˜ (x2) = x2˜ (x2) γ ˜ (x2),
m m−1 2m−1 m−1
P K − P
Orthogonal polynomials on thereal line corresponding to a perturbed chain sequences 7
or equivalently,
˜ (x) = x˜ (x) γ ˜ (x), m 1. (3.5)
m m−1 2m−1 m−1
P K − P ≥
Similarly, for n = 2m+1,
S˜ (x) = xS˜ (x) γ S˜ (x), m 0,
2m+1 2m 2m+2 2m−1
− ≥
which implies,
x˜ (x2) = x˜ (x2) γ x˜ (x2),
m m 2m+2 m−1
K P − K
or equivalently,
˜ (x) = ˜ (x) γ ˜ (x), m 0. (3.6)
m m 2m+2 m−1
K P − K ≥
Using (3.5) and (3.6), it is easy to find the three term recurrence relations for ˜ (x) and
n
P
˜ (x). For this, first ˜ (x) is eliminated. From (3.6), it can be seen that,
n n
K P
˜ (x) = ˜ (x)+γ ˜ (x), m 0.
m m 2m+2 m−1
P K K ≥
Using this in (3.5), gives,
˜ ˜ ˜
(x) = [x (γ +γ )] (x) γ γ (x), m 1. (3.7)
m 2m−1 2m+2 m−1 2m−1 2m m−2
K − K − K ≥
with ˜ (x) = 0 and (using (3.6)) ˜ (x) = 1. Similarly, (3.5) gives
−1 0
K K
˜ ˜ ˜
x (x) = (x)+γ (x),
m−1 m 2m−1 m−1
K P P
Using this in (3.6) yields,
˜ (x) = [x (γ +γ )]˜ (x) γ γ ˜ (x), m 1, (3.8)
m+1 2m+1 2m+2 m 2m−1 2m+2 m−1
P − P − P ≥
with the initial conditions ˜ (x) = 1 and (using (3.5)) ˜ (x) = x γ , thus proving the
0 1 1
P P − (cid:3)
theorem.
Corollary 3.1. Consider the OPS Pˆ (x) ∞ satisfying (3.8) but for m 0. Then
{ n }n=0 ≥
Pˆ (x) ∞ is associated with the generalised complementary chain sequence ϑˆ (0) ∞ .
{ n }n=0 { n }n=1
Proof. From the recurrence relation
ˆ (x) = [x (γ +γ )]ˆ (x) γ γ ˆ (x), m 0, (3.9)
m+1 2m+1 2m+2 m 2m−1 2m+2 m−1
P − P − P ≥
ˆ ˆ
with P (x) = 0 and P (x) = 1, the chain sequence is given by
−1 0
∞
γ γ
2n−1 2n+2
(cid:26)(γ +γ )(γ +γ )(cid:27)
2n−1 2n 2n+1 2n+2 n=1
with the parameter sequence k′ ∞ = γ /(γ γ ) ∞ . The result now follows
since k′ = 1 g , n 0. { n}n=0 { 2n+2 2n+1 2n+2 }n=0 (cid:3)
n − n ≥
˜ ∞ ˆ ∞
TheOPS (x) canbeseentobeco-recursivewithrespect totheOPS (x)
{Pn }n=1 {Pn }n=1
arising from the initial conditions ˜ (x) = 1 and ˜ (x) = ˆ (x) + γ . The co-recursive
0 1 1 2
P P P
polynomials have been investigated in the past; see for example, [8], and later [12] in
which the structure and spectrum of the generalised co-recursive polynomials have been
studied.
8 Kiran KumarBehera and A.Swaminathan
Further, from (3.8), the associated chain sequence is a˜2/b(1)b(1) ∞ with the first few
{ n n n+1}n=1
terms as
a˜2 γ a˜2 γ γ
1 = 4 = (1 k )k ; 2 = 3 6 = (1 k )k
b(1)b(1) (γ3 +γ4) − 0 1 b(1)b(1) (γ3 +γ4)(γ5 +γ6) − 1 2
1 2 2 3
a˜2 γ γ
3 = 5 8 = (1 k )k
b(1)b(1) (γ5 +γ6)(γ7 +γ8) − 2 3
3 4
Proceeding as above, we obtain the minimal parameter sequence k ∞ where k = 0
{ n}n=0 0
and k = γ /b(1) = 1 g , n 1. which shows that the OPS P˜ (x) ∞ is associated
with tnhe co2mn+p2lemne+n1tary−chanin se≥quence ω˜ (0) ∞ . { n }n=0
n }n=1
Viewingthegeneralisedcomplementarychainsequences asperturbationsoftheminimal
parametersorsimplyatransformationoftheoriginalchainsequence, wegiveanimportant
consequence of Theorem 3.1.
Corollary 3.2. The kernel polynomial system (x) remains invariant under gener-
n
{K } ∞
alised complementary chain sequence if the sequence γ satisfies,
{ n}n=1
γ γ = γ γ , n 1.
2n+1 2n−1 2n+2 2n
− − ≥
(2)
Proof. The proof follows from a comparison of (3.7) and the expressions for b and
n+1
a(2). (cid:3)
n
Corollary 3.2 is important because it is known [9, Ex. 7.2, p. 39], that the relation
between the monic orthogonal polynomials and the kernel polynomials is not unique.
That is, for fixed t R, though (x) will lead to a unique kernel polynomial system
n
∈ {P }
(t;x) , there are infinite number of other monic orthogonal polynomial systems which
n
{K }
has the same (t;x) as their kernel polynomial system. Hence generalised comple-
n
{K }
mentary chain sequences can be used to construct two orthogonal polynomials systems
having the same kernel polynomial systems.
The following theorem unifies the recurrence relations for the polynomials and the
associated kernel polynomials for both the chain sequence as well as its generalised com-
plementary chain sequence.
Theorem 3.2. Consider the recurrence relation,
(x) = (x ξ ) (x) η (x), n 1,
n n n−1 n n−2
T − T − T ≥
with (x) = 0 and (x) = 1. Then,
−1 0
T T
˜ ˆ
(i) (x) (x)( (x)), n 1, if and only if,
n n n
T ≡ P ≡ P ≥
(1)
ξ = γ (= γ +γ ), ξ = b , n 1, and
1 1 1 2 n+1 n+1 ≥
(a2 )(2)(a2)(2)
η = γ γ , η = n−1 n , n 2.
2 1 4 n+1 (a2)(1) ≥
n
(ii) (x) ˜ (x), n 1, if and only if,
n n
T ≡ K ≥
(1) (1) (2)
ξ = γ +γ , ξ = b +b b , n 1, and
1 1 4 n+1 n+1 n+2 − n+1 ≥
η = γ γ , η = (a2)(2), n 1.
1 1 2 n+1 n ≥
Let the zeros of (x) ∞ and ˜ (x) ∞ be denoted as
{Pn }n=0 {Pn }n=0
0 < x < x < < x < x and 0 < x˜ < x˜ < < x˜ < x˜
n,1 n,2 n,n−1 n,n n,1 n,2 n,n−1 n,n
··· ···
Orthogonal polynomials on thereal line corresponding to a perturbed chain sequences 9
For fixed n, by interlacing of zeros of (x) and ˜ (x) it is understood that x are
n n n,j
P P
mutuallyseparatedbyx˜ forj = 1,2, ,n. Further, inthepresent case, itisinteresting
n,j
···
tonotefrom(3.1)and(3.8), thatthesumoftherootsof (x)isgivenbyγ +γ + +γ
n 2 3 2n
P ···
while that for ˜ (x) is γ +γ + +γ .
n 1 3 2n
P ···
Remark 3.2. It is clear that if γ = γ , interlacing of the zeros of (x) and ˜ (x)
1 2 n n
{P } {P }
can never occur.
For γ = γ , we have the following result.
1 2
6
Proposition 3.1. For fixed n, the zeros x n and x˜ n cannot interlace if (γ
{ n,j}j=1 { n,j}j=1 1−
γ ) and (x x˜ ) have the same sign for some j = 1,2, ,n.
2 n,j n,j
− ···
Proof. Suppose γ > γ and x > x˜ for some j = 1,2, ,n. If the zeros of (x) and
1 2 n,j n,j n
··· P
˜ (x) interlace, then n x˜ < n x which is a contradiction.
PnThe case γ1 < γ2 anPdjx=n1,j <n,jx˜n,jPcaj=n1ben,jproved similarly. (cid:3)
The last result in this section shows that while the generalised complementary chain
sequence of associated with ˆ (x) yields an OPS, that associated with the associated
n
{P }
ˆ
polynomials (1) (x) leads to a KOPS.
n
{P }
Theorem 3.3. Consider the OPS ˆ(1)(x) ∞ . Then the generalised complementary
{Pn }n=0
chain sequence associated with ˆ(1)(x) leads to a KOPS (x) satisfying the relation
n n
P {Q }
(x) = (x γ γ ) (x) γ γ (x), n 0 (3.10)
n+1 2n+3 2n+4 n 2n+2 2n+3 n−1
Q − − Q − Q ≥
with (x) = 0 and (x) = 1.
−1 0
Q Q
ˆ(1) ∞
Proof. It is clear from (3.9) that (x) satisfy
{Pn }n=1
ˆ(1) (x) = (x γ γ )ˆ(1)(x) γ γ ˆ(1) (x), n 1
Pn+1 − 2n+3 − 2n+4 Pn − 2n+2 2n+3Pn−1 ≥
with ˆ(1)(x) = 0 and ˆ(1)(x) = 1. The associated chain sequence is
P−1 P0
∞
γ γ
2n+1 2n+4
(cid:26)(γ +γ )(γ +γ )(cid:27)
2n+1 2n+4 2n+3 2n+5 n=1
with the (non-minimal) parameter sequence γ /(γ +γ ) ∞ . Hence the OPS
{ 2n+4 2n+3 2n+4 }n=0
(x) associated with the generalised complementary chain sequence satisfy (3.10).
n
{QTo pr}ove that (x) is a KOPS, consider the polynomials (x) ∞ given by
{Qn } {Un }n=1
x (x) = (x)+γ (x), n 0
n n+1 2n+3 n
Q U U ≥
The first thing that we require is that (0) = γ (0) so that choosing (0) =
n+1 2n+3 n 1
U − U U
γ , we have (0) = ( 1)n+1γ γ γ γ .
3 n+1 2n+3 2n+1 5 3
−Suppose nowU that (−x) ∞ satisfy the·r·e·currence relation
{Un }n=1
(x) = (x σ ) (x) η (x), n 1,
n+1 n+1 n n n−1
U − U − U ≥
with (x) = 1 and (x) = x γ and where the coefficients σ and η are to be
0 1 3 n n
U U − { } { }
determined. One way for the recurrence relation to hold, is that we must have (0) =
n+1
U
σ (0) µ (0) which implies γ σ µ = γ γ , n 1. One possible
n+1 n n n−1 2n+1 n+1 n 2n+3 2n+1
− U − U − ≥
choice for σ and µ satisfying the above equations is
n+1 n
σ = γ +γ and µ = γ γ , n 1.
n+1 2n+3 2n+2 n 2n+1 2n+2
≥
Since µ > 0 for n 1, by Favard’s Theorem [9, Theorem 4.4, p. 21] (x) ∞ becomes
a OPS nand (x)≥∞ its corresponding KOPS [9, eqn. 7.3, p. 35].{Un }n=1 (cid:3)
{Qn }n=1
10 Kiran KumarBehera and A.Swaminathan
Corollary 3.3. The following holds
x2 (x) γ [ (1) (x)+γ (1)(x)] =
Qn − 2 Kn+1 2n+3Kn
(x)+(γ +γ ) (x)+γ γ (x), n 1. (3.11)
n+2 2n+3 2n+4 n+1 2n+2 2n+3 n
P P P ≥
Proof. Comparing with the recurrence relation (3.1), it can be observed that (x) ∞
are co-recursive with respect to (x) ∞ and arise due to change in the in{Uitnial v}anlu=e1
{Kn }n=1
(1)
(x) = (x)+γ . Hence (x) = (x)+γ (x), n 1 [8], so that
1 1 2 n n 2 n
U K U K K ≥
x (x) = ( (x)+γ (x))+γ ( (1) (x)+γ (1)(x)).
Qn Kn+1 2n+3Kn 2 Kn+1 2n+3Kn
Since x (x) = (x)+γ (x), n 1, (3.11) follows. (cid:3)
n n+1 2n+2 n−1
K P P ≥
Remark 3.3. The polynomials in the left hand side of (3.11) are quasi-orthogonal of
order 2 on [0, ]. For details regarding quasi-orthogonality, the reader is referred to [5]
∞
and the references therein.
4. Concluding remarks
˜
(A) The monic Jacobi matrix of the polynomials (x) and (x) are given respectively
n n
P P
by,
γ 1 0 γ 1 0
2 1
··· ···
γ γ γ +γ 1 γ γ γ +γ 1
 2 3 3 4   1 4 3 4 
··· ···
JP =  0 γ4γ5 γ5 +γ6 ···  JP˜ =  0 γ3γ6 γ5 +γ6 ··· 
0 0 γ γ 0 0 γ γ
 6 7   5 8 
 ... ... ... ·..·.·   ... ... ... ·.·..· 
   
The respective LU decomposition of the above Jacobi matrices are then given by,
1 0 0 0 γ 1 0 0
2
··· ···
γ 1 0 0 0 γ 1 0
 3   4 
··· ···
LP =  0 γ5 1 0 ··· , UP =  0 0 γ6 1 ··· 
0 0 γ 1 0 0 0 γ
 7   8 
 .. .. .. .. ·.·.·   .. .. .. .. ·..·· 
 . . . . .   . . . . . 
   
and,
1 0 0 0 γ 1 0 0
1
··· ···
γ 1 0 0 0 γ 1 0
 4   3 
··· ···
LP˜ =  0 γ6 1 0 ··· , UP˜ =  0 0 γ5 1 ··· 
0 0 γ 1 0 0 0 γ
 8   7 
 ... ... ... ... ·.·..·   ... ... ... ... ·..·.· 
   
It may be observed that L can be obtained from U by deleting the first column
P˜ P
of U while U can be obtained from L by adding the first column of U , with γ
P P˜ P P 2
replacedbyγ ,asthefirstcolumnofL . TheJacobimatrixanditsLUdecomposition
1 P
for the polynomial system ˆ (x) ∞ can be obtained similarly.
{Pn }n=1
The monic Jacobi matrix associated with the three canonical transformations,
the Christoffel, the Geronimus and the Uvarov transformations, are found in [6],
wherein the procedure is given using the Darboux transformation. Hence, it will
be interesting to study the application of Darboux transformation in the case of
(generalised) complementary chain sequences.