We research behavior and sharp bounds for the zeros of infinite sequences of polynomials orthogonal with respect to a Geronimus perturbation of a positive Borel measure on the real line.

1. Zeros of OP generated by a Geronimus perturbation of measures
Edmundo J. Huertas∗, Am´
ılcar Branquinho∗ and Fernando R. Rafaeli‡
∗
FCTUC, CMUC - Universidade de Coimbra, Portugal – ‡UNESP, Universidade Estadual Paulista, Brasil.
Scientific Research under FCT Project PEst-C/MAT/UI0324/2011, and Grant SFRH/BPD/91841/2012
Abstract
c,N
In this contribution we consider the sequences of polynomials {Qn }n≥0, orthogonal with respect
to the modified measure dvN (x) ( ⇒ inner product)
1
1
dµ(x) + N δ(x − c) ⇒ f, g νN =
dµ(x) + N f (c)g(c),
f (x)g(x)
dνN (x) =
(x − c)
(x − c)
E
where N ∈ R+, dµ(x) is a positive Borel measure supported on E ⊆ R, δ(x − c) is the Dirac delta
function at x = c ∈ E, and f, g are polynomials with real coefficients. This contribution is focused
/
on the behavior of zeros of monic orthogonal polynomial sequences (MOPS in the sequel) associated
with a particular transformation of measures called the Geronimus canonical transformation on
R. We first provide several representations of these polynomials in terms of the MOPS {Pn(x)}n≥0,
We also provide two new connection formulas in terms of the polynomials Qc (x) and the monic
n
c,[1]
Kernel polynomials Pn (x), and in terms of only two consecutive polynomials of the MOPS
{Pn(x)}n≥0. These connection formulas will be useful to obtain results about monotonicity, asympc,N
totics, speed of convergence and the corresponding electrostatic model for the zeros of {Qn (x)}n≥0.
3.2. Connection formula 2
c,N
˜ c,N
˜ c,N
The Geronimus perturbed orthogonal polynomials {Qn (x)}n≥0, with Qn (x) = κnQn (x),
can be represented as
c,[1]
c
˜ c,N
Qn (x) = Qc (x) + N Bn(x − c)Pn−1(x),
n
E
dµ[1](x) = (x − c)dµ(x) ⇒ f, g c,[1] =
and
f (x)g(x)(x − c)dµ(x),
E
1
1
dν(x) =
dµ(x) ⇒ f, g ν =
dµ(x).
f (x)g(x)
(x − c)
(x − c)
E
We analyze the behavior of the zeros in term of the positive real parameter N of the perturbation.
Next we find the second order differential equation (also known as the holonomic equation) that
such polynomials satisfy, and the electrostatic interpretation of their zeros in terms of a logarithmic
potential. As an example, we apply our results to the Laguerre and Jacobi classical measures.
c,N
The monic Geronimus perturbed orthogonal polynomials of the sequence {Qn (x)}n≥0 can be
represented as
c,N
Qn (x) = Pn(x) + Λc (N )Pn−1(x),
n
with
c (N ) = πn−1 − rn−1 − π
Λn
n−1,
c
1 + N Bn
and πn−1, rn−1 given by
Pn(c)
πn−1 = πn−1(c; N ) =
,
Pn−1(c)
where a(x; n) and b(x; n) are polynomials in the variable x, whose fixed degree do not depend on
n. On the other hand, let
n
c
Kn(x, y) =
k=0
Qc (x)Qc (y) Qc (x)Qc (y) − Qc (y)Qc (x) 1
n
n
n+1
k
k
= n+1
(x − y)
||Qc ||2
||Qc ||2
n ν
ν
k
Fn(c)
rn−1 = rn−1(c; N ) =
.
Fn−1(c)
3. Asymptotic behavior of the zeros with N
c,[k]
c,[k] c
c,N
c,N
Let xn,s, xn,s , yn,s, and yn,s , s = 1, . . . , n be the zeros of Pn(x), Pn (x), Qc (x), and Qn (x),
n
respectively, all arranged in an increasing order, and assume that C0(E) = [a, b]. Next we use a
Interlacing Lemma proved by Bracciali et al. (see [1] for details), to obtain results about monoc,N
tonicity, asymptotics, and speed of convergence for the zeros yn,s in terms of the mass N . Thus,
it can be proved that we are in the hypothesis of the Interlacing Lemma, and from the Connection
c
formula 2 and the positivity of Bn, we immediately conclude the following results:
Theorem
If C0(E) = [a, b] and c < a, then
denotes the kernel polynomials corresponding to the MOPS {Qc (x)}n≥0, and whose corresponding
n
confluent form is
n
[Qc ]′(c)Qc (c) − [Qc ]′(c)Qc (c)
[Qc (c)]2
n
n
n+1
c
k
Kn(c, c) = n+1
=
> 0.
c ||2
2
||Qn ν
Qc ν
k=0
k
The measure dν(x) = dνN =0(x) constitutes a linear rational modification of µ, and the corresponding MOPS {Qc (x)}n≥0 has been extensively studied in the literature (see, among others, [2,§2.4.2],
n
[3,§2.7]). These polynomials can be represented as
c,[1]
c,[1]
c,N
c,N
c,N
c
c
c
c < yn,1 < yn,1 < xn−1,1 < yn,2 < yn,2 < · · · < xn−1,n−1 < yn,n < yn,n.
c,N
Moreover, each yn,s is a decreasing function of N and, for each s = 1, . . . , n − 1,
c,N
lim yn,1 = c,
N →∞
as well as
c,N
lim N [yn,1 − c] =
N →∞
c (x) = P (x) − Fn(c) P
Qn
n
n−1(x),
Fn−1(c)
−Qc (c)
n
c,[1]
c
BnPn−1(c)
,
c,[1]
−Qc (xn−1,s)
n
c,[1]
c,[1]
c c,[1]
Bn(xn−1,s − c)[Pn−1]′(xn−1,s)
.
Corollary
c,N
If C0(E) = [a, b] and c ∈ [a, b], the following expressions hold. If c < a, then the smallest zero yn,1
/
c,N
c,N
c,N
yn,1 > a for N < N0, yn,1 = a for N = N0, and yn,1 < a for N > N0,
3.1. Connection formula 1
c,N
Let {Qn (x)}n≥0 and {Qc (x)}n≥0 be the MOPS corresponding to the measures dνN and dν(x)
n
respectively. Then, the following connection formula holds
c,N
where Qn (c) =
where
N0 = N0(n, c, a) =
Qc (c)
n
c (c, c) .
1 + N Kn−1
−Qc (a)
n
c,[1]
c
Kn−1 (c, c) (a − c)Pn−1(a)
Observe that there is a counterpart of these results when c > b.
c,N
where
In turn, for k = 1, 2, the above expressions are given only in terms of Λc (N ), and the coefficients
n
βn, γn, σ(x), a(x; n) and b(x; n) of the three term recurrence relation and the structure relation
satisfied by {Pn(x)}n≥0:
Ck (x; n)B2(x; n) γn−1 + Dk (x; n)Λc (N ) c
Dk (x; n) − Ck (x; n)Λc (N )
n−1
n
c
, ηk (x; n) =
,
ξk (x; n) =
∆(x; n) γn−1
∆(x; n) γn−1
1
c (N ) b(x; n) ,
C1(x; n) =
a(x; n) − Λn
σ(x)
γn−1
1
c (N )b(x; n − 1) a(x; n − 1) + (x − βn−1)
b(x; n) + Λn
D1(x; n) =
σ(x)
b(x; n − 1)
γn−1
−Λc (N )
1
(x − βn−1)
n
c (N )
A2(n) =
, B2(x; n) = Λn−1
,
c (N ) +
γn−1
Λn−1
γn−1
Λc (N ) a(x; n) b(x; n − 1)
(x − βn−1)
1
n−1
+
C2(x; n) = −
c (N ) +
σ(x)
γn−1
γn−1
Λn−1
γn−1
Λc (N ) σ(x) − b(x; n)
D2(x; n) = n−1
+ b(x; n − 1) ·
σ(x)
γn−1
1
a(x; n − 1) (x − βn−1)
(x − βn−1)
+
c (N ) +
b(x; n − 1)
γn−1
Λn−1
γn−1
,
,
The electrostatic interpretation means that, the equilibrium position for the zeros of the Geronic,N
mus perturbed polynomial {Qn (x)}n≥0 occurs under the presence of a total external potential
V (x) = υshort(x) + υlong (x), where υshort(x) = (1/2) ln u(x; n) represents a short range potential
(or varying external potential) corresponding to unit charges located at the zeros of the polynomial
u(x; n), and υlong (x) is said to be a long range potential, which is associated with the measure
c,N
dµ(x). The polynomial u(x; n) plays a remarkable role in the behavior of the zeros of Qn (x). As
an example, we show below total external potentials VJ (x) and VL(x) when the measure dµ(x) is
the classical Jacobi and Laguerre measures respectively. In these two simple cases the polynomial
u(x; n) has degree 1.
1
1
VJ (x) = ln uJ (x; n) − ln(1 − x)α+1(1 + x)β+1, with
2
2
uJ (x; n) = 4n(n + α)(n + β)(n + α + β) + (2n + α + β)(2n + α + β − 1)Λn
satisfies
c,N
We provide the following first representation for the MOPS Qn (x):
c,N
c (x) − N Qc,N (c)K c (x, c),
Qn (x) = Qn
n
n−1
c,[1]
lim yn,s+1 = xn−1,s ,
N →∞
c,[1]
c,N
lim N [yn,s+1 − xn−1,s] =
N →∞
where the functions Fn(s), s ∈ C E, are the Cauchy integrals of {Pn(x)}n≥0, namely, the well
known functions of the second kind associated with {Pn(x)}n≥0.
The key concept to find several of our results is that the polynomials {Pn(x)}n≥0 are the monic
kernel polynomials of parameter c of the sequence {Qc (x)}n≥0.
n
c,N
c,N
with
2. Connection formulas
σ(x)[Pn(x)]′ = a(x; n)Pn(x) + b(x; n)Pn−1(x)
c,N
[Qn (x)]′′ + R(x; n)[Qn (x)]′ + S(x; n)Qn (x) = 0,
−Qc (c)Pn−1(c)
n
c
c
Bn =
= Kn−1(c, c) > 0.
Pn−1 2
µ
Notice that the coefficient Λc (N ) does not depend on the variable x.
n
If dµ(x) is semiclassical, then the MOPS {Pn(x)}n≥0 satisfy a three term recurrence relation
xPn(x) = Pn+1(x) + βnPn(x) + γnPn−1(x), and a structure relation as
c,N
{Qn (x)}n≥0 satisfies the second order linear differential equation
c
[η1(x; n)]′
c
c
R(x; n) = − ξ1 (x; n) + η2(x; n) + c
,
η1(x; n)
c
c
c
c
ξ1 (x; n)[η1(x; n)]′ − [ξ1 (x; n)]′η1(x; n)
c
c
c
c
S(x; n) = ξ1 (x; n)η2(x; n) − η1(x; n)ξ2 (x; n) +
.
c(x; n)
η1
3.3. Connection formula 3
f (x)g(x)dµ(x),
Theorem (The Holonomic Equation) The Geronimus perturbed MOPS
c
with κn = 1 + N Bn and
c,[1]
{Pn (x)}n≥0 and {Qc (x)}n≥0, corresponding respectively to the measures ( ⇒ inner products)
n
dµ(x) ⇒ f, g µ =
4. Electrostatic model
> 0.
· (2n + (α + β))2x + (2n + α + β)(2n + α + β − 1)Λn ,
1
1
VL(x) = ln uL(x; n) − ln xα+1e−x, with
2
2
uL(x; n) = n(n + α) + Λn [x − (2n + α) + Λn] .
References
[1] C. F. Bracciali, D. K. Dimitrov, and A. Sri Ranga, Chain sequences and symmetric generalized
orthogonal polynomials, J. Comput. Appl. Math. 143 (2002), 95–106.
[2] W. Gautschi, Orthogonal Polynomials: Computation and Approximation, in Numerical Mathematics and Scientific Computation Series, Oxford University Press. New York. 2004.
[3] M. E. H. Ismail, An electrostatics model for zeros of general orthogonal polynomials, Pacific
J. Math. 193 (2000), 355-369.
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