On some
asymptotic properties of classical
Hermite
polynomials modified by a rational factor
Luis Alejandro Molano Molano∗
Universidad Pedagógica y Tecnológica de Colombia. Escuela de Matemáticas y Estadística, Duitama, Colombia.
Abstract. In this paper we study some asymptotic properties of the sequence of
monic polynomials orthogonal with respect to the measure dµ = , where a,b > 0 and a =6 b. In this
way we study the outer relative asymptotic with respect to the classical
Hermite polynomials; besides, Mehler-Heine type formulas are analyzed.
Keywords:
Asymptotics properties, perturbed Hermite polynomials, Christoffel and
Geronimus perturbations.
MSC2010: 33C25, 33C45, 33C47, 42C05.
Sobre algunas propiedades asintóticas de polinomios de Hermite clásicos modificados por un factor racional
Resumen
En este artículo estudiamos algunas propiedades asintóticas
de la sucesión de polinomios mónicos ortogonales con respecto a la medida ,
donde a,b > 0 y a =6 b. En este sentido, estudiamos la
asintótica relativa exterior con respecto a los polinomios clásicos de Hermite,
además son analizadas fórmulas tipo Mehler-Heine.
Palabras clave: Propiedades asintóticas, polinomios de Hermite perturbados, perturbaciones tipo Christoffel y Geronimus.
1. Introduction
Let U and V be
quasi-definite linear functionals and let {Pn}n∈N and(U,V{R)nis}na∈coherentN be their sequences of monic
orthogonal polynomials (SMOP), respectively. pair if there exists a sequence (an), an =6
0, such that
This concept
is introduced in [7] where it is studied its connection with polynomials
orthogonal with respect to Sobolev inner products like
Where µ0 and µ1 are positive
Borel measures supported on an infinite subset I ⊆R, with U and V as the
associated functionals, respectively. In this way, among others, it is studied
an algebraic connection between the Sobolev polynomials and the sequence {Rn}n∈N
, in such a way
that the coefficients of connection are independent of the degree; an algorithm is presented for to compute
Fourier coefficients using as basis
the Sobolev polynomials. Likewise, if U and V are symmetric, (U,V ) is a symmetric
coherent pair if there
exists a sequence (an), an =6 0, such that
In [8] all
coherent pairs and symmetric coherent pairs are determined and it is shown that
at least one of the functionals has to be classical (Hermite or Gegenbauer in
the symmetric case); moreover, if ξ > 0, the symmetric coherent pair
is obtained. In connection with this particular
case, in [2] the outer relative asymptotics of Sobolev polynomials orthogonal
with respect to (1) is found; besides, in [9] MehlerHeine type formulas are
established with respect to rational modification of the Hermite polynomials.
Under the same assumptions, (U,V ) is a symmetric (1,1)−coherent pair if there exist sequences (an)n∈N and (bn)n∈N , bn 6= 0, such that the respective SPOM satisfies
About this
subject, in [4] is presented the algebraic relation between the Sobolev
polynomials and the polynomials {Rn}n∈N ,; besides,
the particular case where V is classical is studied, and the
respective symmetric (1,1)−coherent companion is found. In particular, we
focus on the symmetric (1,1)−coherent pair
and we will
study the asymptotic behavior of the orthogonal polynomials associated with dµ1. Thus, the structure of this
manuscript is as follows: In the section 2 we present some basic facts about of
asymptotic behavior of Hermite orthogonal polynomials. In section 3 we present
an algebraic connection between the Hermite polynomials and the SMOP associated
with dµ1, as well as
the asymptotic behavior of the respective connection coefficients. Finally, in
section 4 some asymptotics properties are studied.
2. Preliminaries
Frompolynomials,now
on,orthogonaland as it iswithusual,respect{Hn}nto∈N thewill weightrepresente− thex2 onsequence(−∞,∞of).monicThe Hermiteclassical
Hermite linear functional will be denoted by H and hH,p(x)i will be the application of H on any polynomial p. The norm
of the monic Hermite polynomials is defined as
.
On the
other hand the sequence {Hn}n∈N is defined via
the three terms recurrence relation
with the initial conditions H0(x) = 1 and H −1(x) = 0. With respect to the asymptotic behavior of
Hermite polynomials we present the next results.
Theorem 2.1 (See [10]).
uniformly
on compact sets of C\R.
Theorem 2.2 (Mehler-Heine). (See [1]). For j ∈Z fixed,
And
uniformly on compact sets of the complex plane, where Jα represents Bessel’s function of the
first kind defined by
Theorem 2.3 (Mehler-Heine). (See [10]).
uniformly
on compact subsets of C and uniformly on j ∈N∪{0}.
Theorem 2.4 (See [11]). For j ∈Z fixed,
holds
uniformly on compact subsets of Here,
is the
conformal mapping of C\[−1,1] onto the exterior of the closed unit
disk.
If i = 0 and if p and q are
non-negative integers such that n > p − 1, then
and as a
consequence,
The above proves the next
Corollary
2.5. For j ∈Z fixed, and non-negative integers p and q such that n > p − 1,
holds
uniformly on compact subsets of .
The zeros
of Hn are real, simples and symmetric;
that is, for every n, Hn(t) = 0 is
equivalent to be the positive zeros of Hn in increasing order.
It is well
known that the zeros of Hn and Hn−1 are interlaced and for k fixed, xn,k → 0 when n →∞. Besides,
given that Jα has a
countably infinite set of real and positive zeros if α > −1, as a
consequence of Mehler-Heine formulas and the Hurwitz’s theorem,
if n →∞ and k ≥ 1 then
and, where ck > 0 and {jα,k}n∈N are the zeros
of Jα when α > −1.
On the
other hand, let Hn(a,b) be
the sequence of monic polynomials orthogonal with respect to the positiven definite on∈Nlinear functional Hba, defined as
where a,b > 0, and a =6 b. As it is
usual, let k.k(a,b) be the induced norm. If c > 0, then {Hnc}
will be the sequence of
monic polynomials orthogonal with respect to the positive definite functional Hc defined by
the respective induced norm. On
the algebraic connection between the sequence {Hnc} and the classical Hermite polynomials we get
the next result.
Lemma 2.6
(See [2]). There exists a sequence of real numbers (σn) such that
With
respect to asymptotic behavior and Mehler-Heine type formulas for the sequence {Hnc}n∈N , we get the
next Theorem 2.7 (See [9]).
Let Lα be the classical Laguerre
functional, α > −1, and let {Lαn(x)} be the
respective SMOP. We present the next result about asymptotics behavior of
ratios of Laguerre polynomials that will be necessary in our work. The proof
can be see in [5].
Lemma 2.8. For x ∈C\R+,
In this
paper also will be important to deal with rational perturbations of the Lα and the
asymptotic behavior of the associated SPOM. About this topic, in [6] is made an
exhaustive study of asymptotic behavior of orthogonal polynomials associated to
this kind of perturbations. Indeed, given c1, c2 < 0, let nLn(α,c1,c2)(x)o be the
SMOP associated to positive definite linear functional defined on the space of polynomials as
and about
the asymptotic behavior of the sequence nL(nα,c1,c2)(x)o,we get the next
Theorem 2.9 (See [6]).
3. Some
basic results
Given that H is a symmetric linear functional, it
is well known that there is a relation between the classical Laguerre and
Hermite polynomials, namely
H2n(x) = L−n1/2(x2) and H2n+1(x) = xLn1/2(x2). (22)
In this way,
the next result is an extension of the above relations. Lemma
3.1. For every n ∈N,
Proof. Given that is symmetric, there exists an unique
quasi-definite linear functional v, with {Pn} as the associated SPOM and such that
and
(see [3]), moreover,
. We will
see that
. Indeed,
with
the change of variable u = x 2 we get
On the
other hand, from (22) we have , and given
that (see (18))
in the same
way, in the odd case we get
Then we
have deduced the next