Abstract
In this paper, we study an electrostatic model for zeros of polynomials orthogonal with respect to inner product
where is the classical Laguerre measure with and
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Introduction
In pioneering works of Heine and Stieltjes, (see [4] and [6]), a nice electrostatic interpretation of the zeros of the classical sequences of orthogonal polynomials was established. To illustrate these ideas, consider the two-parameter sequence of Jacobi polynomials , orthogonal with respect to the weight (where supported on It is well known that the zeros of are real, simple and these lie into the interval of orthogonality . Given if two charges of strength and are put at and respectively, movable unit charges are distributed freely in and assuming that the interaction obeys the logarithmic potential law, then there exists exactly one equilibrium position, the global minimum of total energy. Such equilibrium is reached precisely in the vector of zeros associated with This interpretation can also be made in terms of the family of Laguerre polynomials with as weight of orthogonality, where and supported on . It is enough fix one charge with strength at the origin and with the additional condition that the mean arithmetic of the positions of the movable charges is uniformly bounded. So the unique global minimum of total energy coincides with the vector of zeros of the rescaled polynomial where is the constant of uniform boundedness. The cornerstone that supports such results is the second-order linear differential equation
(where and are polynomials with and ) that satisfy the families of classical polynomials. In [5], an electrostatic model for zeros of polynomials orthogonal with respect to weights supported on intervals bounded or unbounded is presented, which generalizes the work of Heine and Stieltjes. There, the total energy contains external fields which are closely related to the respective weight function, and in this way, without to limit the arithmetic mean or rescale the variable, the zeros of still provide the unique equilibrium position, if a component given for the external field is added to the total energy. Inspired by the above ideas, in this paper, we propose an electrostatic model for zeros of the monic polynomials , orthogonal with respect to inner product
where This is, we work with the classical Laguerre weight perturbed by a rational factor. Such perturbations are known in the literature as canonical Geronimus transformations, and a study of this kind of polynomials can be seen in [2], where is obtained relative asymptotics of these polynomial as well as Mehler–Heine type formula and a Plancherel–Rotach type formula for the rescaled polynomials. In this way, the structure of this manuscript is as follows. In "Preliminaries", we present some auxiliary results about the sequences and In "Holonomic equation", we deduce using standard techniques, a second-order linear differential equation satisfied for every and in "An electrostatic model", we discuss a electrostatic interpretation of the zeros of these polynomials.
Preliminaries
Let be the sequence of classical Laguerre monic polynomials, orthogonal with respect to the inner product
where with In the next proposition, some properties of classical Laguerre polynomials which we will use in the sequel are summarized (see [1] or [7]).
Proposition 1
For every the polynomials hold
-
1.
(three-term recurrence relation).
(1)with and
-
2.
(Structure relation)
(2) -
3.
(3)
-
4.
satisfies the differential equation
(4)
In the other hand, let be the sequence of monic polynomials, orthogonal with respect to inner product
To give an efficient algebraic relation between families and , in [2] the connection formula has been found
with
The constant has the explicit representation
where
and is clear that and so Even more, it is known his asymptotic behaviour, namely
Holonomic equation
Using standard techniques, we will deduct a second-order linear differential equation that satisfy. In effect, taking derivatives in (5), multiplying by the factors and respectively, and by adding the resulting equations, we have
and using (4) we get
And considering (5), we find the equation
On the other hand, taking derivatives in (5) and multiplying by we have
and using (2) and (1), we obtain respectively
and
and as a consequence
Then, from (5) and (8), we have a system of two equations with unknowns and namely
then if
clearly
and substituting in (7) we get
or equivalently
We summarize the results in the next
Proposition 2
The n-th monic polynomial is a solution of the second-order linear differential equation
where
and
with
Remark 1
Note that can be expressed as
and taking into account (6) is clear that if then and as a consequence
An electrostatic model
Now, we give an electrostatic interpretation for the zeros of Let be their zeros arranged in a increasing order. Given that such zeros are real and simples, it is easy check that for
Now we evaluate (10) in for to obtain
or equivalently
and using (11) we get
so, from (12) we have
To describe an electrostatic model, consider a fixed positive charge of strength at the origin a fixed negative charge of strength at and movable unit charges distributed in and suppose that such unit charges are in presence of an external field We will suppose that the charges interact with each other through a logarithmic potential under an external field. So the total energy of the system (with ) is given by the sum of the following components: the mutual interaction of these charges, the external fields created by fixed charges and moreover the external field . Mathematically this is
and it is known that the electrostatic equilibrium of the system, that is, a configuration of positions where the movable unit charges are in equilibrium, occurs in stationary points of the total energy We will prove that effectively, provides a critical point for the total energy . Indeed, form (14) and for
But from (13) which completes the proof.
Remark 2
Note that the external field is the same that in [5] is considered for to model the electrostatic behaviour of zeros of classical Laguerre polynomials. Furthermore, note that in our model, there is a fixed charge that depends of and his strength is negative. In this way, in [3], an electrostatic model with such features for Jacobi–Koornwinder orthogonal polynomials is obtained.
References
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L. A. M. Molano is equally contributed to this work.
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Molano, L.A.M. An electrostatic model for zeros of classical Laguerre polynomials perturbed by a rational factor. Math Sci 8, 120 (2014). https://doi.org/10.1007/s40096-014-0120-y
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DOI: https://doi.org/10.1007/s40096-014-0120-y