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1. Introduction
Let {P n } nЄ ℕ be a sequence of monic polynomials orthogonal with respect to a weight function m en (0, ∞). Consider the weight function ρω where
where k, p ≥ 1, ζi,ηj < 0 and mi,qj ∈ Z+ ∪ {0}. When qj = 0 for every j, the function p produces a perturbation on the weight m, known in the literature as Christoffel transformation, introduced in [1], (see also [2], [3] and [4]). Orthogonal polynomials associated with this kind of perturbations have been widely studied, in particular analytic properties associated to zeros and asymptotic behavior, (see for instance [5], [6], [7], [8], [9], [10] and [11]). On the other hand, if mi = 0 for every i, p produces a very particular case of a Geronimus transformation, introduced in a general way in [12] and [13], and related with mechanical quadrature, numerical analysis and physics problems concerning non-isospectral discrete-time Volterra chains. See [14], [15], [16], [17], [18], [8], [19] and [20] for recent developments in analytic and asymptotic properties as well as electrostatic models. Finally, if in (1) there exist i, j such that m i q j ≠ 0, we get a rational transformation of the weight. In [21] can be seen a deep study about asymptotic behavior of orthogonal polynomials with respect to this kind of perturbation when ω is the classical Laguerre weight. For a general treatment [16], [22], [23], [24], [10] and [11], are highly recommended.
Special cases of transformations outlined above are the focus of this paper. In particular, this contribution has to do with location of zeros of polynomials associated with this kind of perturbation in some very particular cases. In this way, the structure of this manuscript is as follows. In Section 2, we present some basic elements of the theory and auxiliary results. In Section 3 we discuss some algebraic connections and interlacing properties of zeros of polynomials associated to particular cases of Christoffel transformations when the weight is perturbed by linear factors (x - ζ1) and (x - ζ2 ), with ζ1 ≠ ζ2, and ζ1,ζ2 <0. Finally, in section 4 we consider interlacing properties of zeros of polynomials orthogonal with respect to rational perturbations, namely, we consider perturbations with the rational functions
and
with ζ≠ V, and ζ, V < 0.
2. Preliminaries
Let P be the linear space of polynomials with complex coefficients. P n will denote the linear subspace of polynomials of degree at most n. Let u be a linear functional in the algebraic dual space of P. It will be denoted P'. ⟨u, p⟩ is the action of the linear functional u on the polynomial p Є P. For u e P', the sequence {un}n≥0, u n = ⟨u, x n , is said to be the respective moment sequence. We define the Hankel determinant of order n + 1 for Δn = |(ui+j )n i,j=0|. Also, u is so called quasi-definite or regular if Δn ≠ 0 for n ≥ 0, and it is called positive-definite if (u, ϖ(x)) > 0 for every nonzero and non-negative real polynomial ϖ.
Theorem 1 ([2]). u is positive definite if and only if their moments are real and Δ n > 0 for n ≥ 0.
If u is positive-definite, then there exists a positive Borel measure д supported on an infinite set E ⊆ ℝ such that u has an integral representation
Given a quasi-definite linear functional u on the space P(ℝ) of polynomials with real coefficient, a bilinear form ⟨,⟩ u : P(R) x P(ℝ) → ℝ is defined as ⟨p, q⟩ u : = ⟨u,pq⟩. If u is positive definite then the bilinear form is an inner product on P(ℝ) and, it is usual,
represents the induced norm.
Definition 1 A sequence {P n } n≥0 is called an orthogonal polynomial sequence, (OPS in short), with respect to a moment functional u if for n, m ≥ 0, i). P n is a polynomial of degree n; ii). (u,P n P m ) = 0, for n ≠ m, and iii). (u, P 2 n ) ≠ 0.
If the leading coefficient of P n is 1 for every n ≥ 0, then { P n } n≥0 is said to be a monic orthogonal polynomial sequence, (MOPS in short).
proposition 1 ( [ 2]). Let u be a moment functional. u is quasi-definite if and only if there exists an OPS {P n } n≥0 with respect to the functional
Theorem 2 (Favard's theorem) ([2]). Let {P n } n≥0 be a sequence ofmonic polynomials. {Pn} n≥0 is a MOPS with respect to a quasi-definite linear functional u if and only if there exist sequences of numbers {β n } n≥1 and {γ n } n≥1 , with γ n ≠ 0 for n ≥ 1, such that
On the other hand,
If {x n ,j} n j=1 are the zeros of the polynomial P n we enunciate the following result.
Theorem 3 ( [ 2]). Let I be the support of a positive-definite linear functional u and {P n } n≥ 0 the respective MOPS. Then, i). The zeros of P n are real, simple and located in the interior of the convex hull of I. ii). (Interlacing property). The zeros of P n andP n + 1 mutually separate each other, i.e. if {x n,j } n j=1 are the n zeros of the polynomial P n , arranged in an increasing order, then x n+1, j < x n,j < x n + 1,j+1 ≤ j ≤ n.
We will consider the next tool, useful to deduce interlacing properties of zeros.
lemma 1 (See [ 25]). Let r n (x) = (x - x 1 ) … (x - x n ) and r n-1 (x) = (x - y 1 ) … (x - y n-1 ) be polynomials with real and interlacing zeros
Then for any real constant C the polynomial R n (x) = r n (x) + Cr n-1 (x) has n real zeros ξ 1 < ξ2 < … < Ç n which interlace with both the zeros of r n (x) and r n-1 (x) in the next way: if C > 0,
but if C < 0
3. Christoffel Transformations
If u is a positive-definite linear functional, the respective Borel measure dμ is supported on [a, b] and if ζ ∉ (a, b), i = 1,2,...,
k,
the measure dμ* =
is called a canonical Christoffel Transformation. Concerning to the relation between polynomials orthogonal with respect to μ and μ* in the particular case k = 1, we get the next result.
proposition 2
( [
2]). Let {P
n
} be the MOPS with respect to д, supported in the interval [a, b]. If ζ ≤ α, is not a zero of P
n
, for every n ≥ 1 , then
the monic sequence orthogonal with respect to dμ* = (x - ζ)dμ, satisfies
In addition, if
are the real and simple zeros of P
n
[1]
then
In the sequel, let {P n } nЄℕ be a sequence of monic polynomials orthogonal with respect to the inner product
where dμ ω = ω(x)dx, and ω is a weight function on (0, ∞). As before, {x n , i } n i = 1 represents the zeros of P n . In this way, let {P n [k]} be the sequence of monic polynomials orthogonal with respect to
ζ < 0, k ≥ 0. For every n, P n [0] : = P n and μ ω,0 : = μ ω . Also, ║.║ k represents the induced norm for (5), with ║.║0:= ║.║, the latter, the norm induced by (4).
proposition 3 (see [ 7]). For k ≥ 1
moreover
proposition 4
Let
the zeros of P
n
[k]
arranged in
an increasing order. It holds that
with p q Є ℕU {0} and p< q.
proof 1 Notice that from (3), we know how zeros of members of the MOPS associated with a weight on [0, ∞), are interlaced with the zeros of members of the MOPS associated with the weight perturbed by (x - ζ). As a consequence we get
for k ≥ 0.
Next, we are going to obtain explicitly the three terms recurrence relation, (TTRR in short), that the sequence
satisfies. To do that, we expand xP
[k]
n-1
in terms of {P
n
[k]}, namely
It is clear that a nk = 0 for 0 ≤ k < n - 3. Then
According to explicit formulas for coefficients in (2) we get
and
By means of (7) we get
and
In particular, for k = 1 we obtain
moreover
Here we have used (6). From (7), we finally obtain
and
Summarizing, we get the following result.
proposition 5 The sequence {P n [k]} satisfies the TTRR
with c n [k] and λ n [k] defined in (9) and ( 10). In particular, when k = 1, we get
and
Now, let {Q n } be the monic sequence of polynomials orthogonal with respect to
ζ 1 < ζ 2 < 0, and let {р n [1,1] }, {р n [1,2] } be the MOPS associated with the weights (x- Z 1 )ω and (x - ζ 2 )ω respectively. Also, ║.║ [1,1] .and ║ ║ [1,2] will denote the respective induced norms. By using of (6) with k = 1 we get
and
Multiplying on both sides of(16) by (x- ζ 2 ) we get
then we get the following result.
lemma 2 For every n
With
and
Notice that γ n (ζ 1 , ζ 2 ), pn(ζ 1 , ζ 2 ) > 0. Connection formula (17) is relevant since allow us to build every polynomial Q n in terms of known data, that is, in terms of members of the original MOPS {P n }.
On the other hand, if we expand (x - ζ1 )P n [1,1] by means of the basis {P n [1,2]}, we obtain
with
By orthogonality with respect to (x- ζ 1 )ω,α ni = 0 for i = 0,..., n - 2. Formulas (14) and (15) allows us to obtain
and
Then we get the formula
Now, by using the TTRR (13) for the sequence {P n [1,2]} we get the following result.
proposition 6 For every n,
where
and
To find out the sign of each coefficient in the above formulas, we present the following useful result.
lemma 3
For n Є N, ℝ
n
(x) =
is a increasing function on (-∞,0].
proof 2 It is enough to prove that the first derivative of R n (x) is positive on (-∞, 0). Indeed
In this way, and as a consequence, we can deduce the next important information.
corollary 1 For every n, ζ1 < ζ 2 < 0, and γ n [1,2] , η n [1,2] defined in (19) and (20) respectively, it holds γ n [1,2] < 0 and η n [1,2] > 0.
Let
and
be the zeros of P
n
[1,1] and P
n
[1,2] respectively, and all arranged in an increasing order. The connection formula (18) can be written as
and we consider the sequence of monic polynomials {D n } defined as follows:
This family is quasi-orthogonal with respect to the weight (x - ζ 2 )ω on [0∞) in the next sense:
Definition 2 LetR n be a polynomial of exact degree n. If m is a weight function on interval [a,b], and R n satisfies the conditions
k = 0,..., n - r - 1, then R n is quasi-orthogonal of order r with respect to ω on [a, b].
Next, some consequences of this definition.
Theorem 4 (See [ 26]). If {P n } is a OPS with respect to m on [a, b], R n is quasi-orthogonal of order r with respect to m on [a, b] if there exist numbers c nii , i = 1,...,r, c nr ≠ 0, such that
Theorem 5 (See [ 27]). IfR n is quasi-orthogonal of order r with respect to ω on [a, b], the zeros are real, simple and at least n - r lie in (a, b).
Let {φ n,i } n i=1 be the zeros of Dn, arranged in an increasing order. By Lemma 1 since x n,i [1,2] < x n-1,i [1,2] < x n,i+1 [1,2] ,i =1… n - 1, from (22), we get the following result.
lemma 4 For every n ≥ 1 , φ n,1 < x n,1 [1,2] , and
for i = 1 , . . . , n - 1 .
If 0 < φ n ,1 < x n,1 [1,2] , and taking into account Lemma 4, we can use formula (21) and Lemma 1 to prove the following interlacing properties.
proposition 7 If 0 < φ n,1 then
for i = 1 , . . . , n.
On the other hand, if φ n,1 < 0, notice that if we use (21) for x = 0 and x = x n,1 [1,2] we get
and from Lemma 4 we know that φ n,1 < 0 < x n,1 [1,2] < φ n,2 , then D n does not change of sign in[0, x n,1 [1,2] ] thus P n [1,1] (0) P n [1,1] (x n,1 [1,2] ) >0 and as a consequence x n,1 [1,2] < x n,1 [1,1] Analogously, we can see that, from Lemma 4, and for i = 2,...,n-1,
It is straightforward show that P n [1,1] (x n,n [1,2] ) < 0 . Thus, we complete the proof of the following proposition.
proposition 8 If φ n,1 < 0, then
for i = 1 , . . . , n.
4. Rational Transformations
Let {T n } and {G n } be the monic sequences of polynomials orthogonal with respect to
and
respectively. Also, let ║ ║.ζ v and ║ ║ [V] be the the respective induced norms. Again we consider the MOPS {P n [1]} associated to the weight (x- ζ) ω on [0, ∞).
If (x - V)P n is expanded in terms of the basis {G n } we get
where, immediately, α n,j = 0, j = 1,...,n-2, and
Thus
Let {g n,i } i n =1 be the zeros of G n arranged in an increasing order. By Lemma 1 applied to (25) we obtain for i = 1 , . . . , n.
Now, we adopt the notation {G n (., k)} to represent the monic sequence of polynomials orthogonal with respect to
k Є ℕU{0}, V < 0. Here, for every n we get G n (.,0) := P n and G n (., 1) := Gn. In addition {g n,i (k)} i n =1 represents the zeros of {G n (.,k)}. As a direct consequence of (26), zeros of {G n (.,k)} and {G n (., k + 1)} are interlaced as follows:
for i = 1 , . . . , n. In this way, we get the following result.
proposition 9 For k, m Є ℕ U{0}, k < m, and for i = 1,..., n
Now, we are going to expand (x - ζ)T n by using the basis {G n } and (x - V )p n [1] by using {T n }. On the one hand, we get
and evaluating in ζ, we have
In [18] the next connection formula is presented:
with
where the sequence {F n } is very known in the literature and so-called the Cauchy integrals of {P n }, or functions of the second kind associated with {P n }. In the genesis of what is known today as orthogonal polynomials, the functions of the second kind allowed to find the relationship between continued fractions and orthogonality. The nice work [28] is highly recommended.
In the same way, we obtain
and evaluating in V we get
Now, multiplying (30) by (x - ζ), and using formula (28), we get
This is a connection formula that allow us to build every polynomial Pn [1] by means of polynomials of the sequence {G n } . We summarize in the next proposition.
proposition 10 For every
where
and
Let {x
[1]
n
,i}
ii=1
and {t
n,i
}
n
i=1
be the zeros of P
n
[1] and T
n
respectively, as before, arranged in an increasing order. Since
> 0 in (28), by Lemma 1 we obtain
for i = 1 ,... ,n. In the same way, since -
> 0 in (30) we get
for i = 1 . . . n. Finally, by means of the inequalities (3), (26), (31) and (32) we can prove the following result.
proposition 11 For every n Є ℕ the zeros of G n T n and p [1] are interlaced as follows:
for i = 1, . . . ,n.
5. Conclusions
In general, we have considered the inner product
on the space of real polynomials.
■ We have built an appropriate connection formula to prove interlacing of zeros of polynomials of equal degree that belong to sequences of monic polynomials orthogonal with respect to the Christoffel transformations (x - ζ1)ω and (x- ζ 2 )ω, respectively, with ζ 1 < ζ 2 < 0. As far as we know, the proof is new.
■ For different values of k, we have exhibited the natural interlacing of zerosof polynomials of equal degree and orthogonal with respect to Rational transformations as
■ We have considered the canonical transformations
and (x - ζ)ω, with ζ, V < 0and ζ ≠ V. In this way, we have obtained interlacing properties for zeros of polynomials of equal degree, orthogonal with respect to each of them. As far as we know, the proof is unpublished.
