Acta Applicandae Mathematicae

, Volume 121, Issue 1, pp 105–135

Higher Order Coherent Pairs

Article

DOI: 10.1007/s10440-012-9696-0

Cite this article as:
Marcellán, F. & Pinzón, N.C. Acta Appl Math (2012) 121: 105. doi:10.1007/s10440-012-9696-0

Abstract

In this paper, we study necessary and sufficient conditions for the relation
$$\begin{array}{@{}l}P_n^{{[r]}}(x) + a_{n-1,r} P_{n-1}^{{[r]}}(x)= R_{n-r}(x) + b_{n-1,r} R_{n-r-1}(x),\\[5pt]\quad a_{n-1,r}\neq0,\ n\geq r+1,\end{array}$$
where {Pn(x)}n≥0 and {Rn(x)}n≥0 are two sequences of monic orthogonal polynomials with respect to the quasi-definite linear functionals \(\mathcal{U},\mathcal{V}\), respectively, or associated with two positive Borel measures μ0,μ1 supported on the real line. We deduce the connection with Sobolev orthogonal polynomials, the relations between these functionals as well as their corresponding formal Stieltjes series. As sake of example, we find the coherent pairs when one of the linear functionals is classical.

Keywords

Coherent pairs Sobolev inner product Stieltjes functions Semiclassical linear functionals Orthogonal polynomials 

Mathematics Subject Classification (2000)

42C05 

1 Introduction

The notion of coherent pair was introduced by A. Iserles, P.E. Koch, S.P. Nørsett and J.M. Sanz-Serna in 1991 [15]. They state that a pair of positive Borel measures (μ0,μ1) supported on the real line is, with our terminology, a (1,0)-coherent pair of order 1 if and only if there exist nonzero constants {an,1}n≥1 such that their corresponding sequences of monic orthogonal polynomials (SMOP) {Pn(x)}n≥0 and {Rn(x)}n≥0 satisfy
$$ R_{n}(x) = \frac{P'_{n+1}(x)}{n+1} +a_{n,1}\frac{P'_{n}(x)}{n}, \quad a_{n,1}\neq0,\ n\geq1.$$
(1.1)
Moreover, this condition of coherence is a sufficient condition for the existence of a relation
$$ Q_{n+1}(x;\lambda) + c_{n,1}(\lambda) Q_{n}(x;\lambda) = P_{n+1}(x) + \frac{n+1}{n}\, a_{n,1}P_{n}(x), \quad n\geq1,$$
(1.2)
where {cn,1(λ)}n≥1 are rational functions in λ>0 and {Qn(x;λ)}n≥0 is the SMOP associated with the Sobolev inner product
$$ \bigl\langle p(x),q(x)\bigr \rangle_{\lambda, 1}=\int_{-\infty}^\infty p(x)q(x)\,d\mu_0 + \lambda\int_{-\infty}^\infty p'(x)q'(x) \,d\mu_1, \quad\lambda>0,\ p,q\in\mathbb{P}.$$
(1.3)
Besides, they study the case when the measure μ0 is classical (Laguerre and Jacobi). Furthermore, they introduce the notion of symmetrically coherent pair, when the two measures μ0 and μ1 are symmetric (i.e., invariant under the transformation x↦−x) and the subscripts in (1.1) are changed appropriately.

In 1995, F. Marcellán, T. Pérez, J.C. Petronilho, and M. Piñar (see [20]) showed that if a pair of positive definite linear functionals \((\mathcal {U},\mathcal {V})\) is a (1,0)-coherent pair of order 1, then both are semiclassical, \(\mathcal {V}\) is of class at most 1 and \(\mathcal {U}\) is of class at most 6. Moreover, they proved that there exist polynomials \(\widetilde{\sigma}_{2}(x)\) and \(\widetilde{\tau}_{2}(x)\) such that \(\widetilde{\sigma}_{2}(x)\mathcal {V}=\widetilde {\tau}_{2}(x)\mathcal {U}\), with \(\mathrm{deg} (\widetilde{\sigma}_{2}(x))\leq2\) and \(\mathrm{deg} (\widetilde{\tau}_{2}(x) )\leq3\).

On the other hand, F. Marcellán and J. Petronilho [18] studied (1.1) when \(\mathcal {U}\) and \(\mathcal {V}\), with respective SMOP {Pn(x)}n≥0 and {Rn(x)}n≥0, are quasi-definite linear functionals and they solved the problem when one of the functionals is classical, i.e., Hermite, Laguerre, Jacobi or Bessel.

Finally, in 1997, H.G. Meijer [28] determined all (1,0)-coherent pairs \((\mathcal {U},\mathcal {V})\) of quasi-definite linear functionals of order 1. He proved that at least one of the functionals must be classical. Moreover, he showed that there are only two cases:
  • If \(\mathcal {U}\) is a classical linear functional, then there exist polynomials \(\sigma_{\mathcal {U}}(x)\), \(\tau_{\mathcal {U}}(x)\) and \(\rho_{\mathcal {U}}(x)\), with \(\mathrm{deg} (\sigma_{\mathcal {U}}(x) )\leq2\) and \(\mathrm{deg} (\tau_{\mathcal {U}}(x) )=\mathrm{deg} (\rho_{\mathcal {U}}(x) )=1\), such that \(D(\sigma_{\mathcal {U}}(x)\mathcal {U})=\tau_{\mathcal {U}}(x)\mathcal {U}\) and \(\sigma_{\mathcal {U}}(x)\mathcal {U}=\rho_{\mathcal {U}}(x)\mathcal {V}\). Here \(D\mathcal {U}\) denotes the derivative of a linear functional \(\mathcal {U}\), which is the linear functional defined by \(\langle D\mathcal {U}, p(x) \rangle = - \langle \mathcal {U}, p'(x) \rangle, \forall p\in\mathbb{P}\).

  • If \(\mathcal {V}\) is a classical linear functional, then there exist polynomials \(\sigma_{\mathcal {V}}(x)\), \(\tau_{\mathcal {V}}(x)\) and \(\rho_{\mathcal {V}}(x)\), with \(\mathrm{deg} (\sigma_{\mathcal {V}}(x) )\leq2\) and \(\mathrm{deg} (\tau_{\mathcal {V}}(x) )=\mathrm{deg} (\rho_{\mathcal {V}}(x) )=1\), such that \(D(\sigma_{\mathcal {V}}(x)\mathcal {V})=\tau_{\mathcal {V}}(x)\mathcal {V}\) and \(\sigma_{\mathcal {V}}(x)\mathcal {U}=\rho_{\mathcal {V}}(x)\mathcal {V}\).

H.G. Meijer also determined all symmetrically (1,0)-coherent pairs of order 1, providing similar results to those obtained in the non-symmetric case. Indeed, they correspond to Hermite and Gegenbauer cases. Some analytic properties of Sobolev orthogonal polynomials associated with such pairs of measures have been studied. In particular, their asymptotic behavior (see [26]) as well as the location of their zeros (see [27]). Later on, in 2004, A. Delgado and F. Marcellán [11] extended the notion of a coherent pair to generalized coherent pairs (in our terminology, (1,1)-coherent pair of order 1), by studying the relation
$$ R_{n}(x) +b_{n,1}R_{n-1}(x)= \frac{P'_{n+1}(x)}{n+1} + a_{n,1}\frac {P'_{n}(x)}{n}, \quad n\geq1,$$
(1.4)
with an,1≠0 for all n≥1. They verified that this condition of generalized coherence is a necessary and sufficient condition for the relation (1.2). Also, they determined all (1,1)-coherent pairs of order 1 of linear functionals (bn,1 can be zero). They proved that at least one of the quasi-definite linear functionals (either \(\mathcal {U}\) or \(\mathcal {V}\)) must be semiclassical of class at most 1, generalizing the results obtained by H. G. Meijer for (1,0)-coherent pairs of order 1. Moreover, they showed that there are only two cases:
  • If \(\mathcal {U}\) is a semiclassical linear functional given by \(D(\sigma _{\mathcal {U}}(x)\mathcal {U})=\tau_{\mathcal {U}}(x)\mathcal {U}\) with \(\mathrm{deg} (\sigma_{\mathcal {U}}(x) )\leq\nobreak 3\) and \(\mathrm{deg} (\tau_{\mathcal {U}}(x) )\leq2\), then there exists a constant \(C_{\mathcal {U}}\) such that \(\sigma_{\mathcal {U}}(x)\mathcal {U}= (x-C_{\mathcal {U}})\mathcal {V}\).

  • If \(\mathcal {V}\) is a semiclassical linear functional given by \(D(\sigma _{\mathcal {V}}(x)\mathcal {V})=\tau_{\mathcal {V}}(x)\mathcal {V}\) with \(\mathrm{deg} (\sigma_{\mathcal {V}}(x) )\leq3\) and \(\mathrm{deg} (\tau_{\mathcal {V}}(x) )\) ≤2, then there exists a constant \(C_{\mathcal {V}}\) such that \(\sigma_{\mathcal {V}}(x)\mathcal {U}= (x-C_{\mathcal {V}})\mathcal {V}\).

Finally, a generalization of this situation to symmetrically coherent pairs is stated by A. Delgado and F. Marcellán in 2005 (see [12]).
In 2001, another generalization of coherent pairs was introduced by F. Marcellán, A. Martínez-Finkelshtein, and J. Moreno-Balcázar in [22]. A pair of positive measures on the real line (μ0,μ1) is said to be a k-coherent pair (a (k+1,0)-coherent pair of order 1 according to our terminology), k∈ℕ, if their corresponding SMOP {Pn(x)}n≥0 and {Rn(x)}n≥0 satisfy
$$ R_n(x) =\frac{P'_{n+1}(x)}{n+1} + \sum_{j=1}^{k+1}a_{n-j+1,n}\frac {P'_{n-j+1}(x)}{n-j+1}, \quad n\geq k+1,$$
(1.5)
with ank,n≠0. Some nontrivial examples of k-coherent pairs are presented as well as the following relation between Sobolev polynomials {Qn(x;λ)}n≥0 associated with the inner product (1.3) and the polynomials {Pn(x)}n≥0 associated with the first measure of this product is stated as a necessary condition for (k+1,0)-coherence of order 1 We get (1.2) when k=0.

K.H. Kwon, J.H. Lee, and F. Marcellán [17] studied k-coherent pairs for k=1, but they called them generalized coherent pairs. They concluded that if \((\mathcal {U},\mathcal {V})\) is a generalized coherent pair of (quasi-definite) linear functionals, then \(\mathcal {U}\) and \(\mathcal {V}\) must be semiclassical (\(\mathcal {U}\) of class at most 6 and \(\mathcal {V}\) of class at most 2). They also studied the case when either \(\mathcal {U}\) or \(\mathcal {V}\) is classical.

In 1999 and 2000, P. Maroni and R. Sfaxi ([24] and [25]) introduced the notion of coherent pair associated withϕ(x) with indexs, where ϕ(x) is a monic polynomial of degree t and s≥0. In such case, a pair ({Rn(x)}n≥0,{Pn(x)}n≥0) of SMOP with respect to the pair of quasi-definite linear functionals \((\mathcal {V},\mathcal {U})\) satisfies
$$ \phi(x)R_{n}(x) = \sum_{k=n-s}^{n+t}a_{n,k}\frac{P'_{k+1}(x)}{k+1}, \quad a_{n,n-s}\neq0,\ n\geq s.$$
(1.6)
If Rn(x):=Pn(x) for all n∈ℕ, {Pn(x)}n≥0 is said to be a diagonal sequence associated withϕ(x) with indexs. They obtained necessary and sufficient conditions for (1.6) and they obtained results for the dual sequences of {Pn(x)}n≥0 and {Rn(x)}n≥0. In 2006, Sfaxi and Alaya [30] continued this study from another point of view.
Notice that from the three-term recurrence relation that {Rn(x)}n≥0 satisfies and expressing ϕ(x)Rn(x) as a linear combination of polynomials Rnt−1,…,Rn+t, we get
$$R_{n+t}(x)+\sum_{k=n-t-1}^{n+t-1}b_{n,k} R_{k}(x) = \frac {P'_{n+t+1}(x)}{n+t+1}+\sum _{k=n-s}^{n+t-1} a_{n,k}\frac {P'_{k+1}(x)}{k+1},\quad a_{n,n-s}\neq0,\ n\geq s.$$
This is a relation of (t+s,2t+1)-coherence of order 1.
Later on, in 2003 and 2004 M. Alfaro, F. Marcellán, A. Peña and M.L. Rezola (see [1] and [2]) analyzed the following algebraic relation between two SMOP {Pn(x)}n≥0 and {Rn(x)}n≥0
$$R_{n}(x) + b_{n-1,0}R_{n-1}(x)= P_{n+1}(x)+ a_{n-1,0}P_{n}(x), \quad n\geq1,$$
that we will call (1,1)-coherence of order 0. It yields the relation \((x-C^{P} )\mathcal {U}=\xi(x-C^{R} )\mathcal {V}\), with CP and CR constants. Under some conditions, a pair of quasi-definite linear functionals \((\mathcal {U},\mathcal {V})\) is a (1,1)-coherent pair of order 0 if and only if these functionals satisfy the above relation of relational type.
Afterwards, in 2006 J. Petronilho [29] extended this problem as follows
$$R_{n}(x) + \sum_{i=1}^Nb_{n-i,n,0} R_{n-i}(x) = P_n(x) + \sum _{i=1}^M a_{n-i,n,0} P_{n-i}(x),\quad n\geq\min\{M,N\},$$
where ani,n,0=0 and bni,n,0=0 if ni<0, (with our terminology, (M,N)-coherence of order 0). He proved that if a pair of quasi-definite linear functionals \((\mathcal {U},\mathcal {V})\) is a (M,N)-coherent pair of order 0 then, under some conditions, these functionals satisfy \(\phi(x)\mathcal {U}=\psi(x)\mathcal {V}\), where ϕ(x) and ψ(x) are polynomials of degree N and M, respectively. On the other hand, there the case (M,N)=(2,1) carefully studied. In 2010 and 2011, M. Alfaro, F. Marcellán, A. Peña and M.L. Rezola ([3] and [4]) studied the case (M,N)=(2,0), with an−2,n,0≠0 for n≥2, and stated the relation \(\mathcal {U}=h(x)\mathcal {V}\) between their quasi-definite linear functionals, where h(x) is a polynomial of degree 2.
To complete this historical overview, in 2008, M.N. de Jesus and J. Petronilho [10] proposed the more general case, the so called (M,N)-coherent pairs of order (r,s), where the derivatives of order r and s of two SMOP {Pn(x)}n≥0 and {Rn(x)}n≥0 are related by
$$\sum_{i=0}^M a_{n+r-i,n,r}P_{n+r-i}^{(r)}(x) = \sum_{i=0}^Nb_{n+s-i,n,s} R_{n+s-i}^{(s)}(x), \quad n\geq0,$$
M and N are non-negative integers and an+ri,n,r, bn+ri,n,r are complex parameters satisfying some natural conditions. They proved that if \(\mathcal {U}\) and \(\mathcal {V}\) are the corresponding quasi-definite linear functionals associated with {Pn(x)}n≥0 and {Rn(x)}n≥0 then, for 0≤sr, there exist four polynomials γM+s+i(x) and φN+r+i(x) of degree M+s+i and N+r+i, respectively, i=0,1, such that
$$D^{r-s} \bigl(\gamma_{M+s+i}(x)\mathcal {V}\bigr)=\varphi_{N+r+i}(x)\mathcal {U}, \quad i=0,1,$$
where \(D\mathcal {U}\) denotes the distributional derivative of the linear functional \(\mathcal {U}\). Therefore, if r=s then \(\mathcal {U}\) and \(\mathcal {V}\) are related by a relation of rational type. Besides, they concluded that if r=s+1 and {Pn(x)}n≥0≠{Rn(x)}n≥0, then there exist polynomials γ(x), ϕ(x), and ψ(x) of degrees at most 2(M+s), M+N+2s+2 and M+N+2s+1, respectively, such that and hence, \(\mathcal {U}\) and \(\mathcal {V}\) are semiclassical linear functionals of classes at most 3M+N+4s and M+N+2s, respectively. When r=s+1 and {Pn(x)}n≥0={Rn(x)}n≥0, \(\mathcal {U}\) and \(\mathcal {V}\) coincide up to a constant factor and are semiclassical of class at most max{M+s−2,N+s}.

Finally, in 2010, A. Branquinho and M.N. Rebocho [7], stated that if a pair of positive Borel measures (μ0,μ1) is a (1,0)-coherent pair of order 2, then each of them is semiclassical and μ1 is a rational modification of μ0. Also, they concluded that if {Pn(x)}n≥0={Rn(x)}n≥0 then, under some conditions, ({Pn(x)}n≥0, {Pn(x)}n≥0) is a (N+2,N)-coherent pair of order 1 if and only if \(\mathcal {U}\) is a semiclassical linear functional of class at most N.

The concept of coherent pair of measures has been generalized to the case of measures supported on curves of the complex plane. In the pioneering contribution, [8] an approach to coherent pairs of measures supported on the unit circle has been done.

In this work, we focus our attention on (1,1)-coherent pairs of order r of quasi-definite linear functionals \((\mathcal {U},\mathcal {V})\), that is, their corresponding SMOP {Pn(x)}n≥0 and {Rn(x)}n≥0 satisfy
$$P_n^{[r]}(x) +a_{n-1,r} P_{n-1}^{[r]}(x) = R_{n-r}(x) +b_{n-1,r} R_{n-r-1}(x), \quad a_{n-1,r}\neq0,\ n\geq r+1. $$
(3.13)
If bn,r=0 for all nr+1, then \((\mathcal {U},\mathcal {V})\) is said to be a (1,0)-coherent pair of order r.
The structure of the manuscript is as follows. In Sect. 2 we review some of the standard facts on orthogonal polynomials. In Sect. 3 we introduce the notion of (1,0) and (1,1)-coherent pair of order r of positive Borel measures supported on the real line and we state their relation with Sobolev orthogonal polynomials. In Sect. 4 we extend our study to (1,0) and (1,1) -coherent pairs of order r of quasi-definite linear functionals. From Lemma 22 which states the relation
$$D^r\bigl[\gamma_{n,r}(x)\mathcal {V}\bigr]=(-1)^r \varphi_{n+r,r}(x)\mathcal {U},\quad n\geq1, $$
(4.2)
for \((\mathcal {U},\mathcal {V})\) a (1,1)-coherent pair of order r of functionals, where γn,r(x) is a monic polynomial of degree n and φn+r,r(x) is a polynomial of degree at most n+r,1 we obtain Theorem 23 and Theorem 28. The first one states that if \((\mathcal {U},\mathcal {V})\) is a (1,1) (or (1,0))-coherent pair of order r, then for r,k∈ℕ and kr, where \(\tilde{\sigma}_{r+1}(x)\), \(\tilde{\sigma}^{{D^{k}\mathcal {V},\mathcal {U}}}_{r+1}(x)\), \(\tilde{\sigma}^{\mathcal {V},D^{k}\mathcal {V}}_{r+1}(x)\), \(\tilde{\tau }_{r+1}(x)\), \(\tilde{\tau}^{D^{k}\mathcal {V},\mathcal {U}}_{r+1}(x)\) and \(\tilde{\tau }^{\mathcal {V},D^{k}\mathcal {V}}_{r+1}(x)\) are polynomials whose degree are given either explicitly or by an upper bound. The proof of this theorem is given for k=0,1,2,3 and rk. On the other hand, we show that if \(\mathcal {U}\) is a classical linear functional given by \(D(\sigma(x)\mathcal {U})=\tau(x)\mathcal {U}\) with deg(σ(x))≤2 and deg(τ(x))=1, and \(\mathcal {V}\) is a quasi-definite linear functional, such that \(\langle \mathcal {U},\sigma^{r}(x)\rangle =1=\langle \mathcal {V},1\rangle\), then \((x-C^{P^{[r]}} )\sigma^{r}(x)\mathcal {U}=\xi (x-C^{R} )\mathcal {V}\) and \(P_{n}^{[r]}(x) \neq R_{n-r}(x)\), nr+1, where ξ, \(C^{P^{[r]}}\), and CR are constants, is a necessary and sufficient condition for \((\mathcal {U},\mathcal {V})\) to be a (1,1)-coherent pair of order r with ar,rbr,r and an,rbn,r≠0, nr. Besides, in this case \(\mathcal {V}\) is a semiclassical linear functional of class at most 2.

In Sect. 5 we deduce some relations between the formal Stieltjes series associated with the linear functionals in a (1,1) (or (1,0))-coherent pair of order r.

2 Basic Background

2.1 Linear Functionals

We will denote by ℙ the linear space of polynomials in one variable with complex coefficients and ℙn denotes the linear subspace of polynomials of degree at most n. Let \(\mathcal {U}\) be a linear functional in ℙ. \(\langle \mathcal {U}, p(x) \rangle\) will denote the image of polynomial p(x) by \(\mathcal {U}\).

Every sequence of monic polynomials {Pn(x)}n≥0, with deg(Pn(x))=n, is a basis for ℙ. There exists a unique sequence of linear functionals {℘n}n≥0, called the dual basis of {Pn(x)}n≥0, such that 〈℘n,Pm(x)〉=δn,m, n,m∈ℕ, where δn,m denotes the Kronecker Delta. So, each linear functional \(\mathcal {U}\) in ℙ can be expressed as
$$\mathcal {U}= \sum_{n\geq0} \lambda_n\wp_n, \quad\text{where } \lambda_n= \bigl\langle \mathcal {U},P_n(x) \bigr\rangle.$$
δa will denote the Delta Dirac linear functional ata, a∈ℂ, defined by
$$\bigl\langle\delta_a, p(x) \bigr\rangle= p(a), \quad\forall p\in \mathbb{P}.$$
Let \(\mathcal {U}\) be a linear functional in ℙ and q(x)∈ℙ∖{0}. We define the linear functionals \(q(x)\mathcal {U}\) and \((q(x))^{-1}\mathcal {U}\) by
$$ \bigl\langle q(x)\mathcal {U}, p(x) \bigr\rangle= \bigl \langle \mathcal {U}, q(x)p(x) \bigr\rangle, \qquad\bigl\langle\bigl(q(x)\bigr)^{-1}\mathcal {U}, p(x) \bigr\rangle= \biggl\langle \mathcal {U}, \frac {p(x)-L_q(x;p)}{q(x)}\biggr\rangle,$$
(2.1)
for all p∈ℙ, where Lq(x;p) denotes the interpolatory polynomial of p(x) at the zeros of q(x) taking into account their multiplicity. The above operations are not commutative. Indeed, \((x-a)(x-a)^{-1}\mathcal {U}=\mathcal {U}\) but
$$ (x-a)^{-1}(x-a)\mathcal {U}=\mathcal {U}-\langle \mathcal {U}, 1 \rangle\delta_a.$$
(2.2)

Proposition 1

Let\(\mathcal {U}\)be a linear functional andq∈ℙ, then forr∈ℕ
$$ D^r\bigl(q(x)\mathcal {U}\bigr) = \sum_{k=0}^r\binom{r}{k} q^{(k)}(x) D^{r-k}\mathcal {U}.$$
(2.3)
Finally, the formal Stieltjes series of the linear functional \(\mathcal {U}\) is
$$ S_\mathcal {U}(z) = -\sum _{n\geq0} \frac{u_n}{z^{n+1}}.$$
(2.4)
Its rth derivative, r∈ℕ, is given by \(S_{\mathcal {U}}^{(r)}(z) = (-1)^{r+1}\sum_{n\geq0} (n+1)_{r}\frac{u_{n}}{z^{n+1+r}}\), where (n+1)r denotes the Pochhammer symbol,
$$(a)_n = \frac{\Gamma(a+n)}{\Gamma(a)} = a(a+1)\cdots(a+n-1), \quad n\geq1,\quad\text{and} \quad(a)_0=1.$$
The Pochhammer symbol satisfies \((a+b)_{n} = \sum_{k=0}^{n} \binom{n}{k}(a)_{n-k}(b)_{k}\) and (−a)n=(−1)n(an+1)n.

2.2 Orthogonal Polynomials

A linear functional \(\mathcal {U}\) is said to be quasi-definite or regular (see [9]) if the Hankel matrix \(H=(u_{i+j})_{i,j=0}^{\infty}\) associated with the moments of the functional is quasi-definite, i.e., \(\Delta_{n}=\det(H_{n})=\det ((u_{i+j})_{i,j=0}^{n} )\neq0\), for all n∈ℕ. Hence, there exists a sequence of polynomials {Pn(x)}n≥0 such that
  1. (i)

    deg(Pn(x))=n, for all n∈ℕ,

     
  2. (ii)

    \(\langle \mathcal {U}, P_{n}(x)P_{m}(x)\rangle= k^{P}_{n}\,\delta_{n,m}\), \(k^{P}_{n} \neq0\) and n,m∈ℕ.

     
{Pn(x)}n≥0 is said to be a sequence of orthogonal polynomials (SOP) with respect to the linear functional \(\mathcal {U}\). This sequence is unique up to multiplicative constants. If all polynomials of the sequence are monic, {Pn(x)}n≥0 is called the sequence of monic orthogonal polynomials (SMOP) with respect to the linear functional \(\mathcal {U}\).

Moreover, if the leading principal submatrices of H are positive definite, then \(\mathcal {U}\) is said to be positive definite. In this case there exists a positive Borel measure μ supported on the real line such that \(\langle \mathcal {U}, p(x)\rangle= \int_{\mathbb {R}}p(x) d\mu(x)\), for all p∈ℙ.

Let {Pn(x)}n≥0 be a SMOP with respect to a quasi-definite linear functional \(\mathcal {U}\). Then,
$$ P_n(x) = \frac{1}{\Delta_{n-1}} \, \left\vert \arraycolsep=5pt\begin{array}{@{}ccccc@{}}u_0 & u_1 & \cdots& u_{n-1} & u_n \\u_1 & u_2 & \cdots& u_n & u_{n+1} \\\vdots& \vdots& \ddots& \vdots& \vdots\\u_{n-1} & u_n & \cdots& u_{2n-2} & u_{2n-1} \\1 & x & \cdots& x^{n-1} & x^n \\\end{array}\right\vert, \quad n\geq1, \qquad P_0(x)=1.$$
(2.5)
Another characterization of orthogonal polynomials is given by the Favard Theorem: (see [9]). A sequence of monic polynomials {Pn(x)}n≥0is a SMOP with respect to a quasi-definite linear functional\(\mathcal {U}\) (that is unique ifu0=1) if and only if there exist sequences of complex numbers\(\{\alpha _{n}^{P}\}_{n\geq0}\)and\(\{\beta_{n}^{P}\}_{n\geq0}\), \(\beta_{n}^{P}\neq0\), n≥2, such that they satisfy the three-term recurrence relation (TTRR)
$$ P_{n}(x)= \bigl(x- \alpha_{n}^{P}\bigr) P_{n-1}(x) - \beta^{P}_{n}P_{n-2}(x),\quad n\geq2, \qquad P_{1}(x)=x-\alpha_{1}^{P},\qquad P_0(x)=1,$$
(2.6)
\(\alpha_{n}^{P} = \frac{ \langle \mathcal {U}, xP_{n-1}^{2}(x) \rangle }{ \langle \mathcal {U}, P_{n-1}^{2}(x) \rangle}\), \(\beta_{n+1}^{P} =\frac{ \langle \mathcal {U}, P_{n}^{2}(x) \rangle}{ \langle \mathcal {U},P_{n-1}^{2}(x) \rangle}\neq0\), n≥1. Moreover, the linear functional\(\mathcal {U}\)is positive definite if and only if\(\alpha _{n}^{P}\)is real and\(\beta_{n+1}^{P} > 0\), forn≥1.
Given a SMOP {Pn(x)}n≥0, for all n∈ℕ we can define the nth reproducing kernel associated with\(\mathcal {U}\)
$$K_n(x,y;\mathcal {U})= \sum_{j=0}^n\frac{P_j(y)}{ \langle \mathcal {U}, P_{j}^2(x)\rangle} \, P_j(x),$$
(2.7)
that satisfy \(\langle \mathcal {U}, K_{n}(x,y;\mathcal {U}) p(x) \rangle= p(y)\), for all p∈ℙn.

Finally, we give some results relating the dual basis of a SMOP and its respective linear functional and derivatives.

Proposition 2

[23]

Let {Pn(x)}n≥0be the SMOP with respect to a quasi-definite linear functional\(\mathcal {U}\)and let {℘n}n≥0be its corresponding dual basis. Thenr∈ℕ, where\(\{\wp_{n}^{[r]} \}_{n\geq r}\)is the dual basis of the sequence of monic polynomials\(\{\frac{(n-r)!}{n!}P_{n}^{(r)}(x)\allowbreak =P_{n}^{[r]}(x) \}_{n\geq r}\).

Corollary 3

Under the conditions of the previous proposition we get
$$D^r\wp_n^{[r]}= (-1)^r\frac{n!}{(n-r)!}\, \frac{P_n(x)}{\langle \mathcal {U},P^2_n(x)\rangle} \, \mathcal {U}, \quad n\geq r.$$
In particular, if\(\{P_{n}^{[r]}(x) \}_{n\geq r}\)is the SMOP with respect to the linear functional\(\mathcal {U}^{[r]}\), then
$$D^r\mathcal {U}^{[r]}= (-1)^r r! \frac{ \langle \mathcal {U}^{[r]}, 1 \rangle }{\langle \mathcal {U}, P^2_r(x)\rangle} \,P_r(x)\mathcal {U}.$$

2.3 Semiclassical and Classical Linear Functionals

Let σ(x),τ(x) be non-zero polynomials such that deg(σ(x))=k≥0 and deg(τ(x)) =l≥1, σ(x)=akxk+… and τ(x)=blxl+… . (σ(x),τ(x)) is said to be an admissible pair if they satisfy either k−1≠l or, k−1=l and nal+1+bl≠0 for all n∈ℕ.

A linear functional \(\mathcal {U}\) is said to be semiclassical if it is quasi-definite and there exists an admissible pair of polynomials (σ(x),τ(x)) such that the following distributional Pearson equation holds
$$ D\bigl(\sigma(x)\mathcal {U}\bigr)=\tau(x)\mathcal {U}.$$
(2.10)
Under these conditions, the class of\(\mathcal {U}\) is defined by the non-negative integer
$$ s:= \min\max\bigl\{\mathrm {deg}\bigl(\sigma(x)\bigr) - 2,\mathrm{deg}\bigl(\tau(x)\bigr) - 1 \bigr\},$$
(2.11)
where the minimum is taken among all admissible pairs of polynomials (σ(x),τ(x)) such that (2.10) holds. If σ(x) is monic, the admissible pair that determines the class is unique. We also say that the SMOP associated with a semiclassical linear functional is a semiclassical SMOP of classs if the class of \(\mathcal {U}\) is s (see [23]). Examples of semiclassical SMOP have been extensively studied in the literature (see in [5, 6] and [16]). Its role in the theory of polynomials orthogonal with respect to weighted Sobolev inner products has been emphasized in [21].

Proposition 4

[24]

Let\(\mathcal {U}\)and\(\mathcal {V}\)be two quasi-definite linear functionals. If these functionals are related by a expression of rational type, i.e., there exist nonzero polynomialsp(x) andq(x) such that
$$ p(x)\mathcal {U}= q(x)\mathcal {V},$$
(2.12)
then, \(\mathcal {U}\)is a semiclassical linear functional if and only if so is\(\mathcal {V}\). Moreover, if the class of\(\mathcal {U}\)iss, then the class of\(\mathcal {V}\)is at mosts+deg(p(x))+deg(q(x)).
A semiclassical linear functional \(\mathcal {U}\) of class s=0, i.e.,
$$ D\bigl(\sigma(x)\mathcal {U}\bigr)=\tau(x)\mathcal {U}\quad \text{with }\mathrm{deg}\bigl(\sigma(x)\bigr)\leq2 \text{ and } \mathrm {deg}\bigl(\tau(x)\bigr)=1,$$
(2.13)
is said to be a classical linear functional and its SMOP associated is called classical SMOP. Under linear transformations of the variable and some conditions on the parameters, we get the classical SMOP with respect to definite positive linear functionals,where the weight function w(x) is positive and integrable. The associated linear functional is given by \(\langle \mathcal {U}, p(x)\rangle=\int_{a}^{b} p(x) w(x) dx\), for all p∈ℙ.

A characterization of classical orthogonal polynomials is the following

Theorem 5

[13]

Let\(\mathcal {U}\)be a quasi-definite linear functional with SMOP {Pn(x)}n≥0. The following statements are equivalent
  1. (i)

    {Pn(x)}n≥0is a classical SMOP and\(\mathcal {U}\)satisfies (2.13).

     
  2. (ii)

    The sequence of monic polynomials\(\{P_{n}^{[1]}(x)=P'_{n}(x)/n\}_{n\geq1}\)is a SMOP with respect to a linear functional\(\mathcal {U}^{[1]}\). Indeed, \(\mathcal {U}^{[1]}=\sigma(x)\mathcal {U}\).

     
Besides, \(\{P_{n}^{[1]}(x)\}_{n\geq1}\)is also a classical SMOP of the same type as {Pn(x)}n≥0because\(\mathcal {U}^{[1]}\)satisfies the distributional differential equation
$$ D \bigl(\sigma(x)\mathcal {U}^{[1]} \bigr)= \bigl(\tau(x)+\sigma'(x) \bigr)\mathcal {U}^{[1]}.$$
(2.14)

Corollary 6

[14]

In the conditions of Theorem 5, the following statements are equivalent
  1. (i)

    {Pn(x)}n≥0is a classical SMOP and\(\mathcal {U}\)satisfies (2.13).

     
  2. (ii)
    Forr∈ℕ, the sequence of monic polynomials\(\{P_{n}^{[r]}(x)=\frac{(n-r)!}{n!} \, P_{n}^{(r)}(x)\}_{n\geq r}\), where\(P_{n}^{(r)}(x)\)denotes therth derivative ofPn(x), is a SMOP associated to the linear functional
    $$ \mathcal {U}^{[r]}=\sigma^r(x)\mathcal {U}.$$
    (2.15)
     
Moreover, \(\{P_{n}^{[r]}(x)\}_{n\geq r}\)is also a SMOP of the same type as {Pn(x)}n≥0because\(\mathcal {U}^{[r]}\)satisfies
$$ D \bigl(\sigma(x)\mathcal {U}^{[r]} \bigr)= \bigl(\tau(x)+r\sigma'(x) \bigr)\mathcal {U}^{[r]}.$$
(2.16)

Remark 7

For r∈ℕ, rn, and for the classical monic orthogonal polynomials (Hermite {Hn(x)}n≥0, Laguerre \(\{L_{n}^{(\alpha )}(x)\}_{n\geq0}\) and Jacobi \(\{P_{n}^{(\alpha,\beta)}(x)\}_{n\geq0}\)), we get

Remark 8

Other characterizations of classical orthogonal polynomials have been presented in [19]. Notice that the basic tool for their proofs is the algebraic theory of linear functionals in the linear space of polynomials with complex coefficients.

3 Sobolev Orthogonal Polynomials and Coherent Pairs of Order r of Measures

We will consider the Sobolev inner product
$$ \bigl\langle p(x),q(x) \bigr \rangle_{\lambda, r}=\int_{-\infty}^\infty p(x)q(x) \,\,d\mu_0 + \lambda\int_{-\infty}^\infty p^{(r)}(x)q^{(r)}(x) \,d\mu_1,$$
(3.1)
where μ0, μ1 are positive Borel measures supported on the real line, λ∈ℝ+ and p,q∈ℙ, from now on, polynomials with real coefficients. In these conditions, we make the following assumptions and notations as in Table 2:
Thus,
$$w_{n,m} =\begin{cases}u_{n+m} + \lambda \frac{n!}{(n-r)!}\frac{m!}{(m-r)!} \,v_{n+m-2r} & n\geq r\text{ and } m\geq r,\\u_{n+m} & n<r \text{ or } m<r.\end{cases}$$
Notice that Qn(x;λ,r)=Pn(x), for nr. Besides, as the coefficients of Qn(x;λ,r) are rational functions of λ, with the degree of the denominator greater than or equal to the degree of the numerator, then
$$ O_n^{(r)}(x;r) =\frac{n!}{(n-r)!} \, R_{n-r}(x), \quad n\geq r,$$
(3.2)
where
$$O_n(x;r) = \underset{\lambda\rightarrow\infty}{\lim}Q_n(x;\lambda,r)$$
is a monic polynomial of degree n.
Furthermore, for nr,
$$\frac{(n-r)!}{n!} \,O_n(x;r)=\frac{(n-r)!}{n!}\,P_n(x) + \sum_{k=0}^{n-1}a_{k,n,r} \frac{(k-r)!}{k!} \,P_k(x),$$
with
$$a_{k,n,r} = \frac{(n-r)!}{n!}\frac{k!}{(k-r)!}\frac{\langle O_n(x;r),P_k(x)\rangle_{\mu_0}}{\|P_k(x)\|_{\mu_0}^2}.$$
Taking into account \(\langle O_{n}(x;r),\,x^{k}\rangle_{\mu_{0}}\,=\int_{\mathbb {R}}O_{n}(x;r)x^{k} d\mu_{0}= 0\), for k<min{n,r}, then
$$ \frac{(n-r)!}{n!} \,O_n(x;r)=\frac{(n-r)!}{n!} \,P_n(x) + \sum_{k=r}^{n-1}a_{k,n,r} \frac{(k-r)!}{k!} \,P_k(x), \quad n\geq r.$$
(3.3)
If the above expression is derived r times, we get
$$ R_{n-r}(x)=O_n^{[r]}(x;r)= P_n^{[r]}(x) + \sum _{k=r}^{n-1} a_{k,n,r}P_k^{[r]}(x), \quad n\geq r,$$
(3.4)
where we use the notation
$$p^{[r]}(x) = \frac{(n-r)!}{n!} \,p^{(r)}(x),$$
where p(x) a monic polynomial of degree n and r∈ℕ.

Definition 9

The pair of measures (μ0,μ1) is said to be a (1,0)-coherent of orderr if in (3.4) ak,n,r=0 for k=r,…,n−2 y an−1,r=an−1,n,r≠0, nr+1, i.e., if their respective SMOP satisfy
$$ R_{n-r}(x)= P_n^{[r]}(x) +a_{n-1,r} P_{n-1}^{[r]}(x) \quad\text{with }a_{n-1,r}\neq0, \,\, n\geq r+1.$$
(3.5)
In this case, we say that either \((\mathcal {U},\mathcal {V})\) or ({Pn(x)}n≥0,{Rn(x)}n≥0) is a (1,0)-coherent pair of order r.
Note that if (μ0,μ1) is a pair of (1,0)-coherent measures of order r, then (3.3) becomes
$$ O_n(x;r)=P_n(x) + \frac{n}{n-r}\, a_{n-1,r} P_{n-1}(x)\quad a_{n-1,r}\neq0, \, n\geq r+1.$$
(3.6)
On the other hand, for n∈ℕ
$$O_n(x;r)= Q_{n}(x;\lambda ,r) + \sum_{k=0}^{n-1}c_{k,n,r}(\lambda)Q_{k}(x;\lambda ,r),$$
where So, ck,n,r(λ)=0 for k=0,…,n−2,
$$ c_{n-1,r}(\lambda):=c_{n-1,n,r}(\lambda)=\frac{n}{n-r}\,a_{n-1,r}\frac {\|P_{n-1}(x)\|_{\mu_0}^2}{\|Q_{n-1}(x;\lambda ,r)\|_{\lambda,r}^2}\neq0, \quad n\geq r+1,$$
(3.7)
and
$$ O_n(x;r)= Q_{n}(x;\lambda ,r) +c_{n-1,r}(\lambda)Q_{n-1}(x;\lambda ,r), \quad c_{n-1,r}(\lambda)\neq0, \, n\geq r+1.$$
(3.8)
We have thus proved the following

Proposition 10

If (μ0,μ1) is a pair of (1,0)-coherent measures of orderrgiven by (3.5), then there exist constantscn−1,r(λ)≠0 such thatholds. Besides, cn−1,r(λ) is given by (3.7).
Conversely, if there exist constants dn−1,r(λ) and an−1,r≠0 such that then, applying 〈⋅,q(x)〉λ,r with q∈ℙn−2, we get
$$\int_\mathbb {R}\biggl(\frac{(n-r)!}{n!}P_n^{(r)}(x)+ a_{n-1,r} \frac {(n-r-1)!}{(n-1)!}P_{n-1}^{(r)}(x) \biggr)q^{(r)}(x) d\mu_1= 0, \quad\forall\, q\in \mathbb{P}_{n-2},$$
i.e.,
$$ \int_\mathbb {R}\bigl(P_n^{[r]}(x) +a_{n-1,r} P_{n-1}^{[r]}(x) \bigr) p(x) d\mu_1= 0, \quad\forall\, p\in\mathbb{P}_{n-r-2}.$$
(3.11)
So,
$$P_n^{[r]}(x) + a_{n-1,r} P_{n-1}^{[r]}(x)= R_{n-r}(x) + \sum_{k=0}^{n-r-1}b_{k,n,r} R_k(x),$$
with bk,n,r=0 for k=0,…,nr−2, and
$$ b_{n-1,r} :=b_{n-r-1,n,r} = \frac{ \langle P_n^{[r]}(x),R_{n-r-1}(x) \rangle_{\mu_1}}{\|R_{n-r-1}(x)\|_{\mu_1}^2} \, + \,a_{n-1,r}, \quad n\geq r+1.$$
(3.12)
Therefore,
$$ P_n^{[r]}(x) +a_{n-1,r} P_{n-1}^{[r]}(x) = R_{n-r}(x) +b_{n-1,r} R_{n-r-1}(x),\quad a_{n-1,r}\neq0,\ n\geq r+1.$$
(3.13)
Finally, from (3.10) and (3.13), for nr+1, we have
$$ Q^{(r)}_n(x;\lambda,r) + d_{n-1,r}(\lambda)Q^{(r)}_{n-1}(x;\lambda,r) = \frac{n!}{(n-r)!}\bigl[R_{n-r}(x) + b_{n-1,r} R_{n-r-1}(x) \bigr].$$
(3.14)

Definition 11

The pair of measures (μ0,μ1) is said to be (1,1)-coherent of orderr if their respective SMOP satisfy (3.13). In this case, we say that either \((\mathcal {U},\mathcal {V})\) or ({Pn(x)}n≥0,{Rn(x)}n≥0) is a (1,1)-coherent pair of order r.

Remark 12

\((\mathcal {U},\mathcal {V})\) is a (1,0)-coherent pair of order r if and only if \((\mathcal {U},\mathcal {V})\) is a (1,1)-coherent pair of order r and bn−1,r=0 for all nr+1 (in (3.13)).

Proposition 13

Letnr+1. If there exist constantsan−1,r≠0 anddn−1,r(λ)≠0 such that (3.10) holds, then (μ0,μ1) is a (1,1)-coherent pair of orderrgiven by (3.13) wherebn−1,ris (3.12).

Conversely, (μ0,μ1) is a (1,1)-coherent pair of orderrgiven by (3.13), then there are constantsdn−1,r(λ)≠0 given by
$$ d_{n-1,r}(\lambda)=\frac{\frac{n}{n-r}\,a_{n-1,r} \|P_{n-1}(x)\|_{\mu _0}^2 + \lambda\frac{n!}{(n-r)!}\,\frac{(n-1)!}{(n-r-1)!} \,b_{n-1,r}\|R_{n-r-1}(x)\|_{\mu_1}^2 }{\|Q_{n-1}(x;\lambda ,r)\|_{\lambda,r}^2},$$
(3.15)
nr+1, such that (3.10) is satisfied.

Therefore, (3.13) and (3.10) (≈(3.9)2) are equivalent.

Proof

We already have shown (3.10) implies (3.13). Now, we going to see the converse. If (3.13) holds, then we have (3.11). So,
$$\biggl\langle P_n(x) + \frac{n}{n-r}\, a_{n-1,r}P_{n-1}(x) ,q(x) \biggr\rangle_{\lambda,r}=0 \quad\text{for all } q\in\mathbb{P}_{n-2}.$$
On the other hand,
$$P_n(x) + \frac{n}{n-r}\, a_{n-1,r} P_{n-1}(x)= Q_{n}(x;\lambda ,r) + \sum_{k=0}^{n-1}d_{k,n,r}(\lambda)Q_{k}(x;\lambda ,r),$$
where
$$d_{k,n,r}(\lambda) =\frac{\langle P_n(x) + \frac{n}{n-r}\, a_{n-1,r}P_{n-1}(x),Q_{k}(x;\lambda ,r)\rangle_{\lambda,r}}{\| Q_{k}(x;\lambda ,r)\|_{\lambda,r}^2}.$$
Thus dk,n,r(λ)=0, k=0,…,n−2 and if we denote dn−1,r:=dn−1,n,r, we obtain (3.10). Besides, using (3.13) we have (3.15). □
Let come back to (1,0)-coherence of order r. We will see that the sequence {cn,r(λ)}nr in (3.9) has an additional property. From (3.9) and (3.5) we have

Remark 14

If ({Pn(x)}n≥0,{Rn(x)}n≥0) is a (1,0)-coherent pair of order r given by (3.5), then from (3.16), (3.7) and (3.9) we can obtain the Sobolev SMOP {Qn(x;λ,r)}n≥0.

Finally, if we replace (3.16) in (3.7) we obtain
$$ c_{n-1,r}(\lambda)= \frac{B_{n-1,r}}{E_{n-1,r}(\lambda) -c_{n-2,r}(\lambda)}, \quad n\geq r+2,$$
(3.17)
where with initial condition
$$ c_{r,r}(\lambda)=\frac{(r+1)a_{r,r}\|P_{r}(x)\|_{\mu_0}^2}{(r!)^2\|R_{0}(x)\|_{\mu_1}^2\,\lambda+\|P_{r}(x)\|_{\mu_0}^2}.$$
(3.20)
Note that En,r(λ) is a polynomial in λ of degree 1, for nr+1. Besides, if (μ0,μ1) is a (1,0)-coherent pair of measures of order r, then using (3.17)–(3.20) we can get the sequence {cn−1,r(λ)}nr+1, and we can obtain the Sobolev orthogonal polynomials Qn(x;λ,r) (see Remark 14) taking into account the telescopic relation (3.9). For this, let remind that Qn(x;λ,r)=Pn(x), for nr.

Proposition 15

In the conditions of Proposition 10, the sequence {cn,r(λ)}nris given by
$$ c_{m+r,r}(\lambda) =\frac{f_{m,r}(\lambda)}{f_{m+1,r}(\lambda)}, \quad m\geq0,$$
(3.21)
where {fm,r(λ)}m≥0is a sequence of orthogonal polynomials with respect to a positive Borel measure supported on ℝ.

Proof

From (3.17)–(3.20) and by induction on m, it is easy to verify (3.21) and to check that fm,r(λ) is a polynomial in λ of degree m.

On the other hand, from (3.17) and (3.21) for m≥2, where \(f_{m,r}(\lambda)=\tilde{k}_{m}\tilde {f}_{m,r}(\lambda )\), Em+r−1,r(λ)=em+r−1,r,1λ+em+r−1,r,2 and \(\tilde{f}_{m,r}(\lambda)\) is a monic polynomial. As \(\tilde {k}_{m}=\frac {e_{m+r-1,r,1}}{B_{m+r-1,r}}\,\tilde{k}_{m-1}\neq0\), then with So, using the Favard Theorem, \(\{\tilde{f}_{m,r}(\lambda)\}_{m\geq0}\) is a SMOP with respect to a positive linear functional and therefore {fm,r(λ)}m≥0 is a sequence of orthogonal polynomials associated with a positive Borel measure supported on the real line. □
Now, in the same way, but for (μ0,μ1) a (1,1)-coherent pair of order r, we will compute an explicit expression for {dn,r(λ)}nr. From (3.10), (3.13), and (3.14) we get

Note that if bn−2,r=0, then (3.16) and (3.22) coincide and so cn−2,r(λ)=dn−2,r(λ), which is the case when there is (1,0)-coherence of order r.

Remark 16

If ({Pn(x)}n≥0,{Rn(x)}n≥0) is a (1,1)-coherent pair of order r (respectively, a (1,0)-coherent pair of order r), then from Qn(x;λ,r)=Pn(x) for nr, (3.22), (3.15) and (3.10) (respectively, (3.16), (3.7) and (3.9)) we can obtain the Sobolev SMOP {Qn(x;λ,r)}n≥0.

Finally, if we substitute (3.22) in (3.15), we get
$$ d_{n-1,r}(\lambda)= \frac{F_{n-1,r,1}(\lambda)}{G_{n-1,r,1}(\lambda )-d_{n-2,r}(\lambda)H_{n-1,r,1}(\lambda)}, \quad n\geq r+2,$$
(3.23)
where
$$ \begin{aligned}F_{n-1,r,1}(\lambda) &= \biggl[\frac{n!}{(n-r)!}\,\frac {(n-1)!}{(n-r-1)!} \,b_{n-1,r} \bigl\|R_{n-r-1}(x)\bigr\|_{\mu_1}^2\biggr]\lambda \\&\quad{} + \frac{n}{n-r}\,a_{n-1,r} \bigl\|P_{n-1}(x)\bigr\|_{\mu_0}^2,\\G_{n-1,r,1}(\lambda) &= \biggl[ \biggl(\frac{(n-1)!}{(n-r-1)!}\biggr)^2 \bigl(\bigl\|R_{n-r-1}(x)\bigr\|^2_{\mu_1}+ b_{n-2,r}^2 \bigl\|R_{n-r-2}(x)\bigr\|^2_{\mu_1}\bigr) \biggr]\lambda \\&\quad {}+ \bigl\|P_{n-1}(x)\bigr\|_{\mu_0}^2+ \biggl(\frac{n-1}{n-r-1}\, a_{n-2,r} \biggr)^2 \bigl\|P_{n-2}(x)\bigr\|_{\mu_0}^2,\\H_{n-1,r,1}(\lambda) &= \biggl[\frac{(n-1)!}{(n-r-1)!}\, b_{n-2,r}\,\bigl\|R_{n-r-2}(x)\bigr\|_{\mu_1}^2 \biggr]\lambda \\&\quad{} + \frac{n-1}{n-r-1}\, a_{n-2,r}\,\bigl\|P_{n-2}(x)\bigr\|_{\mu_0}^2,\end{aligned}$$
(3.24)
with initial condition
$$ d_{r,r}(\lambda)=\frac{ [(r+1)!r! b_{r,r} \| R_{0}(x)\|_{\mu _1}^2 ]\lambda+ (r+1)a_{r,r}\|P_{r}(x)\|_{\mu_0}^2}{[(r!)^2\|R_{0}(x)\|_{\mu_1}^2 ]\lambda+\|P_{r}(x)\|_{\mu_0}^2}.$$
(3.25)
Note that Fn,r,1(λ), Gn,r,1(λ) and Hn,r,1(λ), for nr+1, are polynomials in λ of degree 1 and so dn,r(λ) is a rational function in λ for nr. Besides, if (μ0,μ1) is a (1,1)-coherent pair of measures of order r given by (3.13), then from (3.23)–(3.25) we can obtain the sequence {dn,r(λ)}nr and using (3.10) we can compute the Sobolev orthogonal polynomials Qn(x;λ,r), nr+1, without calculating their norms (see Remark 16). For this remember that Qn(x;λ,r)=Pn(x) for nr.

Also note that when there is (1,0)-coherence of order r, then the decompositions (3.17) and (3.23) coincide.

Remark 17

The sequence {dn,r(λ)}nr, given by Proposition 13 and (3.23), is such that
$$ d_{m+r,r}(\lambda) =\frac{g_{m+1,r}(\lambda)}{h_{m+1,r}(\lambda)}, \quad m\geq0,$$
(3.26)
where gm+1,r(λ) and hm+1,r(λ) are polynomials in λ of degree at most m+1. This is easy to check by induction on m.

4 Coherent Pairs of Order r of Linear Functionals

In this section we assume that \(\mathcal {U}\) and \(\mathcal {V}\) are quasi-definite linear functionals and we study necessary and sufficient conditions for \((\mathcal {U},\mathcal {V})\) be a (1,1)-coherent pair of order r, i.e., their respective SMOP {Pn(x)}n≥0 and {Rn(x)}n≥0 satisfy the algebraic relation
$$ P_n^{[r]}(x) +a_{n-1,r} P_{n-1}^{[r]}(x) = R_{n-r}(x) +b_{n-1,r} R_{n-r-1}(x), \quad a_{n-1,r}\neq0,\, n\geq r+1.$$
(4.1)
Many of the results of this section remain valid if bn−1,r=0 for all nr+1, i.e., if \((\mathcal {U},\mathcal {V})\) is a (1,0)-coherent pair of order r.

From (4.1), we get \(\langle \mathcal {V},P_{n}^{[r]}(x) \rangle= - a_{n-1,r} \langle \mathcal {V}, P_{n-1}^{[r]}(x)\rangle\) for nr+23 and \(\langle \mathcal {V}, P_{r+1}^{[r]}(x) \rangle= (b_{r,r}-a_{r,r})\langle \mathcal {V},1\rangle\). From now on we assume that ar,rbr,r.

Remark 18

If \((\mathcal {U},\mathcal {V})\) is a (1,1)-coherent pair of order r given by (4.1) then, ar,rbr,r if and only if \(P_{n}^{[r]}(x) \neq R_{n-r}(x)\) for all nr+1.

Remark 19

If ({Pn(x)}n≥0,{Rn(x)}n≥0) is a (1,1)-coherent pair of order r given by (4.1), then for nr+1

Lemma 20

If\((\mathcal {U},\mathcal {V})\)is a (1,1)-coherent pair of orderrgiven by (4.1), then there exists a monic polynomialγn,r(x) of degreen≥1 such that
$$\bigl\langle\gamma_{n,r}(x)\mathcal {V}, P_{m+r}^{[r]}(x)\bigr\rangle=0, \quad m\geq n+1.$$
If\((\mathcal {U},\mathcal {V})\)is a (1,0)-coherent pair of orderr, this equality holds formn.

Proof

Let
$$\gamma_{n,r}(x)=R_n(x) + \sum _{j=0}^{n-1} A_{j,n,r}R_j(x).$$
From Remark 19 we get Thus, since br,rar,r, we can choose real numbers A0,n,r,…,An−1,n,r, not all zero, such that \(\langle \gamma_{n,r}(x)\mathcal {V}, P_{n+r+1}^{[r]}(x) \rangle= 0\), for n≥1. Also, from (4.1)
$$\bigl\langle\gamma_{n,r}(x)\mathcal {V}, P_{m+r+1}^{[r]}(x)\bigr\rangle= -a_{m+r,r} \bigl\langle\gamma_{n,r}(x)\mathcal {V},P_{m+r}^{[r]}(x)\bigr\rangle, \quad n\leq m-1.$$
Therefore, \(\langle\gamma_{n,r}(x)\mathcal {V}, P_{m+r}^{[r]}(x) \rangle=0\), for mn+1. □

Remark 21

In Lemma 20, we can choose A1,n,r=A2,n,r=⋯=An−1,n,r=0 and
$$A_{0,n,r}=\frac{(-1)^{n+1}(b_{n+r,r}-a_{n+r,r})\langle \mathcal {V}, R_{n}^2(x)\rangle}{a_{n+r,r}a_{n+r-1,r}\cdots a_{r+2,r}a_{r+1,r}(b_{r,r}-a_{r,r})\langle \mathcal {V},1 \rangle}.$$
So, γn,r(x)=Rn(x)+A0,n,r, for n≥1.

Lemma 22

If\((\mathcal {U},\mathcal {V})\)is a (1,1)-coherent pair of orderrgiven by (4.1), then
$$ D^r\bigl[\gamma_{n,r}(x)\mathcal {V}\bigr]=(-1)^r \varphi_{n+r,r}(x)\mathcal {U},\quad n\geq1,$$
(4.2)
whereγn,r(x) is a monic polynomial of degreenandφn+r,r(x) is a polynomial of degree at mostn+r.4
Moreover,
$$ \varphi_{n+r,r}(x)=\sum _{k= 0}^n \frac{(k+r)!}{k!}\frac{\langle\gamma_{n,r}(x)\mathcal {V},P_{k+r}^{[r]}(x)\rangle}{\langle \mathcal {U}, P^2_{k+r}(x)\rangle}\,P_{k+r}(x), \quad n\geq1.$$
(4.3)

Proof

Let γn,r(x) be the polynomial introduced in Lemma 20. Let {℘n}n≥0 be the dual basis of the SMOP  {Pn(x)}n≥0, and let \(\{\wp _{n+r}^{[r]}\}_{n\geq0}\) be the dual basis of the monic polynomials \(\{P_{n+r}^{[r]}(x)\}_{n\geq0}\). Since \(\gamma_{n,r}(x)\mathcal {V}= \sum_{k\geq0}\lambda_{k+r,n,r} \wp_{k+r}^{[r]}\), where λk+r,n,r = \(\langle\gamma_{n,r}(x)\mathcal {V}, P_{k+r}^{[r]}(x)\rangle\), and, from Lemma 20, λk+r,n,r=0 for kn+1 and n≥1. Thus, \(\gamma_{n,r}(x)\mathcal {V}= \sum_{k= 0}^{n}\lambda_{k+r,n,r} \wp_{k+r}^{[r]}\) and, as a consequence, for n≥1. So, if we denote \(\varphi_{n+r,r}(x)=\sum_{k= 0}^{n}\lambda_{k+r,n,r}\frac{(k+r)!}{k!}\frac{P_{k+r}(x)}{\langle \mathcal {U},P^{2}_{k+r}(x)\rangle}\), then the proof is completed. □

The following theorem states that if \((\mathcal {U},\mathcal {V})\) is a (1,1)-coherent pair of order r, then the linear functionals \(\mathcal {U}\), \(\mathcal {V}\), \(D\mathcal {V}\) are related by a expression of rational type.

Theorem 23

If\((\mathcal {U},\mathcal {V})\)is a (1,1)-coherent pair of orderrgiven by (4.1), then forr∈ℕ
  1. (1)
    There exist polynomials\(\tilde{\sigma}_{r+1}(x)\)and\(\tilde{\tau }_{r+1}(x)\)such that
    $$ \tilde{\sigma}_{r+1}(x)\mathcal {V}=\tilde{\tau}_{r+1}(x)\mathcal {U},$$
    (4.4)
    with
    $$\mathrm{deg}\bigl(\tilde{\sigma}_{r+1}(x)\bigr)=2^r \quad \text{\textit{and}} \quad\mathrm{deg}\bigl(\tilde{\tau}_{r+1}(x)\bigr)\leq2^r+2r.$$
    If\((\mathcal {U},\mathcal {V})\)is a (1,0)-coherent pair of orderr, then\(\mathrm{deg}(\tilde{\tau}_{r+1}(x))\leq2^{r}+2r-1\).
     
  2. (2)
    Forr≥1, there exist polynomials\(\tilde{\sigma }^{D\mathcal {V},\mathcal {U}}_{r+1}(x)\)and\(\tilde{\tau}^{D\mathcal {V},\mathcal {U}}_{r+1}(x)\)such that
    $$ \tilde{\sigma}^{{D\mathcal {V},\mathcal {U}}}_{r+1}(x)D\mathcal {V}=\tilde{\tau}^{D\mathcal {V},\mathcal {U}}_{r+1}(x)\mathcal {U},$$
    (4.5)
    with
    $$\mathrm{deg} \bigl(\tilde{\sigma}^{D\mathcal {V},\mathcal {U}}_{r+1}(x) \bigr) =2^{r-1}+1 \quad\text{\textit{and}} \quad\mathrm{deg} \bigl(\tilde{\tau}^{D\mathcal {V},\mathcal {U}}_{r+1}(x) \bigr) \leq2^{r-1}+2r.$$
    If\((\mathcal {U},\mathcal {V})\)is a (1,0)-coherent pair of orderr, then\(\mathrm{deg}(\tilde{\tau}^{D\mathcal {V},\mathcal {U}}_{r+1}(x))\leq2^{r-1}+2r-1\).
     
  3. (3)
    Forr≥2, there exist polynomials\(\tilde{\sigma }^{D^{2}\mathcal {V},\mathcal {U}}_{r+1}(x)\)and\(\tilde{\tau}^{D^{2}\mathcal {V},\mathcal {U}}_{r+1}(x)\)such that
    $$ \tilde{\sigma }^{{D^2\mathcal {V},\mathcal {U}}}_{r+1}(x)D^2\mathcal {V}=\tilde{\tau}^{D^2\mathcal {V},\mathcal {U}}_{r+1}(x)\mathcal {U},$$
    (4.6)
    with
    $$\mathrm{deg} \bigl(\tilde{\sigma}^{D^2\mathcal {V},\mathcal {U}}_{r+1}(x) \bigr) =2^{r-2}+2 \quad\text{\textit{and}} \quad\mathrm{deg} \bigl(\tilde{\tau}^{D^2\mathcal {V},\mathcal {U}}_{r+1}(x) \bigr) \leq2^{r-2}+2r.$$
    If\((\mathcal {U},\mathcal {V})\)is a (1,0)-coherent pair of orderr, then\(\mathrm{deg}(\tilde{\tau}^{D^{2}\mathcal {V},\mathcal {U}}_{r+1}(x))\leq2^{r-2}+2r-1\).
     
  4. (4)
    Forr≥3, there exist polynomials\(\tilde{\sigma }^{D^{3}\mathcal {V},\mathcal {U}}_{r+1}(x)\)and\(\tilde{\tau}^{D^{3}\mathcal {V},\mathcal {U}}_{r+1}(x)\)such that
    $$ \tilde{\sigma }^{{D^3\mathcal {V},\mathcal {U}}}_{r+1}(x)D^3\mathcal {V}=\tilde{\tau}^{D^3\mathcal {V},\mathcal {U}}_{r+1}(x)\mathcal {U},$$
    (4.7)
    with
    $$\mathrm{deg} \bigl(\tilde{\sigma}^{D^3\mathcal {V},\mathcal {U}}_{r+1}(x) \bigr) =2^{r-3}+3 \quad\text{\textit{and}} \quad\mathrm{deg} \bigl(\tilde{\tau}^{D^3\mathcal {V},\mathcal {U}}_{r+1}(x) \bigr) \leq2^{r-3}+2r.$$
    If\((\mathcal {U},\mathcal {V})\)is a (1,0)-coherent pair of orderr, then\(\mathrm{deg}(\tilde{\tau}^{D^{3}\mathcal {V},\mathcal {U}}_{r+1}(x))\leq2^{r-3}+2r-1\).
     
  5. (5)
    Forr≥1, there exist polynomials\(\tilde{\sigma }^{\mathcal {V},D\mathcal {V}}_{r+1}(x)\)and\(\tilde{\tau}^{\mathcal {V},D\mathcal {V}}_{r+1}(x)\)such that
    $$ \tilde{\tau}^{\mathcal {V},D\mathcal {V}}_{r+1}(x)D\mathcal {V}=\tilde{\sigma}^{\mathcal {V},D\mathcal {V}}_{r+1}(x)\mathcal {V},$$
    (4.8)
    with
    $$\mathrm{deg} \bigl(\tilde{\sigma}^{\mathcal {V},D\mathcal {V}}_{r+1}(x) \bigr) \leq 2^{r}+2r-1 \quad\text{\textit{and}} \quad\mathrm{deg} \bigl(\tilde{\tau}^{\mathcal {V},D\mathcal {V}}_{r+1}(x) \bigr) \leq2^r+2r.$$
    If\((\mathcal {U},\mathcal {V})\)is a (1,0)-coherent pair of orderr, then\(\mathrm{deg}(\tilde{\sigma}^{\mathcal {V},D\mathcal {V}}_{r+1}(x) ) \leq 2^{r}+2r-2\)and\(\mathrm{deg} (\tilde{\tau}^{\mathcal {V},D\mathcal {V}}_{r+1}(x))\leq2^{r}+2r-1\).
     
  6. (6)
    Forr≥2, there exist polynomials\(\tilde{\sigma }^{\mathcal {V},D^{2}\mathcal {V}}_{r+1}(x)\)and\(\tilde{\tau}^{\mathcal {V},D^{2}\mathcal {V}}_{r+1}(x)\)such that
    $$ \tilde{\tau }^{\mathcal {V},D^2\mathcal {V}}_{r+1}(x)D^2\mathcal {V}=\tilde{\sigma}^{\mathcal {V},D^2\mathcal {V}}_{r+1}(x)\mathcal {V},$$
    (4.9)
    with
    $$\mathrm{deg} \bigl(\tilde{\sigma}^{\mathcal {V},D^2\mathcal {V}}_{r+1}(x) \bigr)\leq 2^{r-1}+2r-1 \quad\text{\textit{and}} \quad\mathrm{deg} \bigl(\tilde{\tau}^{\mathcal {V},D^2\mathcal {V}}_{r+1}(x) \bigr) \leq2^{r-1}+2r+1.$$
    If\((\mathcal {U},\mathcal {V})\)is a (1,0)-coherent pair of orderr, then\(\mathrm{deg}(\tilde{\sigma}^{\mathcal {V},D^{2}\mathcal {V}}_{r+1}(x) ) \leq 2^{r-1}+2r-2\)and\(\mathrm{deg} (\tilde{\tau}^{\mathcal {V},D^{2}\mathcal {V}}_{r+1}(x)) \leq2^{r-1}+2r\).
     
  7. (7)
    Forr≥3, there exist polynomials\(\tilde{\sigma }^{\mathcal {V},D^{3}\mathcal {V}}_{r+1}(x)\)and\(\tilde{\tau}^{\mathcal {V},D^{3}\mathcal {V}}_{r+1}(x)\)such that
    $$ \tilde{\tau }^{\mathcal {V},D^3\mathcal {V}}_{r+1}(x)D^3\mathcal {V}=\tilde{\sigma}^{\mathcal {V},D^3\mathcal {V}}_{r+1}(x)\mathcal {V},$$
    (4.10)
    with
    $$\mathrm{deg} \bigl(\tilde{\sigma}^{\mathcal {V},D^3\mathcal {V}}_{r+1}(x) \bigr)\leq 2^{r-2}+2r-1 \quad\text{\textit{and}} \quad\mathrm{deg} \bigl(\tilde{\tau}^{\mathcal {V},D^3\mathcal {V}}_{r+1}(x) \bigr) \leq2^{r-2}+2r+2.$$
    If\((\mathcal {U},\mathcal {V})\)is a (1,0)-coherent pair of orderr, then\(\mathrm{deg}(\tilde{\sigma}^{\mathcal {V},D^{3}\mathcal {V}}_{r+1}(x) ) \leq 2^{r-2}+2r-2\)and\(\mathrm{deg} (\tilde{\tau}^{\mathcal {V},D^{3}\mathcal {V}}_{r+1}(x)) \leq2^{r-2}+2r+1\).
     

Proof

From Lemma 22 and Proposition 1 we get
$$\sum_{k=0}^r \binom{r}{k}\gamma_{n,r}^{(k)}(x) D^{r-k}\mathcal {V}= D^r\bigl[\gamma_{n,r}(x) \mathcal {V}\bigr] = (-1)^r \varphi_{n+r,r}(x)\mathcal {U}, \quad n\geq1,$$
where γn,r(x) is a monic polynomial of degree n and φn+r,r(x) is a polynomial of degree at most n+r. In order to simplify the notation we write \(\gamma_{n}^{(k)}\) and φr+n instead of \(\gamma_{n,r}^{(k)}(x)\) and φr+n,r(x), respectively. Then, taking n=1,2,…,r+1 we obtain the system of linear equations
$$ \Gamma\mathbf{d} = (-1)^r\Phi \mathcal {U},$$
(4.11)
where Γ is a matrix of size (r+1)×(r+1), d and Φ are vectors of size (r+1)×1 such that
  1. (1)
    Solving (4.11) for \(\mathcal {V}\) and \(\mathcal {U}\) we get (4.4) where
    $$\tilde{\sigma}_{r+1}(x)=\sigma_{r+1,r+1}(x) \quad\text{and}\quad \tilde{\tau}_{r+1}(x)=\tau_{r+1,r+1}(x),$$
    with
    $$ \begin{aligned}[c]\sigma_{n,k}&=\binom{r}{k-1}\sigma_{00}\sigma_{11}\sigma_{22}\cdots\sigma_{k-2,k-2} \bigl(\gamma_n^{(k-1)}\sigma_{k-1,k-1}-(k-1)!\,\sigma_{n,k-1} \bigr),\\\tau_{n,k}&=\sigma_{k-1,k-1}\tau_{n,k-1}-\sigma_{n,k-1}\tau_{k-1,k-1},\end{aligned}$$
    (4.12)
    for k=2,…,r+1 and n=k,…,r+1, and initial conditions
    $$ \sigma_{00}=1, \qquad\sigma_{11}=\gamma_1, \qquad\sigma_{n1}=\gamma_n, \qquad \tau_{11}=(-1)^r\varphi_{r+1}, \qquad \tau_{n1}=(-1)^r\varphi_{r+n}.$$
    (4.13)
     
  2. (2)
    For r≥1, solving (4.11) for \(D\mathcal {V}\) and \(\mathcal {U}\) we obtain (4.5) where
    $$\tilde{\sigma}^{D\mathcal {V},\mathcal {U}}_{r+1}=r!\,\sigma_{r+1,r}-\gamma_{r+1}^{(r)}\sigma_{r,r} \quad\text{and}\quad \tilde{\tau}^{D\mathcal {V},\mathcal {U}}_{r+1}=r!\,\tau_{r+1,r}-\gamma_{r+1}^{(r)}\tau_{r,r} ,$$
    with σn,k and τn,k given by (4.13) and (4.12) for k=2,…,r and n=k,…,r+1.
     
  3. (3)
    For r≥2, solving (4.11) for \(D^{2}\mathcal {V}\) and \(\mathcal {U}\) we get (4.6) where with σn,k and τn,k given by (4.13) and (4.12) for k=2,…,r−1 and n=k,…,r+1.
     
  4. (4)
    For r≥3, solving (4.11) for \(D^{3}\mathcal {V}\) and \(\mathcal {U}\) we obtain (4.7) where with σn,k and τn,k given by (4.13) and (4.12) for k=2,…,r−2 and n=k,…,r+1.
     
  5. (5)
    For r≥1, solving (4.11) for \(\mathcal {V}\) and \(D\mathcal {V}\) we get (4.8) where with σn,k and τn,k given by (4.13) and (4.12) for k=2,…,r and n=k,…,r+1.
     
  6. (6)
    For r≥2, solving (4.11) for \(\mathcal {V}\) and \(D^{2}\mathcal {V}\) we obtain (4.9) where with σn,k and τn,k given by (4.13) and (4.12) for k=2,…,r−1 and n=k,…,r+1.
     
  7. (7)
    For r≥3, solving (4.11) for \(\mathcal {V}\) and \(D^{3}\mathcal {V}\) we get (4.10) where with σn,k and τn,k given by (4.13) and (4.12) for k=2,…,r−2 and n=k,…,r+1.
     
Finally, the degrees of all polynomials are calculated using the respective recursive formulas and taking into account that γn,r(x) is a monic polynomial of degree n and deg(φn+r,r(x))≤n+r. We obtain for k=1,…,r+1 and n=k,…,r+1 Note that if \((\mathcal {U},\mathcal {V})\) is a (1,0)-coherent pair of order r, then deg(τnk(x))≤n+r+2k−1−2, because deg(φn+r,r(x))≤n+r−1. □

Corollary 24

Letr∈ℕ. If\((\mathcal {U},\mathcal {V})\)is a (1,1)-coherent pair of orderrgiven by (4.1), then there exist polynomialsσr(x), πr(x) andτr(x) such that
$$ \begin{aligned} \sigma_{r}(x)\mathcal {V}&= \tau_{r}(x)\mathcal {U},\\\sigma_{r}(x)D\mathcal {V}&= \pi_{r}(x)\mathcal {U},\\\tau_{r}(x)D\mathcal {V}&= \pi_{r}(x)\mathcal {V},\end{aligned}$$
(4.16)
with
$$ \mathrm{deg}\bigl(\sigma_{r}(x)\bigr)=2^r, \qquad\mathrm{deg}\bigl(\pi_{r}(x)\bigr)\leq2r+2^r-1, \qquad\mathrm{deg}\bigl(\tau_{r}(x)\bigr)\leq2r+2^r.$$
(4.17)
If\((\mathcal {U},\mathcal {V})\)is a (1,0)-coherent pair of orderr, then deg(πr(x))≤2r+2r−2 and deg(τr(x))≤2r+2r−1.

Proof

From the proof of Theorem 23 it is easy to check that (4.16) holds if where, for k=2,…,r and n=k,…,r+1, σn,k and τn,k are given by (4.13) and (4.12), and their degrees are given by (4.14) and (4.15). □

From the previous theorem we can state the following

Conjecture

If\((\mathcal {U},\mathcal {V})\)is a (1,1)-coherent pair of orderrgiven by (4.1), then forr,k∈ℕ
  • There exist polynomials\(\tilde{\sigma}_{r+1}(x)\)and\(\tilde {\tau }_{r+1}(x)\)such that
    $$ \tilde {\sigma}_{r+1}(x)\mathcal {V}=\tilde{\tau}_{r+1}(x)\mathcal {U},$$
    (4.18)
    with
    $$\mathrm{deg}\bigl(\tilde{\sigma}_{r+1}(x)\bigr)=2^r \quad \text{\textit{and}} \quad\mathrm{deg}\bigl(\tilde{\tau}_{r+1}(x)\bigr)\leq2^r+2r.$$
    If\((\mathcal {U},\mathcal {V})\)is a (1,0)-coherent pair of orderr, then\(\mathrm{deg} (\tilde{\tau}_{r+1}(x) )\leq2^{r}+2r-1\).
  • Forkr, there exist polynomials\(\tilde{\sigma}^{D^{k}\mathcal {V},\mathcal {U}}_{r+1}(x)\)and\(\tilde{\tau}^{D^{k}\mathcal {V},\mathcal {U}}_{r+1}(x)\)such that
    $$ \tilde{\sigma }^{{D^k\mathcal {V},\mathcal {U}}}_{r+1}(x)D^k\mathcal {V}=\tilde{\tau}^{D^k\mathcal {V},\mathcal {U}}_{r+1}(x)\mathcal {U},$$
    (4.19)
    with
    $$\mathrm{deg} \bigl(\tilde{\sigma}^{D^k\mathcal {V},\mathcal {U}}_{r+1}(x) \bigr) =2^{r-k}+k \quad\text{\textit{and}} \quad\mathrm{deg} \bigl(\tilde{\tau}^{D^k\mathcal {V},\mathcal {U}}_{r+1}(x) \bigr) \leq2^{r-k}+2r.$$
    If\((\mathcal {U},\mathcal {V})\)is a (1,0)-coherent pair of orderr, then\(\mathrm{deg}(\tilde{\tau}^{D^{k}\mathcal {V},\mathcal {U}}_{r+1}(x) ) \leq 2^{r-k}+2r-1\).
  • Forkr, there exist polynomials\(\tilde{\sigma}^{\mathcal {V},D^{k}\mathcal {V}}_{r+1}(x)\)and\(\tilde{\tau}^{\mathcal {V},D^{k}\mathcal {V}}_{r+1}(x)\)such that
    $$ \tilde{\tau }^{\mathcal {V},D^k\mathcal {V}}_{r+1}(x)D^k\mathcal {V}=\tilde{\sigma}^{\mathcal {V},D^k\mathcal {V}}_{r+1}(x)\mathcal {V},$$
    (4.20)
    with
    $$\mathrm{deg} \bigl(\tilde{\sigma}^{\mathcal {V},D^k\mathcal {V}}_{r+1}(x) \bigr)\leq 2^{r-(k-1)}+2r-1 \quad\text{\textit{and}} \quad\mathrm{deg} \bigl (\tilde{\tau }^{\mathcal {V},D^k\mathcal {V}}_{r+1}(x) \bigr) \leq2^{r-(k-1)}+2r+k-1.$$
    If\((\mathcal {U},\mathcal {V})\)is a (1,0)-coherent pair of orderr, then\(\mathrm{deg}(\tilde{\sigma}^{\mathcal {V},D^{k}\mathcal {V}}_{r+1}(x) ) \leq 2^{r-(k-1)}+2r-2\)and\(\mathrm{deg} (\tilde{\tau}^{\mathcal {V},D^{k}\mathcal {V}}_{r+1}(x)) \leq2^{r-(k-1)}+2r+k-2\).

When one of the quasi-definite linear functionals is classical, next we analyze its companion coherent measure. From Proposition 4, Corollary 6 and Theorem 25, we state in Corollary 27 that if \(\mathcal {U}\) is a classical linear functional and \(\mathcal {V}\) is a quasi-definite linear functional, then \((\mathcal {U},\mathcal {V})\) is a (1,1)-coherent pair of order r (with ar,rbr,r and an,rbn,r≠0 for nr) if and only if \(\mathcal {U}\) and \(\mathcal {V}\) are related by a expression of rational type and, therefore, \(\mathcal {V}\) is a semiclassical linear functional of class at most 2. The general result is as follows

Theorem 25

(See [1])

Let\(\{P_{n}^{[r]}(x)\}_{n\geq r}\)and {Rn(x)}n≥0be two SMOP with respect to the linear functionals\(\mathcal {U}^{[r]}\)and\(\mathcal {V}\), normalized by\(\langle \mathcal {U}^{[r]}, 1\rangle= 1 = \langle \mathcal {V}, 1\rangle\), r∈ℕ. The following statements are equivalent
  1. (i)
    There exist sequences {an,r}nrand {bn,r}nrwithar,rbr,randan,rbn,r ≠0, nr, such that\(\{P_{n}^{[r]}(x)\}_{n\geq r}\)and {Rn(x)}n≥0are connected by
    $$ P_n^{[r]}(x) +a_{n-1,r} P_{n-1}^{[r]}(x) = R_{n-r}(x) +b_{n-1,r} R_{n-r-1}(x), \quad a_{n-1,r}\neq0,\ n\geq r+1,$$
    (4.1)
    i.e., \((\mathcal {U},\mathcal {V})\)is a (1,1)-coherent pair of orderr, ar,rbr,randan,rbn,r≠0 fornr.
     
  2. (ii)
    \(P_{n}^{[r]}(x) \neq R_{n-r}(x)\)fornr+1, and there exist constantsξ, \(C^{P^{[r]}}\)andCRsuch that
    $$ \bigl(x-C^{P^{[r]}} \bigr)\mathcal {U}^{[r]}=\xi \bigl(x-C^R \bigr)\mathcal {V}.$$
    (4.21)
     

Proof

(i) ⟹ (ii): From (4.1) we get and \(\langle \mathcal {V}, P_{r+1}^{[r]}(x) \rangle= b_{r,r}-a_{r,r}\). Then, \(\langle \mathcal {V}, P_{n}^{[r]}(x) \rangle\neq0\) for nr+1, and therefore, \(P_{n}^{[r]}(x) \neq R_{n-r}(x)\) for nr+1.
Now, we consider the respective TTRR of the SMOP \(\{P_{n}^{[r]}(x)\}_{n\geq r}\) and {Rn(x)}n≥0
$$ \begin{aligned}&P_{n}^{[r]}(x)=\bigl(x- \alpha_{n}^{P^{[r]}} \bigr) P^{[r]}_{n-1}(x)- \beta^{P^{[r]}}_{n}P_{n-2}^{[r]}(x) \quad n\geq r+2,\\&P^{[r]}_{r+1}(x)=x-\alpha_{r+1}^{P^{[r]}}, \qquad P^{[r]}_r(x)=1,\\\end{aligned}$$
(4.22)
$$ \begin{aligned}&R_{n}(x)= \bigl(x- \alpha_{n}^{R} \bigr)R_{n-1}(x) - \beta^{R}_{n}R_{n-2}(x)\quad n\geq2,\\&R_{1}(x)=x-\alpha_{1}^{R}, \qquad R_0(x)=1.\end{aligned}$$
(4.23)
Let A be a constant given by
$$ A = \frac{ \beta_{r+2}^{P^{[r]}}(a_{r+1,r}-b_{r+1,r}) }{b_{r+1,r}(a_{r,r}-b_{r,r}) } - \alpha_{r+1}^{P^{[r]}}.$$
(4.24)
We can express the linear functional \((x+A)\mathcal {U}^{[r]}\) as \((x+A)\mathcal {U}^{[r]}=\sum_{j\geq0} \rho_{j} \frac{R_{j}(x)}{\|R_{j}(x)\|^{2}_{\mathcal {V}}} \mathcal {V}\), where \(\rho_{j}= \langle(x+A)\mathcal {U}^{[r]}, R_{j}(x) \rangle\). We will show by induction on j that
$$ \rho_j =\bigl\langle(x+A)\mathcal {U}^{[r]},R_j(x) \bigr\rangle= 0, \quad j\geq2.$$
(4.25)
From (4.22) for n=r+1 and n=r+2 we get
$$ \bigl\langle \mathcal {U}^{[r]} , \bigl( P^{[r]}_{r+1}(x)\bigr)^2 \bigr\rangle= \bigl \Vert P^{[r]}_{r+1}(x)\bigr \Vert ^2_{\mathcal {U}^{[r]}} = \beta^{P^{[r]}}_{r+2}.$$
(4.26)
So, from (4.1), for n=r+1 and n=r+2, and (4.22), for n=r+1, we obtain Now let us assume that ρi=0 for 2≤ij−1. Then completing the proof of (4.25). Therefore, \((x+A)\mathcal {U}^{[r]}= \sum_{j=0}^{1} \rho_{j} \frac{R_{j}(x)\mathcal {V}}{\|R_{j}(x)\|^{2}_{\mathcal {V}}}\), where
$$\rho_0 \overset{(4.22)}{=} \bigl\langle \mathcal {U}^{[r]}, P^{[r]}_{r+1}(x) + \alpha_{r+1}^{P^{[r]}}+A \bigr\rangle= \alpha_{r+1}^{P^{[r]}} + A \overset{(4.24)}{=} \frac{ \beta_{r+2}^{P^{[r]}}(a_{r+1,r}-b_{r+1,r}) }{ b_{r+1,r}(a_{r,r}-b_{r,r}) }$$
and, from (4.1) for n=r+1 and (4.22) for n=r+1, Therefore, since \(R_{1}(x)=x-\alpha_{1}^{R}\) and \(\|R_{1}(x)\|^{2}_{\mathcal {V}}=\beta _{2}^{R}\), we obtain \((x-C^{P^{[r]}} )\mathcal {U}^{[r]}=\xi (x-C^{R} )\mathcal {V}\) where
$$ \begin{aligned}C^{P^{[r]}}&= \alpha_{r+1}^{P^{[r]}} - \frac{ \beta_{r+2}^{P^{[r]}}(a_{r+1,r}-b_{r+1,r}) }{ b_{r+1,r}(a_{r,r}-b_{r,r}) } ,\\C^R&= \alpha_1^R - \frac{ \beta_2^R(a_{r+1,r}-b_{r+1,r}) }{a_{r+1,r}(a_{r,r}-b_{r,r}) } \, ,\\\xi&= \frac{ \beta_{r+2}^{P^{[r]}} a_{r+1,r} }{ \beta_2^R b_{r+1,r}} .\end{aligned}$$
(4.27)
(ii) ⟹ (i) We suppose that the linear functionals \(\mathcal {U}^{[r]}\) and \(\mathcal {V}\) satisfy (4.21). Then
$$\mathcal {U}^{[r]} - \bigl\langle \mathcal {U}^{[r]} , 1 \bigr\rangle \delta_{C^R} + \bigl(C^R - C^{P^{[r]}} \bigr) \bigl(x-C^R \bigr)^{-1}\mathcal {U}^{[r]} = \xi\bigl[\mathcal {V}- \langle \mathcal {V}, 1 \rangle\delta_{C^R} \bigr].$$
But since \(\langle \mathcal {U}^{[r]} , 1 \rangle= 1 = \langle \mathcal {V}, 1 \rangle\),
$$ \mathcal {V}= \frac{1}{\xi}\bigl\{ \bigl[ 1+ \bigl(C^R-C^{P^{[r]}} \bigr)\bigl(x-C^R \bigr)^{-1} \bigr]\mathcal {U}^{[r]} + (\xi-1 )\delta_{C^R} \bigr\}.$$
(4.28)
Now we consider the Fourier expansion
$$P^{[r]}_n(x) = R_{n-r}(x) + \sum _{j=0}^{n-r-1} \xi_{j,n-r,r}^{P^{[r]}}R_{j}(x),$$
where Here \(\{ P^{[r],(1)}_{n}(x) \}_{n\geq r}\) denotes the associated SMOP of the first kind for the SMOP \(\{ P^{[r]}_{n}(x) \}_{n\geq r}\). Then, for nr+1 where \(K_{n-r-1}(x, C^{R}; \mathcal {V})\), nr+1, denotes the reproducing kernel associated with \(\mathcal {V}\) and, thus
$$ \bigl\langle \mathcal {V},P^{[r]}_n(x) \bigr\rangle= \frac{1}{\xi} \bigl[ (\xi-1)P^{[r]}_n\bigl(C^R\bigr) +\bigl(C^R-C^{P^{[r]}} \bigr) P^{[r],(1)}_{n-1}\bigl(C^R\bigr) \bigr], \quad n\geq r+1.$$
(4.29)
Therefore,
$$ P^{[r]}_n(x) =R_{n-r}(x) + \bigl\langle \mathcal {V}, P^{[r]}_n(x) \bigr \rangle K_{n-r-1}\bigl(x, C^R; \mathcal {V}\bigr), \quad n\geq r+1.$$
(4.30)
In the same way
$$ \bigl\langle \mathcal {U}^{[r]}, R_{n-r}(x) \bigr\rangle= (1-\xi) R_{n-r}\bigl(C^{P^{[r]}}\bigr) + \xi\bigl(C^{P^{[r]}}-C^R \bigr)R_{n-r-1}^{(1)}\bigl(C^{P^{[r]}}\bigr),\quad n\geq r+1,$$
(4.31)
and
$$ R_{n-r}(x) = P^{[r]}_{n}(x)+ \bigl\langle \mathcal {U}^{[r]}, R_{n-r}(x) \bigr\rangle K_{n-1}\bigl(x, C^{P^{[r]}}; \mathcal {U}^{[r]}\bigr), \quad n\geq r+1,$$
(4.32)
where \(K_{n-1}(x, C^{P^{[r]}}; \mathcal {U}^{[r]})\), for nr+1, denotes the reproducing kernel associated with \(\mathcal {U}^{[r]}\).
Moreover, since \(P_{n}^{[r]}(x) \neq R_{n-r}(x)\) for nr+1, from (4.30) and (4.32) we get \(\langle \mathcal {V}, P_{n}^{[r]}(x)\rangle\neq0\) and \(\langle \mathcal {U}^{[r]}, R_{n-r}(x) \rangle \neq0\) for nr+1. Hence, from (4.30) for n and n−1 we obtain for nr+2 Applying the linear functional \(\mathcal {U}^{[r]}\) on both hand sides, we get
$$\frac{ \langle \mathcal {U}^{[r]}, R_{n-r}(x) \rangle}{ \langle \mathcal {U}^{[r]},R_{n-r-1}(x) \rangle} = \frac{ \langle \mathcal {V},P_n^{[r]}(x) \rangle}{ \langle \mathcal {V}, P_{n-1}^{[r]}(x)\rangle} - \frac{ \langle \mathcal {V}, P_n^{[r]}(x) \rangle R_{n-r-1}(C^R)}{\| R_{n-r-1}(x) \|^2}.$$
Thus, for nr+2,
$$P^{[r]}_n(x) - \frac{ \langle \mathcal {V}, P_n^{[r]}(x) \rangle}{\langle \mathcal {V}, P_{n-1}^{[r]}(x) \rangle} P_{n-1}^{[r]}(x)= R_{n-r}(x) - \frac{ \langle \mathcal {U}^{[r]}, R_{n-r}(x) \rangle }{ \langle \mathcal {U}^{[r]}, R_{n-r-1}(x) \rangle} \, R_{n-r-1}(x).$$
Therefore, (i) holds if we take
$$a_{n-1,r} = \frac{ \langle \mathcal {V}, P_n^{[r]}(x) \rangle}{\langle \mathcal {V}, P_{n-1}^{[r]}(x) \rangle} \neq0 \quad\text{and} \quad b_{n-1,r} = \frac{ \langle \mathcal {U}^{[r]}, R_{n-r}(x) \rangle }{ \langle \mathcal {U}^{[r]}, R_{n-r-1}(x) \rangle} \neq0,$$
for nr+2, and since \(P_{r+1}^{[r]}(x) \neq R_{1}(x)\) we can write \(P_{r+1}^{[r]}(x) + a_{r,r}= R_{1}(x)+ b_{r,r}\) with ar,rbr,r and ar,rbr,r≠0. □

Remark 26

Notice that
$$C^{R} = \alpha_{n+1}^{P^{[r]}} - a_{n,r} -\frac{\beta^{P^{[r]}}_{n+1}}{a_{n-1,r}} \quad\text{and}\quad C^{P^{[r]}} = \alpha_{n+1}^{R}- b_{n,r} - \frac{ \beta_{n+1}^{R}}{b_{n-1,r}}, \quad\text{for }n\geq r+2.$$
Indeed, from (4.29) and the TTRR (4.22) and (4.23)), we obtain for nr+3
$$\bigl\langle \mathcal {V}, P^{[r]}_n(x) \bigr\rangle=\bigl(C^R- \alpha_{n}^{P^{[r]}} \bigr) \bigl\langle \mathcal {V},P^{[r]}_{n-1}(x) \bigr\rangle- \beta^{P^{[r]}}_{n}\bigl\langle \mathcal {V}, P^{[r]}_{n-2}(x) \bigr\rangle.$$
Besides, if we apply \(\mathcal {V}\) to (4.1), then we get for nr+2
$$\bigl\langle \mathcal {V}, P^{[r]}_n(x) \bigr\rangle= -a_{n-1,r} \bigl\langle \mathcal {V}, P^{[r]}_{n-1}(x) \bigr \rangle.$$
Thus,
$$0 = \biggl(a_{n-1,r} + C^R - \alpha_{n}^{P^{[r]}}+ \frac{\beta^{P^{[r]}}_{n}}{a_{n-2,r}} \biggr) \bigl\langle \mathcal {V},P^{[r]}_{n-1}(x)\bigr\rangle, \quad n\geq r+3,$$
but, \(\langle \mathcal {V}, P^{[r]}_{n}(x) \rangle\neq0\) for nr+1, then \(C^{R} = \alpha_{n+1}^{P^{[r]}} - a_{n,r} - \frac{\beta ^{P^{[r]}}_{n+1}}{a_{n-1,r}}\)  for nr+2.

In the same way, but from (4.31), we obtain \(C^{P^{[r]}} = \alpha_{n+1}^{R} - b_{n,r} - \frac{ \beta _{n+1}^{R}}{b_{n-1,r}}\) for nr+2.

As a straightforward consequence of Theorem 25, Corollary 6 and Proposition 4, we get

Corollary 27

Let\(\mathcal {U}\)be a classical linear functional given by (2.13) and let\(\mathcal {V}\)be a quasi-definite linear functional, such that\(\langle \mathcal {U}, \sigma^{r}(x)\rangle= 1 = \langle \mathcal {V}, 1\rangle\), r∈ℕ, and corresponding SMOP {Pn(x)}n≥0and {Rn(x)}n≥0. The following statements are equivalent:
  1. (i)

    \((\mathcal {U},\mathcal {V})\)is a (1,1)-coherent pair of orderrgiven by (4.1), withar,rbr,randan,rbn,r≠0 fornr.

     
  2. (ii)
    \(P_{n}^{[r]}(x) \neq R_{n-r}(x)\)fornr+1 and there exist constants\(C^{P^{[r]}}\), CRandξ (see Remark 26) such that
    $$\bigl(x-C^{P^{[r]}} \bigr)\sigma^r(x)\mathcal {U}=\xi \bigl(x-C^R \bigr)\mathcal {V},$$
     
Therefore, \(\mathcal {V}\)is a semiclassical linear functional of class at most 2, taking into account\(\sigma^{r}(x)\mathcal {U}\)is a classical linear functional.
We will analyze every classical case. From (2.2), (2.17) and Table 1, for r∈ℕ we get
  • If \(\mathcal {U}_{L^{(\alpha)}}\) is the Laguerre linear functional and \(\{L^{(\alpha)}_{n}(x)\}_{n\geq0}\) is its SMOP, then σ(x)=x. Assuming \(\langle \mathcal {U}_{L^{(\alpha)}}, x^{r} \rangle\) =1 and then
    $$\mathcal {V}=\xi^{-1} \bigl(x-C^R \bigr)^{-1}\bigl(x-C^{ (L^\alpha )^{[r]}} \bigr)x^r\mathcal {U}_{L^{(\alpha)}} +\delta_{C^R}.$$
  • If \(\mathcal {U}_{P^{(\alpha,\beta)}}\) is the Jacobi linear functional and \(\{P^{(\alpha,\beta)}_{n}(x)\}_{n\geq0}\) is its SMOP, then σ(x)=1−x2. Assuming \(\langle \mathcal {U}_{P^{(\alpha,\beta)}},(1-x^{2} )^{r} \rangle\) =1 and then
    $$\mathcal {V}=\xi^{-1} \bigl(x-C^R \bigr)^{-1}\bigl(x-C^{ (P^{(\alpha,\beta )} )^{[r]}} \bigr) \bigl(1-x^2 \bigr)^r\mathcal {U}_{P^{(\alpha,\beta)}} + \delta_{C^R}.$$
Table 1

Classical SMOP

Classical MOP

Hermite

Laguerre

Jacobi

Pn(x)

Hn(x)

\(L_{n}^{(\alpha)}(x)\)

\(P_{n}^{(\alpha,\beta)}(x)\)

σ(x)

1

x

1−x2

τ(x)

−2x

x+α+1

−(α+β+2)x+βα

(a,b)⊂ℝ

(−∞,∞)

(0,∞)

(−1,1)

w(x)

\(e^{-x^{2}}\)

xαex

(1−x)α(1+x)β

Restriction

α>−1

α>−1,β>−1

Table 2

Basic notations

SMOP

Measure

Linear/Bilinear Functional

Moments

Pn(x)

μ0

\(\mathcal {U}\)

un

Rn(x)

μ1

\(\mathcal {V}\)

vn

Qn(x;λ,r)

〈, 〉λ,r

\(\mathcal {W}\)

wn,m

5 The (Formal) Stieltjes Series and Coherent Pairs of Order r

The following theorem gives some relations for the formal Stieltjes Series associated with the linear functionals constituting either a (1,0)-coherent pair of order r or (1,1)-coherent pair of order r.

Theorem 28

If\((\mathcal {U},\mathcal {V})\)is a (1,1)-coherent pair of orderrgiven by (4.1), then forr∈ℕ,
$$ \varphi_{n+r,r}(z) S_{\mathcal {U}}(z) +(-1)^{r+1} \bigl[\gamma_{n,r}(z)S_{\mathcal {V}}(z)\bigr]^{(r)} \,=\, -A_{n,r}(z), \quad n\geq1,$$
(5.1)
where
$$A_{n,r}(z) = (\mathcal {U}\theta_0\varphi_{n+r,r} )(z)+(-1)^{r+1} [\mathcal {V}\theta_0\gamma_{n,r}]^{(r)}(z)$$
and
$$\mathrm{deg} \bigl(A_{n,r}(z) \bigr)\leq n+r-1,$$
If\((\mathcal {U},\mathcal {V})\)is a (1,0)-coherent pair of orderr, then deg(An,r(z))≤n+r−2, because deg(φn+r,r(x))≤n+r−1. with\((\mathcal {U}\theta_{0}\varphi_{n+r,r} )(z)\)and\([\mathcal {V}\theta_{0}\gamma _{n,r} ]^{(r)}(z)\)given in [23].

Proof

From Lemma 22, for n≥1 there exist polynomials
$$\gamma_{n,r}(x) = \sum_{j=0}^{n}a^{\gamma_{n,r}}_{j}x^j,\quad a^{\gamma_{n,r}}_{n}=1\quad\text{and} \quad\varphi_{n+r,r}(x)=\sum _{j=0}^{n+r} b^{\varphi_{n+r,r}}_{j}x^j$$
such that \(\langle D^{r}[\gamma_{n,r}(x)\mathcal {V}], x^{k} \rangle=\langle(-1)^{r} \varphi_{n+r,r}(x)\mathcal {U}, x^{k} \rangle\), for k∈ℕ. So, for n≥1, \(\frac{k!}{(k-r)!} \sum_{j=0}^{n} a^{\gamma _{n,r}}_{j} v_{k-r+j} = \sum_{j=0}^{n+r} b^{\varphi _{n+r,r}}_{j}u_{k+j}\), where vkr+j=0 if kr+j<0. Thus, multiplying by z−(k+1) and adding for k=0,1,… , we get in each hand side and Therefore, for n≥1  □

Theorem 29

If\((\mathcal {U},\mathcal {V})\)is a (1,1)-coherent pair of orderrgiven by (4.1), then forr∈ℕ,
$$ \sum _{k=0}^r B_{k,n,r}(z)S_{\mathcal {V}}^{(k)}(z)=C_{n,r}(z),\quad n\geq1,$$
(5.2)
i.e., \(S_{\mathcal {V}}(z)\)is the (formal) solution of a non-homogeneous ordinary differential equation of orderrwith polynomial coefficientswith degree
$$\mathrm{deg} \bigl(B_{k,n,r}(z) \bigr) \leq2n+k+1 \quad\text{\textit{and}}\quad \mathrm{deg} \bigl(C_{n,r}(z) \bigr) \leq2n+2r.$$
If\((\mathcal {U},\mathcal {V})\)is a (1,0)-coherent pair of orderr, then deg(Bk,n,r(z))≤2n+kand deg(Cn,r(z))≤2n+2r−2.

Proof

From (5.1), for n≥1, we obtain and subtracting it, we get
$$\varphi_{n+r+1,r}(z) \bigl[\gamma_{n,r}(z)S_{\mathcal {V}}(z)\bigr]^{(r)} - \varphi_{n+r,r}(z) \bigl[\gamma_{n+1,r}(z)S_{\mathcal {V}}(z)\bigr]^{(r)}= C_{n,r}(z) , \quad n\geq1,$$
where
$$C_{n,r}(z) = (-1)^{r+1} \bigl[\varphi_{n+r,r}(z)A_{n+1,r}(z)- \varphi_{n+r+1,r}(z)A_{n,r}(z) \bigr].$$
Finally, using the Leibniz rule, we complete the proof. □

Remark 30

We can get \(S_{\mathcal {V}}\) if we solve (formally) the differential equation (5.2). Hence, from (5.1) we can also obtain \(S_{\mathcal {U}}\).

Footnotes
1

If \((\mathcal {U},\mathcal {V})\) is a (1,0)-coherent pair of order r, then deg(φn+r,r(x))≤n+r−1.

 
2

In (1,0)-coherence of order r, (3.7) and (3.15) coincide.

 
3

If \((\mathcal {U},\mathcal {V})\) is a (1,0)-coherent pair of order r, this equality holds for nr+1.

 
4

If \((\mathcal {U},\mathcal {V})\) is a (1,0)-coherent pair of order r, then the polynomial φn+r,r(x) has degree at most n+r−1 and its expression corresponds to φn+r−1,r(x) in (4.3).

 

Acknowledgements

The authors thank the referees for the careful revision of the manuscript. Their comments and suggestions contributed to improve its presentation. The work of the first author (FM) has been supported by Dirección General de Investigación, Ministerio de Ciencia e Innovación of Spain under grant MTM2009-12740-C03-01.

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Carlos III de MadridLeganés, MadridSpain

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