On linearly related sequences of difference derivatives of discrete orthogonal polynomials

On linearly related sequences of difference derivatives of discrete orthogonal polynomials R. ´ Alvarez-Nodarsea, J. Petronilhob, N. C. Pinz´on-Cort´esc, R. Sevinik-Adıg¨ uzeld aUniversidad de Sevilla, Departamento de An´alisis Matem´atico, IMUS. Apdo. 1160, E-41080, Sevilla, Spain. bUniversity of Coimbra, Department of Mathematics, CMUC. EC Santa Cruz, 3001-501 Coimbra, Portugal. cPontificia Universidad Javeriana. Departamento de Matem ˜ A, aticas. Carrera 7, 40 - 62, Bogot ˜ A, a, Colombia dSel¸cuk University, Department of Mathematics, Faculty of Science. 42075, Konya, Turkey. Abstract Let νbe either ω∈C\ {0}or q∈C\ {0,1}, and let Dνbe the corresponding difference operator defined in the usual way either by Dωp(x) = p(x+ω)−p(x) ωor Dqp(x) = p(qx)−p(x) (q−1)x. Let Uand Vbe two moment regular linear functionals and let {Pn(x)}n≥0and {Qn(x)}n≥0 be their corresponding orthogonal polynomial sequences (OPS). We discuss an inverse problem in the theory of discrete orthogonal polynomials involving the two OPS {Pn(x)}n≥0and {Qn(x)}n≥0assuming that their difference derivatives Dνof higher orders mand k(resp.) are connected by a linear algebraic structure relation such as M X i=0 ai,nDm νPn+m−i(x) = N X i=0 bi,nDk νQn+k−i(x), n ≥0, where M, N, m, k ∈N∪ {0},aM,n 6= 0 for n≥M,bN,n 6= 0 for n≥N, and ai,n =bi,n = 0 for i>n. Under certain conditions, we prove that Uand Vare related by a rational factor (in the ν−distributional sense). Moreover, when m6=kthen both Uand Vare Dνsemiclassical functionals. This leads us to the concept of (M, N)-Dν-coherent pair of order (m, k) extending to the discrete case several previous works. As an application we consider the OPS with respect to the following Sobolev-type inner product hp(x), r(x)iλ,ν =hU, p(x)r(x)i+λhV,(Dm νp)(x)(Dm νr)(x)i, λ > 0, assuming that Uand V(which, eventually, may be represented by discrete measures sup- Email addresses: [email protected] (R. ´ Alvarez-Nodarse), [email protected] (J. Petronilho), [email protected] (N. C. Pinz´on-Cort´es), [email protected] (R. Sevinik-Adıg¨ uzel) Preprint submitted to Elsevier July 2, 2014 arXiv:1402.0773v2 [math.CA] 1 Jul 2014 ported either on a uniform lattice if ν=ω, or on a q-lattice if ν=q) constitute a (M, N)- Dν-coherent pair of order m(that is, an (M, N)-Dν-coherent pair of order (m, 0)), m∈N being fixed. Keywords: Orthogonal polynomials, inverse problems, semiclassical orthogonal polynomials, coherent pairs, Sobolev-type orthogonal polynomials. 2010 Mathematics Subject Classification: 33C45, 41A10, 42C05. 1. Introduction An interesting problem in the theory of orthogonal polynomials is the one associated with linearly related sequences of derivatives of two sequences of polynomials (see e.g. [7,8,9,22] and references therein). To be more precise assume that Uand Vare two regular functionals and suppose that their corresponding orthogonal polynomial sequences (OPS) {Pn(x)}n≥0 and {Qn(x)}n≥0are connected by the following linear structure relation M X i=0 ai,nDmPn+m−i(x) = N X i=0 bi,nDkQn+k−i(x), n ≥0,(1.1) where M, N, m, k ∈N∪{0},ai,n and bi,n are complex parameters with aM,n 6= 0 for n≥M, bN,n 6= 0 for n≥N, and ai,n =bi,n = 0 for i>n, and Djis the (continuous) derivative operator of order j. The pair (U,V) such that the above relation (1.1) holds is said to be a (M, N)-coherent pair of order (m, k). Historically, the notion of coherent pair –i.e. (1,0)-coherent pair of order (1,0), according with the terminology above– arose in the framework of the theory of Sobolev orthogonal polynomials and it was introduced by A. Iserles, P. E. Koch, S. P. Nørsett and J. M. Sanz- Serna in the very influent work [6]. Subsequent extensions of this notion have been widely introduced and studied in recent decades. For a review of the work done on the subject (including an historical perspective) see e.g. the introductory sections in the papers [7,9,14]. For instance, it is known that for a pair of positive definite linear functionals (U,V), coherence of order m—i.e., of order (m, 0)— is a necessary and sufficient condition for the existence of an algebraic structure relation between the OPS with respect to Uand the Sobolev OPS with respect to an appropriate inner product defined in terms of the measures µ0and µ1 associated with Uand V(resp.), such as hp(x), q(x)iλ=ZR p(x)q(x)dµ0+λZR p(m)(x)q(m)(x)dµ1, λ > 0, m ∈N for every polynomials p, q ∈P. Indeed, this fact has been firstly remarked (and proved) in [6] for ordinary coherent pairs (i.e., (M, N, m, k) = (1,0,1,0)), and stated in [13,21] for N= 0 and m= 1 (being Marbitrary and k= 0). The ideas presented in [13,21] have led to the statement of the mentioned structure relation for arbitrary (M, N, m) (see [9] for the case 2 m= 1 and [7] for arbitrary m≥1). On the other hand, when (U,V) is a (M, N)-coherent pair of order (m, k) of regular linear functionals, it is known that these linear functionals are related by an expression of rational type in the distributional sense and, moreover, they are semiclassical when m6=k(see [22] for the case m=k= 0, [8] for the cases m=kand m=k+ 1, and [7] for arbitrary m>k+ 1). The concept of coherent pair was extended to the OPS of a discrete variable by I. Area, E. Godoy, and F. Marcell´an [2,3,4], and also by F. Marcell´an and N. Pinz´on-Cort´ez [15,16]. Here we generalize this concept as follows. Let Uand Vbe two regular linear functionals and let {Pn(x)}n≥0and {Qn(x)}n≥0be their respective sequences of monic orthogonal polynomials (SMOP). (U,V) is a (M, N)-Dν-coherent pair of order (m, k), for either ν= ω∈C\ {0}or ν=q∈C\ {0,1}, if the algebraic relation M X i=0 ai,nDm νPn+m−i(x) = N X i=0 bi,nDk νQn+k−i(x), n ≥0, holds, where M, N, m, k ∈N∪ {0},{ai,n}n≥0,{bj,n}n≥0⊂Cfor 0 ≤i≤Mand 0 ≤j≤N, aM,n 6= 0 for n≥M,bN,n 6= 0 for n≥N,ai,n =bi,n = 0 for i>n, and Dωp(x) = p(x+ω)−p(x) ω, Dqp(x) = p(qx)−p(x) (q−1)x, p ∈P. Marcell´an and N. C. Pinz´on-Cort´es ([15] for ν=ω, [16] for ν=q) showed that if (U,V) is a (1,1)-Dν-coherent pair then they are Dν-semiclassical linear functionals (one of class at most 1 and the other of class at most 5) and they are related by σ(x)U=ρ(x)V, with deg(σ(x)) ≤3,deg(ρ(x)) = 1. Also, they studied the case when Uis Dν-classical. This is a generalization of the results obtained by I. Area, E. Godoy, and F. Marcell´an ([2,4] for ν=ω, [3] for ν=q) for (1,0)-Dν-coherent pairs. They proved that (1,0)-Dν-coherence is a sufficient condition for at least one of the linear functionals to be Dν-classical and each of them to be a rational modification of the other as above with deg(σ(x)) ≤2. Besides, they determined all Dν-coherent pairs of positive definite linear functionals when Uor Vis some specifical Dν-classical linear functional. Notice that from the study of Dν-coherent pairs it is possible to recover the properties of coherent pairs in the continuous case taking limits when ω→0 and q→1. As before, there is an important connection between Dν-Sobolev orthogonal polynomials and Dν-coherent pairs. In fact, we can consider the Sobolev inner product hp(x), r(x)iλ,ν =hU, p(x)r(x)i+λhV,(Dm νp)(x)(Dm νr)(x)i, λ > 0,(1.2) for fixed m∈N, when Uand V(which will be supported on, either a uniform lattice if ν=ω, or a q-lattice if ν=q) constitute a (M, N)-Dν-coherent pair of order m(i.e., order (m, 0)). In this way, K. H. Kwon, J. H. Lee and F. Marcell´an ([12]) showed that the (M, 0)- Dω-coherence of order 1 condition (for them, (M+1-term) generalized Dω-coherence) yields 3 the relation Pn+1(x) + M X j=1 (n+ 1)aj,n n−j+ 1 Pn−j+1(x) = Sn+1(x;λ, ω) + M X j=1 cj,n,λ,ωSn−j+1(x;λ, ω),(1.3) for n≥M, where {cn,λ,ω}n≥Mare rational functions in λ > 0, cM,n,λ,ω 6= 0, aM,n 6= 0, and {Sn(x;λ, ω)}n≥0is the SMOP associated with the inner product (1.2) for m= 1. Conversely, if (1.3) holds, then (U,V) is a (M, M)-Dω-coherent pair. Additionally, they studied (2,0)-D1-coherent pairs of order 1 and they concluded that the linear functionals must be D1-semiclassical (of class ≤6 for Uand of class ≤2 for V), and they are related by a rational factor. Also, they analized the cases when either Uor Vis a D1-classical linear functional. The aim of this work is twofold. On one hand we will extend some recent results concerning the (M, N)-coherent pairs of order (m, k) for the derivative operator and, on the other, we will show that the concept of (M, N)-Dνcoherent pair of order (m, k) for the discrete analogues of the derivative Dνwill play an important role in the study of the Sobolev orthogonal polynomials on linear and q-linear lattices similar to the one that the (M, N)-coherent pair of order (m, k) plays in the theory of Sobolev orthogonal polynomials [7]. More precisely, we prove that the regular linear functionals associated to an (M, N)-Dν- coherent pair (U,V) are related by a rational modification (in the sense of the distribution theory) and, moreover, Uand Vare both Dν-semiclassical when m6=k(see Theorem 3.2 from below). As an application, we study the sequence of Sobolev OPS with respect to the Sobolev-type inner product (1.2), under the assumption that (U,V)isa(M, N)-Dν-coherent pair of positive definite discrete linear functionals (see theorems 4.3 and 4.4). The structure of this paper is as follows. In Section 2, we state the definitions, results and notation which will be useful in the forthcoming sections. In Section 3, we prove that if a pair of regular linear functionals form a (M, N)-Dν-coherent pair of order (m, k), then they are related by an expression of rational type, and in the case when m6=k, they are Dν-semiclassical. In Section 4, we show the relationship between (M, N)-Dν-coherent pairs of order mand Dν-Sobolev orthogonal polynomials (orthogonal with respect to h·,·iλ,ν given in (1.2)). 2. Preliminaries and Notations Let Pbe the linear space of polynomials with complex coefficients and let P0be its topological dual space which coincides with its algebraic dual P∗[18]. For U ∈ P0and n≥0, un=hU, xni ∈ Cis called the moment of order nof U, where hU, p(x)i ∈ Cdenotes the image of p∈Pby U ∈ P0. For π∈P,π(x)U ∈ P0is defined by hπ(x)U, p(x)i=hU, π(x)p(x)i, p ∈P. Also, for a sequence of polynomials {pn(x)}n≥0with deg(pn(x)) = n,n≥0, we can consider its dual basis {pn}n≥0⊂P0(i.e., hpn, pm(x)i=δn,m,m, n ≥0). In this way, any U ∈ P0can 4 be expanded as U=Pn≥0hU, pn(x)ipn. In this paper we will work with the following two linear difference operators on P Dω:P7→ P, Dωp(x) = p(x+ω)−p(x) ω, ω ∈C\ {0}, Dq:P7→ P, Dqp(x) = p(qx)−p(x) (q−1)x, q ∈C\ {0,±1}. Notice that D1= ∆ and D−1=∇are the well-known forward and backward difference operators, respectively, and Dqis the classical q-derivative operator. From now on, νand ν∗denote either ωand −ω, or, qand q−1, respectively. Then, for U ∈ P0, the linear functional DνUis defined by hDνU, p(x)i=− hU, Dν∗p(x)i, p ∈P. Notice that when q→1 and ω→0 we recover the standard derivative operator. Furthermore, when ω→0 and q→1, (Dνp)(x)→d dx p(x) in Pand DνU → DUin P0, where DUis defined by hDU, p(x)i=− hU, p0(x)i,∀p∈P. For the difference operators Dνthe following straightforward properties hold. Dm ω[p(x)U] = m X j=0 m jDj ωpx+ (m−j)ωDm−j ωU, m ≥0,(2.1) Dm q[p(x)U] = m X j=0 m jq qjDj qpqm−jxDm−j qU, m ≥0,(2.2) where the q-binomial coefficient is defined by n jq := (q, q)n (q, q)j(q, q)n−j , n ≥j≥0, and (α;q)ndenotes the q-Pochhammer symbol, which is the q-analogue of the Pochhammer symbol (α)n, defined by (α)0:= 1,(α)n:= α(α+ 1) · · · (α+n−1), n ≥1, (α;q)0:= 1,(α;q)n:= (1 −α)(1 −αq)· · · (1 −αqn−1), n ≥1. Let U ∈ P0and {Pn(x)}n≥0⊂P.{Pn(x)}n≥0is called the sequence of monic orthogonal polynomials (SMOP) with respect to Uif deg(Pn(x)) = nand hU, Pn(x)Pm(x)i= ξnδn,m,ξn6= 0, n, m ≥0. In this case, Uis said to be regular or quasi-definite, and Υn= det [ui+j]n i,j=06= 0, ∀n≥0. When Υn>0, n≥0, Uis called positive definite. An important characterization of OPs is given by the Favard Theorem:{Pn(x)}n≥0is the SMOP with respect to Uif and only if there exist {αn}n≥0,{βn}n≥0⊂C,βn6= 0, n≥1, 5 such that the three-term recurrence relation (TTRR) Pn+1(x)=(x−αn)Pn(x)−βnPn−1(x), n≥0, holds, with P0(x) = 1, P−1(x) = 0. Moreover, Uis positive definite if and only if αn∈Rand βn+1 >0, for n≥0. (see e.g. [5]). Consider {pn}n≥0, the dual basis of the SMOP {Pn(x)}n≥0, then for fixed m≥0, pn=Pn(x) hU, P2 n(x)iU, Dm ν∗en,ν = (−1)mηn,m,ν pn+m, n ≥0,(2.3) where {en,ν }n≥0is the dual basis of monic polynomials {P[m,ν] n(x)}n≥0given by P[m,ν] n(x) := Dm νPn+m(x) ηn,m,ν ,with ηn,m,ω := (n+ 1)m, ηn,m,q := (qn+1;q)m (1 −q)m. A regular linear functional U ∈ P0is called Dν-semiclassical linear functional (see e.g. [17] for ν=ω, [11] for ν=q) if it is regular and there exist σ, τ ∈P, with deg(τ(x)) ≥1, such that Dν(σ(x)U) = τ(x)U.(2.4) In this way, the class of Uis s:= min max {deg σ−2,deg τ−1} ∈ N∪ {0}, where the minimum is taken among all pairs of polynomials (σ, τ), with deg(τ(x)) ≥1, satisfying (2.4). When s= 0, Uis called a Dν-classical functional. Besides, the corresponding SMOP is said to be Dν-semiclassical of class s, or Dν-classical, respectively. Proposition 2.1. The following equivalence hold Dν[σ(x)U] = τ(x)U ⇐⇒ Dν∗[eσ(x)U] = τ(x)U, where eσ(x) := σ(x) + ωτ(x) if ν=ω , qσ(x)+(q−1)xτ(x) if ν=q . Thus, Uis Dν-semiclassical if and only if it is Dν∗-semiclassical. The proof of this proposition is straightforward and will be omitted. Proposition 2.2. If the regular linear functionals U,Vare related by p(x)U=r(x)V, p, r ∈P\ {0},(2.5) then, Uis Dν-semiclassical (respectively Dν∗-semiclassical) if and only if Valso is Dνsemiclassical (respectively Dν∗-semiclassical). Moreover, if the class of Uis s, then the class of Vis at most s+ deg(p(x)) + deg(r(x)). Proof. Let us suppose that Uis a Dω-semiclassical linear functional given by (2.4), then V satisfies 6 Dω[p(x−ω)σ(x)r(x)V](2.5) =Dω[p(x−ω)p(x)σ(x)U] (2.1) =p(x)p(x+ω)Dω[σ(x)U] + p(x)Dω[p(x)] + p(x)Dω[p(x−ω)]σ(x)U (2.4) = (2.5)p(x+ω)τ(x) + Dωp(x) + p(x−ω)σ(x)r(x)V. Therefore, Vis also Dω-semiclassical and the class of Vis at most s+ deg(p(x)) + deg(r(x)). The Dν∗-semiclassical character of Vfollows from Proposition 2.1. The proof of the q-case is similar but using (2.2) instead of (2.1), and in this case V satisfies Dqp(q−1x)σ(x)r(x)V=p(qx)τ(x) + qDqp(x) + p(q−1x)σ(x)r(x)V. A characterization of Dν-semiclassical linear functionals is the following Proposition 2.3 ([1,19,20]).Let {Pn(x)}n≥0be a SMOP with respect to a linear functional Uand let σ(x)be a monic polynomial. Usatisfies (2.4)if and only if there exists an integer s≥0such that σ(x)P[1,ν∗] n(x) = n+deg(σ(x)) X j=n−s λj,nPj(x), n ≥s, and λn−s,n 6= 0, n ≥s+ 1. Equivalently, eσ(x)P[1,ν] n(x) = n+deg(eσ(x)) X j=n−se λj,nPj(x), n ≥s, and e λn−s,n 6= 0, n ≥s+ 1.(2.6) In these equations, σ(x)and eσ(x) are the polynomials appearing in Proposition 2.1, and λj,n and e λj,n are complex parameters for all nand j. 3. Main results Definition 3.1. A pair of regular linear functionals (U,V) is said to be a (M, N)-Dν- coherent pair of order (m, k), with fixed M, N, m, k ∈N∪ {0}, if their corresponding SMOP {Pn(x)}n≥0and {Qn(x)}n≥0satisfy P[m,ν] n(x) + M X i=1 ai,nP[m,ν] n−i(x) = Q[k,ν] n(x) + N X i=1 bi,nQ[k,ν] n−i(x), n ≥0,(3.1) 7 where ai,n, bi,n ∈C,aM,n 6= 0 for n≥M,bN,n 6= 0 for n≥N, and ai,n =bi,n = 0 if i>n. In addition, (U,V) is said to be a (M, N)-Dν-coherent pair of order mif it is a (M, N)-Dν-coherent pair of order (m, 0). In the next theorems, we state the Dν-analogue results obtained in [7,8,14], and we generalize the results stated in [2,4,12,15] for ν=ω, and in [3,16] for ν=q, respectively. Moreover, we give a complete description of the Dν-semiclassical discrete orthogonal polynomials in the framework of (M, N)-Dν-coherence of order (m, k). Theorem 3.2. Let (U,V)be a (M, N)-Dν-coherent pair of order (m, k)given by (3.1)with m≥k. Let LM+N= [li,j]M+N−1 i,j=0 be the following squared matrix of order M+N li,j =   aj−i,j if 0≤i≤N−1and i≤j≤M+i, bj−i+N,j if N≤i≤M+N−1and i−N≤j≤i, 0otherwise, (3.2) with a0,j1=b0,j2= 1,0≤j1≤N−1,0≤j2≤M−1. If det(LM+N)6= 0, then there exist polynomials φM+k+n(x;ν)and ψN+m+n(x;ν), of degrees M+k+nand N+m+n, respectively, such that Dm−k ν∗[φM+k+n(x;ν)V] = ψN+m+n(x;ν)U, n ≥0,(3.3) and there exist polynomials ϕ(x;ν)and ρ(x;ν)such that ϕ(x;ν)U=ρ(x;ν)V.(3.4) Furthermore 1. If k=mthen Uis a Dν-semiclassical linear functional if and only if so is V. 2. If m > k, then Uand Vare both Dν-semiclassical linear functionals. Proof. From (3.1), let a0,n =b0,n = 1 and Rn(x;ν) = M X i=0 ai,nP[m,ν] n−i(x) = N X i=0 bi,nQ[k,ν] n−i(x), n ≥0.(3.5) Let us consider {pn}n≥0,{qn}n≥0,{rn,ν }n≥0,{en,ν }n≥0and {hn,ν }n≥0be the dual bases of the SMOP {Pn(x)}n≥0,{Qn(x)}n≥0and the sequences {Rn(x;ν)}n≥0,{P[m,ν] n(x)}n≥0and {Q[k,ν] n(x)}n≥0, respectively. From hen,ν , Rj(x;ν)i(3.5) = M X i=0 hen,ν , ai,jP[m,ν] j−i(x)i=aj−n,j if n≤j≤n+M, 0 otherwise, 8 hhn,ν , Rj(x;ν)i(3.5) = N X i=0 hhn,ν , bi,jQ[k,ν] j−i(x)i=bj−n,j if n≤j≤n+N, 0 otherwise, it follows that en,ν =X j≥0 hen,ν , Rj(x;ν)irj,ν = n+M X j=n aj−n,jrj,ν , n ≥0,(3.6) hn,ν =X j≥0 hhn,ν , Rj(x;ν)irj,ν = n+N X j=n bj−n,jrj,ν , n ≥0.(3.7) Using (3.6) and (3.7) for 0 ≤n≤N−1 and 0 ≤n≤M−1, respectively, we set LM+N           r0,ν . . . rN−1,ν rN,ν . . . rN+M−1,ν           =           e0,ν . . . eN−1,ν h0,ν . . . hM−1,ν           , where the matrix LM+Nis given by (3.2). By assumption det(LM+N)6= 0, then we can solve this linear system and obtain, for 0 ≤i≤M+N−1, ri,ν =αi,0e0,ν +· · · +αi,N−1eN−1,ν +αi,N h0,ν +· · · +αi,N+M−1hM−1,ν ,(3.8) where αi,j, 0 ≤j≤N+M−1, are some constants. If, for every i≥0, we multiply (3.6) for n=N+iby bN,M+N+i, and (3.7) for n=M+iby aM,M+N+i, and subtracting the resulting equations, we get bN,M+N+ieN+i,ν −aM,M+N+ihM+i,ν =β1,irmin{M,N}+i,ν +· · · +βmax{M,N},irM+N+i−1,ν , i ≥0,(3.9) where βj,i, 1 ≤j≤max{M, N},i≥0, are constants. Additionally, for t≥0 fixed, using (3.6) we can recursively obtain an expression for rM+N+t,ν as a linear combination of ri,ν , 0≤i≤M+N−1, and ej,ν ,N≤j≤N+t, (since aM,M+j6= 0, N≤j≤N+t). Hence, using (3.8), (3.9) becomes ˜αi,0e0,ν +· · · + ˜αi,N+i−1eN+i−1,ν +bN,M+N+ieN+i,ν =˜ βi,0h0,ν +· · · +˜ βi,M−1hM−1,ν +aM,M+N+ihM+i,ν , i ≥0, where ˜αi,j1,˜ βi,j2, for 0 ≤j1≤N+i−1, 0 ≤j2≤M−1, are constants. Applying the mth 9 Besides, for each n≥K, (i) if M > N and aM,n 6= 0, then cK,n,λ,ν 6= 0, (ii) if M < N and bN,n 6= 0, then cK,n,λ,ν 6= 0, (iii) if M=N(= K)and aM,nbN,n 6= 0 then, cK,n,λ,ν 6= 0 iff aK,nhU, P2 n−K+m(x)i+λη2 n−K,m,ν bK,nhV, Q2 n−K(x)i 6= 0. Conversely, if there exist constants {cj,n,λ,ν }n≥0,1≤j≤K, and {ai,n}n≥0,1≤i≤M, with cj,n,λ,ν = 0,n−j+m < 0, and ai,n = 0,n−i+m < 0, such that (4.8)holds, then (U,V)is a (M, K)-Dν-coherent pair of order mgiven by P[m,ν] n(x) + M X i=1 ai,nP[m,ν] n−i(x) = Qn(x) + K X j=1 bj,nQn−j(x), n ≥0,(4.10) (whenever bK,n 6= 0 for n≥K), where bj,n = 0 for n<j≤K, and for n≥0, bj,n =DV,P[m,ν] n(x) + PM i=1 ai,nP[m,ν] n−i(x)Qn−j(x)E hV, Q2 n−j(x)i,1≤j≤min{K, n}.(4.11) Proof. Sn(x;λ, ν) = Pn(x), n<m, follows from hPn(x), xiiλ,ν = 0, i<n<m. On the other hand, substituting (4.6) in (4.7), and then, computing Dν-antiderivatives mtimes (this is, a function F(x) is a Dν-antiderivative of a function f(x) if DνF(x) = f(x), [10]), we obtain for n≥0, Pn+m(x) ηn,m,ν + M X i=1 ai,nPn−i+m(x) ηn−i,m,ν =Tn+m(x;ν) ηn,m,ν + N X i=1 bi,nTn−i+m(x;ν) ηn−i,m,ν + m−1 X j=0 κn,jxj. Taking hxiU,· i,i < m, from (4.5), we get the linear system Pm−1 j=0 κn,juj+i= 0, i= 0, . . . , m −1. Since det [ui+j]m−1 i,j=06= 0, then κn,j = 0, j= 0, . . . , m −1, n≥0. Hence, for n≥0, Pn+m(x) ηn,m,ν + M X i=1 ai,n Pn−i+m(x) ηn−i,m,ν =Tn+m(x;ν) ηn,m,ν + N X i=1 bi,n Tn−i+m(x;ν) ηn−i,m,ν .(4.12) Furthermore, for n≥0, Tn+m(x;ν) ηn,m,ν + N X i=1 bi,n Tn−i+m(x;ν) ηn−i,m,ν =Sn+m(x;λ, ν) ηn,m,ν + n+m X j=1 cj,n,λ,ν ηn,m,ν Sn−j+m(x;λ, ν), 16 where from (4.1), (4.12) and (4.6), for 1 ≤j≤n+m, hSn−j+m(x;λ, ν), Sn−j+m(x;λ, ν)iλ,ν cj,n,λ,ν ηn,m,ν = M X i=1 ai,n ηn−i,m,ν U, Pn−i+m(x)Sn−j+m(x;λ, ν)+λ N X i=1 bi,nV, Qn−i(x)Dm ν[Sn−j+m(x;λ, ν)], then cj,n,λ,ν = 0 for j > i or j > K. Thus, (4.8) and (4.9) hold. Also, for n≥K, cK,n,λ,ν ηn,m,ν = aM,n ηn−M,m,ν hU, P2 n−M+m(x)iδM,K +ληn−N,m,ν bN,nhV, Q2 n−N(x)iδN,K hSn−K+m(x;λ, ν), Sn−K+m(x;λ, ν)iλ,ν , holds, and as a consequence, (i),(ii) and (iii) follow. Finally, P[m,ν] n(x) + M X i=1 ai,nP[m,ν] n−i(x) = Qn(x) + n X j=1 bj,nQn−j(x), n ≥0, with bj,n, for 1 ≤j≤n, given by (4.11). Applying h · , p(x)iλ,ν to both sides of (4.8), for p∈Pn−K+m−1, it follows that 0 = λ*V, Dm ν[Pn+m(x)] + M X i=1 ηn,m,ν ai,n ηn−i,m,ν Dm ν[Pn−i+m(x)]!Dm ν[p(x)]+, i.e., 0 = *V, P[m,ν] n(x) + M X i=1 ai,nP[m,ν] n−i(x)!r(x)+,∀r∈Pn−K−1, thus bj,n = 0, for n−j≤n−(K+ 1), which proves (4.10). Theorem 4.4. Let (U,V)be a (M, N)-Dν-coherent pair of order mgiven by (4.7),K= max{M, N}, and for n≥0, sn,ν =hSn(x;λ, ν), Sn(x;λ, ν)iλ,ν ,eai,n =ηn,m,ν ηn−i,m,ν ai,n,ebi,n =ηn,m,ν bi,n, with ai,n =bi,n = 0 if i>n, and, a0,n =b0,n = 1 for n≥0. Then sn+m,ν cj,n+j,λ,ν =ζj,n,λ,ν − K−j X `=1 c`,n,λ,ν cj+`,n+j,λ,ν sn−`+m,ν ,0≤j≤K, n ≥0,(4.13) with sn,ν =hU, P2 n(x)ifor n<m,c0,n,λ,ν = 1 for n≥0,cj,n,λ,ν = 0 for n<j≤K, and 17 for 0≤j≤K, ζj,n,λ,ν = M X i=jeai,n+jeai−j,nhU, P 2 n+j−i+m(x)i+λ N X i=jebi,n+jebi−j,nhV, Q2 n+j−i(x)i. Proof. Notice that (4.8) and (4.9) hold setting c0,n,λ,ν = 1 for n≥0. Then, from (4.7) and (4.8), (4.9) becomes, for n≥jand 0 ≤j≤K, sn−j+m,ν cj,n,λ,ν = M X i=j M X `=0 eai,nea`,n−jhU, Pn−i+m(x)Pn−j−`+m(x)i − M X i=j K X `=1 eai,nc`,n−j,λ,ν hU, Pn−i+m(x)Sn−j−`+m(x;λ, ν)i +λ N X i=j N X `=0 ebi,neb`,n−jhV, Qn−i(x)Qn−j−`(x)i −λ N X i=j K X `=1 ebi,nc`,n−j,λ,ν hV, Qn−i(x)Dm ν[Sn−j−`+m(x;λ, ν)]i. Since hU, Pn−i+m(x)Sn−j−`+m(x;λ, ν)i= 0 and hV, Qn−i(x)Dm ν[Sn−j−`+m(x;λ, ν)]i= 0, for i < j +`or j+` > K (≥M, N), thus, for n≥jand 0 ≤j≤K, sn−j+m,ν cj,n,λ,ν = M X i=jeai,neai−j,n−jhU, P 2 n−i+m(x)i+λ N X i=jebi,nebi−j,n−jhV, Q2 n−i(x)i − K−j X `=1 c`,n−j,λ,ν M X i=j+`eai,nhU, Pn−i+m(x)Sn−j−`+m(x;λ, ν)i −λ K−j X `=1 c`,n−j,λ,ν N X i=j+`ebi,nhV, Qn−i(x)Dm ν[Sn−j−`+m(x;λ, ν)]i, and from (4.9), the sum of the last two terms is −PK−j `=1 c`,n−j,λ,ν sn−j−`+m,ν cj+`,n,λ,ν . Finally, substituting nby n+j, we get (4.13). Remark 4.5. Notice that Theorem 4.3 allows recursively compute the Dν-Sobolev SMOP {Sn(x;λ, ν)}n≥0and the coefficients {cj,n,λ,ν }n≥0, 1 ≤j≤K. Moreover, Theorem 4.4 gives a recursive equation for computing the sequences {cj,n,λ,ν }n≥0, 1 ≤j≤K, and {hSn(x;λ, ν), Sn(x;λ, ν)iλ,ν }n≥0, and thus, using (4.8) and Sn(x;λ, ν) = Pn(x) for n<m, we can get the Dν-Sobolev SMOP {Sn(x;λ, ν)}n≥0. 18 Acknowledgements We are grateful to Prof. Francisco Marcell´an for his valuable comments and remarks that helped us to improve the paper. This work was supported by Direcci´on General de Investigaci´on, Desarrollo e Innovaci´on, Ministerio de Econom´ıa y Competitividad of Spain, under grants MTM2012-36732-C03 (RAN, NCP-C, JP), Junta de Andaluc´ıa (Spain) under grants FQM262, FQM-7276, and P09-FQM-4643 (RAN), FEDER funds (RAN). The work of J. 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