Rodrigues’ formulas for orthogonal matrix polynomials satisfying second-order difference equations

Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Taylor & Francis in : Integral Transforms and Special Functions, Vol. 25, n. 11, on 2014, available at: https://doi.org/10.1080/10652469.2014.928819 RODRIGUES’ FORMULAS FOR ORTHOGONAL MATRIX POLYNOMIALS SATISFYING SECOND-ORDER DIFFERENCE EQUATIONS ANTONIO J. DUR´ AN AND VANESA S´ ANCHEZ-CANALES Abstract. We develop a method to find discrete Rodrigues’ formulas for orthogonal matrix polynomials which are also eigenfunctions of a second-order difference operator. Using it, we produce Rodrigues’ formulas for two illustrative examples of arbitrary size. 1. Introduction and results It is well known that the orthogonal polynomials of Charlier, Meixner, Krawtchouk and Hahn have a number of very interesting extra properties. Among those properties are the following two: (1) each one of these classical discrete families are eigenvalues of a second order difference operator of the form f−1(x)s−1+f0(x)s0+f1(x)s1, where sldenotes the shift operator sl(f) = f(x+l) and fi,i=−1,0,1, are polynomials of degree not larger than 2 (independent of n) satisfying that deg(P1 l=−1lkfl)≤k, k = 0,1,2; (2) they can be obtained using a discrete Rodrigues’ formula: pn(x) = ∆n w(x) n−1 Y m=0 f−1(x−m)!.w(x),(1.1) where ∆ is the first order difference operator ∆(f) = f(x+ 1) −f(x) and wis the corresponding classical discrete weight. Each one of these characterizations is the result of a different effort, and they are usually associated to the names of O. Lancaster and W. Hahn, respectively. Actually these properties can be seen to follow from the so called Pearson discrete equation for the weight w:f−1(x)w(x) = f1(x−1)w(x−1), which also characterizes 2020 Mathematics Subject Classification. 33E30, 42C05, 47B39. Key words and phrases. Matrix orthogonal polynomials. Difference operators and equations. Charlier polynomials. Meixner polynomials. Partially supported by MTM2012-36732-C03-03 (Ministerio de Econom´ıa y Competitividad), FQM-262, FQM-4643, FQM-7276 (Junta de Andaluc´ıa) and Feder Funds (European Union). 1 2 ANTONIO J. DUR´ AN AND VANESA S´ ANCHEZ-CANALES the four classical discrete weights of Charlier, Meixner, Krawtchouk and Hahn. (For a historical account of this and other related subjects, see for instance [1, 2]). The theory of orthogonal matrix polynomials starts with two papers by M.G. Krein in 1949, [3], [4]. Each sequence of orthogonal matrix polynomials (Pn)nis associated to a weight matrix Wand satisfies that Pn,n≥0, is a matrix polynomial of degree nwith non singular leading coefficient and RPndWP∗ m= Γnδn,m, where Γn,n≥0, is a positive definite matrix. When Γn=I, we say that the polynomials (Pn)nare orthonormal. However, the first examples of orthogonal matrix polynomials (Pn)nwhich are eigenfunctions of a second order difference operator of the form L(·) = s−1(·)F−1(x) + s0(·)F0(x) + s1(·)F1(x)(1.2) (with left eigenvalues) appeared in 2012 (see [5, 6, 7]). Here F−1,F0and F1are matrix polynomials satisfying deg(P1 l=−1lkFl)≤k, k = 0,1,2. These examples have been essentially found by solving an appropriate set of commuting and difference equations. This set includes a matrix analogous to the Pearson equation of the scalar case and is the following one F0W=WF∗ 0,(1.3) F1(x−1)W(x−1) = W(x)F∗ −1(x).(1.4) Under certain boundary conditions, these equations imply that the orthonormal polynomials with respect to Ware eigenfunctions of the second order difference operator (1.2) with Hermitian (left) eigenvalues Λn; that is, L(Pn)=ΛnPn,n≥0. The families of orthogonal matrix polynomials found using these methods are among those that are likely to play in the case of matrix orthogonality the role of the classical discrete families of Charlier, Meixner, Krawtchouk and Hahn in the case of scalar orthogonality. This paper is devoted to the question of the existence of Rodrigues’ formulas for these families of orthogonal matrix polynomials, that is, assuming that Wsatisfies the commuting and difference equations (1.3) and (1.4), is there any efficient and canonical way to produce a sequence of orthogonal matrix polynomials with respect to W? Say in an analogous way as to (1.1) produces the orthogonal polynomials with respect to a classical discrete weight w. Even if F−1and Wcommute in the strong form that F−1(x)W(y) = W(y)F−1(x), x, y in the support of W, orthogonal matrix polynomials which are eigenfunctions of a difference operator like (1.2) do not seem to satisfy, in general, a scalar-type Rodrigues’ formula of the form (1.5) Pn(x) = Cn∆n W(x) n−1 Y m=0 F−1(x−m)!W−1(x), n ≥0, where Cn,n≥0, are nonsingular matrices. (In this paper we consider discrete weight matrices of the form W=P∞ x=0 W(x)δx, but implicitly assume that the function W(x) is an entire function in the whole complex plane so that the right hand side of (1.5) makes sense for x∈C). DISCRETE RODRIGUES’ FORMULAS 3 Instead of (1.5), these orthogonal matrix polynomials seem to satisfy some modified Rodrigues’ formula. The first instance of that modified Rodrigues’ formula appeared in [5]: the expression Pn(x)=∆nax Γ(x−n+ 1) 1 + ab2n+b2x2bx bx 1W−1(x) defines a sequence of orthogonal matrix polynomials with respect to the discrete weight matrix (1.6) W= ∞ X x=0 ax Γ(x+ 1) 1 + b2x2bx bx 1δx. Along this paper, we take 1 Γ(y+1) = 0 if yis a negative integer, so that the function 1/Γ(x) is entire. Similar Rodrigues’ formulas of the form (1.7) Pn(x)=∆n(ξn(x))W−1(x), have been found for other families of orthogonal polynomials of size 2 ×2 (see [5]). In all these examples, the functions ξnare simple enough as to make the Rodrigues’ formula (1.7) useful for the explicit calculation of the sequence of orthogonal polynomials Pnwith respect to W. Under mild conditions on the functions ξnin (1.7), one can easily prove that Pn is orthogonal with respect to any polynomial of degree less than n(by performing a sum by parts). The difficulty to find Rodrigues’ formulas like (1.7) is that, in general, it is rather involved to guarantee that Pndefined by (1.7) is a polynomial of degree nwith nonsingular leading coefficient. In the extant examples, this requirement on Pnhas been checked by a direct computation. As a consequence of this, only examples of small size have been worked out (actually size 2 ×2). The purpose of this paper is to develop a method to explore the existence of Rodrigues’ formulas of the form (1.7) for orthogonal matrix polynomials of arbitrary size (for Rodrigues’ formulas for orthogonal matrix polynomials which are eigenfunctions of second order differential operators see [8]). The key of this method is to exploit the set of commuting and difference equations (1.3) and (1.4) for the weight matrix W, and use the following lemma as the main tool: Lemma 1.1. Let F1,F0and F−1be matrix polynomials satisfying that (1.8) deg 1 X l=−1 lkFl!≤k, k = 0,1,2. Let Wand Rnbe N×Nmatrix functions defined in a discrete set Ω = {0,1,2, . . . , τ}, where τcan be a positive integer or infinity. Assume that Wis nonsingular for x∈Ωand satisfies the equations (1.9) F0W=WF∗ 0, F1(x−1)W(x−1) = W(x)F∗ −1(x). Define the functions Pn,n≥1, by (1.10) Pn= ∆n(Rn)W−1. 4 ANTONIO J. DUR´ AN AND VANESA S´ ANCHEZ-CANALES If for a matrix Λn, the function Rnsatisfies s−1(RnF∗ 1) + s1Rnn+ 1 2∆2F∗ 1−n∆F∗ −1+F∗ −1 +Rn−n∆2F∗ 1+n∆F∗ 1+F∗ 0= ΛnRn (1.11) for x∈Ω, then the function Pnsatisfies (1.12) s−1(Pn)F−1+s0(Pn)F0+s1(Pn)F1= ΛnPn, x ∈Ω. Once we have a weight matrix Wsatisfying the equations (1.3) and (1.4), and an appropriate choice of eigenvalues Λn,n≥1, our method consists in using a solution of the equation (1.11) to produce a sequence of orthogonal matrix polynomials with respect to Wby means of the Rodrigues’ formula (1.10). According to the previous lemma, the function Pngiven by (1.10) satisfies the difference equation (1.12). It turns out that, in general, these difference equations are not enough to guarantee that Pn,n≥1, is a polynomial of degree nwith nonsingular leading coefficient, because the eigenvalues Λn,n≥1, have, in general, non disjoint spectrum. However, the method seems to work very efficiently as long as the weight matrix Wsatisfies the equations (1.3) and (1.4) for a couple of linearly independent sets of coefficients F−1,1, F0,1, F1,1and F−1,2, F0,2, F1,2. In this case, one can do a suitable choice of the eigenvalues Λn,1and Λn,2and real constants a1, a2such that the spectrum of the linear combination a1Λn,1+a2Λn,2is disjoint for n≥1, from where one can deduce that Pnis a polynomial of degree nwith nonsingular leading coefficient. Moreover, the functions Rn,n≥1, provided by this method have a so surprisingly simple expression that it suggests the existence of a certain hidden pattern. The existence of orthogonal matrix polynomials being eigenfunctions of several linearly independent second order difference operators as (1.2) is a new phenomenon of the matrix orthogonality (see [5]; for differential operators see [9], [10], [11], [12], [13], and the references therein). In the scalar case, the classical discrete families of Charlier, Meixner, Krawtchouk and Hahn are eigenfunctions of only one second order difference operator, up to multiplicative constants. Using our method we have found Rodrigues’ formulas for the following two illustrative examples. The first example is the weight matrix (1.13) W1=X x≥0 ax Γ(x+ 1)(I+A)x(I+A∗)xδx, where Ais the N×Nnilpotent matrix (1.14) A= N−1 X i=1 viEi,i+1, a > 0, and vi,i= 1,· · · , N −1, are complex numbers satisfying that (1.15) (N−i−1)a|vi|2|vN−1|2−i(N−i)|vN−1|2+ (N−1)|vi|2= 0. The matrix Ei,j stands for the matrix with entry (i, j) equal to 1 and 0 otherwise. For i=N−1, the condition (1.15) is an identity. For N= 2, this example DISCRETE RODRIGUES’ FORMULAS 5 depends on just one parameter v1and gives the weight matrix (1.6). For N= 2, the condition (1.15) always fulfils. Let us note that since Ais nilpotent of order Nthen (I+A)x(I+A∗)xis a matrix polynomial of degree 2N−2. It was proved in [6] and [7] that this weight matrix satisfies equations (1.3) and (1.4) for a couple of sets of linearly independent coefficients F−1,j, F0,j, F1,j ,j= 1,2 (see (4.1) and (4.3) below). The Rodrigues’ formula provided by our method for this example is the following: Theorem 1.2. Assume that the moduli of the entries |vi|,i= 1,· · · , N −1, of the matrix A(1.14) satisfy (1.15). Then, a sequence of orthogonal polynomials with respect to the weight matrix W1(1.13) can be defined by using the Rodrigues’ formula Pn(x)=∆nax Γ(x−n+ 1)(I+A)xLn,1(I+A∗)xW−1 1(x), n ≥1,(1.16) where Ln,1is the diagonal matrix independent of xand with entries Ln,1= N X i=1 N−1 Y k=i1 + an|vk|2 k(N−k)Eii (1.17) (for i>jwe take Qj l=i= 1). The assumption (1.15) on the parameters seems to be necessary. We have symbolic computational evidence which shows that if (1.15) does not hold then, for any choice of the matrix Ln,1(diagonal or not), the polynomial Pnin (1.16) has degree bigger than nor singular leading coefficient. The second example is the weight matrix W2=X x≥0 ax Γ(x+ 1)(I+A)xΓ ((x+c)I+J) (I+A∗)xδx,(1.18) where Ais defined by (1.14), 0 <a<1, c > 0, Jis the diagonal matrix (1.19) J= N X i=1 (N−i)Ei,i and the complex numbers vi,i= 1,· · · , N −1, satisfy that (N−i−1)a|vi|2|vN−1|2 1−a−i(N−i)|vN−1|2+ (N−1)|vi|2= 0.(1.20) The properties of the matrices A(nilpotent of order N) and J(diagonal) imply that (I+A)xΓ ((x+c)I+J) (I+A∗)x/Γ(x+c) is a matrix polynomial of degree 2N−2. As far as we know, this is the first time this weight matrix appears in the literature. We show that this weight matrix satisfies equations (1.3) and (1.4) for a couple of sets of linearly independent coefficients F−1,j ,F0,j ,F1,j,j= 1,2 (see (4.2) and (4.2) below). The Rodrigues’ formula provided by our method for this example is the following: 6 ANTONIO J. DUR´ AN AND VANESA S´ ANCHEZ-CANALES Theorem 1.3. Assume that the moduli of the entries |vi|,i= 1,· · · , N −1, of the matrix A(1.14) satisfy (1.20). Then, a sequence of orthogonal polynomials with respect to the weight matrix W2(1.18) can be defined by using the Rodrigues’ formula Pn(x)=∆nax Γ(x−n+ 1)(I+A)xΓ ((x+c)I+J)Ln,2(I+A∗)xW−1 2(x), n ≥1, where Ln,2is the diagonal matrix independent of xand with entries Ln,2= N X i=1 N−1 Y k=i1 + an|vk|2 k(N−k)(1 −a)Eii.(1.21) 2. Preliminaries A weight matrix Wis an N×Nmatrix of measures supported in the real line satisfying that (1) W(A) is positive semidefinite for any Borel set A∈R, (2) W has finite moments of every order, and (3) RP(t)dW(t)P∗(t) is nonsingular if the leading coefficient of the matrix polynomial Pis nonsingular. All the examples considered in this paper are discrete weight matrices of the form (2.1) W=X x∈N W(x)δx. For a discrete weight matrix W=Px∈NW(x)δxsupported in a countable set Xof real numbers, the Hermitian sesquilinear form defined by ⟨P, Q⟩=RP dW Q∗takes the form ⟨P, Q⟩=X x P(x)W(x)Q∗(x). If Wdoes not satisfy condition (3) above, we will say that Wis degenerate. That happens, for instance, if Wis supported in finitely many points (as is the case of the discrete classical families of Krawtchouk and Hahn). That condition (3) is necessary and sufficient to guarantee the existence of a sequence (Pn)nof matrix polynomials orthogonal with respect to W,Pnof degree nwith nonsingular leading coefficient. For a discrete weight matrix as (2.1) Condition (3) is fulfilled, in particular, when W(x) is positive definite for infinitely many x∈N. We then say that a sequence of matrix polynomials (Pn)n,Pnof degree nwith nonsingular leading coefficient, is orthogonal with respect to Wif ⟨Pn, Pk⟩= Λnδk,n, where Λnis a nonsingular matrix for n≥0. Since each orthogonal polynomial Pn has degree nwith nonsingular leading coefficient, any matrix polynomial of degree less than or equal to ncan be expressed as a linear combination of Pk, 0 ≤k≤n, with matrix coefficients (multiplying on the left or on the right). That property, together with the orthogonality, defines the sequence of orthogonal polynomials uniquely from Wup to multiplication on the left by a sequence of nonsingular matrices (multiplication by unitary matrices for the orthonormal polynomials). DISCRETE RODRIGUES’ FORMULAS 7 Along this paper, we will use without an explicit mention the usual properties listed below of the first order difference operator ∆ defined by ∆(p) = p(x+ 1) −p(x), ∆ [f(x)g(x)] = f(x)∆g(x) + (∆f(x)) g(x+ 1),(2.2) b−1 X a f(x)∆g(x) = f(x)g(x)b a− b−1 X a (∆f(x)) g(x+ 1),(2.3) ∆nf(x) = n X k=0 (−1)kn kf(x+n−k).(2.4) Given a discrete weight matrix W= b X x=a W(x)δx, supported in {a, a + 1,· · · , b −1, b}(acan be −∞ and bcan be +∞), the matrices of measures ∆Wand ∇Ware defined in the usual way by ∆W= b X x=a−1 (W(x+ 1) −W(x))δx,∇W= b+1 X x=a (W(x)−W(x−1))δx, respectively, where by definition W(b+ 1) = W(a−1) = 0 (if a=−∞ or b=∞, we take a−1 = −∞ or b+ 1 = +∞, respectively). It is worth to note that the support of ∆Wis {a−1, a, a + 1,· · · , b −1, b}and that it is different to the support of W, except when a=−∞. In the same way, the support of ∇Wis {a, a + 1,· · · , b −1, b, b + 1}and it is different to the support of W, except when b= +∞. 3. Proof of Lemma 1.1 The proof is a matter of computation. We start denoting En= ∆nRnto simplify the notation, so Pn=EnW−1. From (1.9) we deduce W−1(x)F0(x) = F∗ 0(x)W−1(x), W−1(x+ 1)F1(x) = F∗ −1(x+ 1)W−1(x), W−1(x−1)F−1(x) = F∗ 1(x−1)W−1(x). (3.1) Then, the equation (3.2) s−1(Pn)F−1+s0(Pn)F0+s1(Pn)F1= ΛnPn is equivalent to En(x−1)W−1(x−1)F−1(x) + En(x)W−1(x)F0(x) + En(x+ 1)W−1(x+ 1)F1(x) = ΛnEn(x)W−1(x). Applying (3.1) and multiplying by W(x) on the right, we have (3.3) En(x−1)F∗ 1(x−1) + En(x)F∗ 0(x) + En(x+ 1)F∗ −1(x+ 1) = ΛnEn(x). 8 ANTONIO J. DUR´ AN AND VANESA S´ ANCHEZ-CANALES Denote now (3.4) H2=F1, H1=F−1−F1, H0=F1+F0+F−1. The assumptions (1.8) imply that (3.5) deg(Hi)≤i, i = 0,1,2. Using (3.4) we obtain that equation (3.3) is equivalent to ∆∇(EnH∗ 2) + ∆(EnH∗ 1) + EnH∗ 0= ΛnEn. Using (3.5) and the wellknown identity ∆n(FG) = n X k=0 n k∆kF(x+n−k)∆n−kG(x), we get EnH∗ 0= (∆nRn)H∗ 0= ∆n(RnH∗ 0), EnH∗ 1= (∆nRn)H∗ 1= ∆n(Rn(H∗ 1−n∆H∗ 1)) + ∆n−1(−nRn∆H∗ 1), EnH∗ 2= (∆nRn)H∗ 2= ∆nRnn+ 1 2∆2H∗ 2−n∆H∗ 2+H∗ 2 + ∆n−1Rnn2∆2H∗ 2−n∆H∗ 2+ ∆n−2Rnn 2∆2H∗ 2. So, equation (3.2) is equivalent to ∇∆n+1 Rnn+ 1 2∆2H∗ 2−n∆H∗ 2+H∗ 2+∇∆nRnn2∆2H∗ 2−n∆H∗ 2 +∇∆n−1Rnn 2∆2H∗ 2+ ∆n+1 (Rn(H∗ 1−n∆H∗ 1)) + ∆n(Rn(H∗ 0−n∆H∗ 1)) =Λn∆nRn. this can be rewritten as ∆n∆∇(RnH∗ 2)+∆Rnn+ 1 2∆2H∗ 2−n∆H∗ 2−n∆H∗ 1+H∗ 1 +Rnn 2∆2H∗ 2−n∆H∗ 1+H∗ 0−ΛnRn= 0. Replacing the coefficients H0,H1and H2by F−1,F0and F1according with (3.4), we can express this equation in terms of the shift operators ∆ns−1(RnF∗ 1) + s1Rnn+ 1 2∆2F∗ 1−n∆F∗ −1+F∗ −1 +Rn−n∆2F∗ 1+n∆F∗ 1+F∗ 0−ΛnRn= 0. (3.6) Equation (3.6) obviously holds if s−1(RnF∗ 1) + s1Rnn+ 1 2∆2F∗ 1−n∆F∗ −1+F∗ −1 +Rn−n∆2F∗ 1+n∆F∗ 1+F∗ 0−ΛnRn= 0. DISCRETE RODRIGUES’ FORMULAS 15 We have found a couple of linearly independent sets of coefficients F−1,1,F0,1,F1,1 and F−1,2,F0,2,F1,2such that the orthogonal polynomials with respect to W2are eigenfunctions of the associated second order difference operator. Lemma 4.2. The orthogonal matrix polynomials with respect to W2(4.24) are eigenfunctions of the second order difference operator D(·) = s−1(·)F−1+s0(·)F0+s1(·)F1 for these two sets of coefficients F−1,1= (I+A)−1x, F1,1=aIx+a(I+A)(cI+J), F0,1=−J−aI + (I+A)−1x, and F−1,2=[I−(I+A)−1]x2+(N−1)(a−1) a|vN−1|2I−Jx, F1,2=[I−(I+A)−1]x2+a(I+A)(N−1)(a−1) a|vN−1|2I−J(cI +J+A∗) +(a−1) N−1 |vN−1|2−N+aI+ (a−2)J+aA(cI +J)−(I+A)−1x, F0,2=−F−1,2−F1,2. Proof. We only sketch the proof. For i= 1, we can proceed as follows. It is enough to prove that the operator Dis symmetric with respect to the weight matrix W2, and then use Lemma 1.1 in [6] to deduce that the orthogonal polynomials with respect to W2are eigenfunctions of D. Using Theorem 2.1 of [6], in order to prove the symmetry of Dwith respect to W2, it is enough to prove that s−1(F1,1W2) = W2F∗ −1,1, F0,1W2=W2F∗ 0,1, W2(0)F∗ −1,1(0) = 0. This can be proved by a careful computation using (4.10). For i= 2, we can proceed as follows. We write G2=F−1,2and G1=F1,2−F−1,2. Using Theorem 2 of [7], it is enough to prove that G2W2=W2G∗ 2,∆(G2W2) = G1W2. This can be proved in a similar way as Theorem 3 of [7] but using here the identities G2(x)(I+A)x= (I+A)xe J2x, G1(x)(I+A)x= (I+A)xa(I+A)e J2(cI +J+A∗)+(a−1)xe J2+axA e J2, where e J2=(N−1)(a−1) a|vN−1|2I−J, instead of the identities (5.8) and (5.9) (used in [7]). □ To produce the Rodrigues’ formula for this example, we consider Rn,2(x) = ax Γ(x−n+ 1)(I+A)xΓ ((x+c)I+J)Ln,2(I+A∗)x, where the diagonal matrix Ln,2is given by (1.21). 16 ANTONIO J. DUR´ AN AND VANESA S´ ANCHEZ-CANALES Theorem 1.3 states that an n-th orthogonal polynomial with respect to W2is given by the formula Pn= ∆n(Rn,2)W−1 2. 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