Zeros of Jacobi-Sobolev orthogonal polynomials following non-coherent pair of measures

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1 Volume 29, N. 3, pp , 2010 Copyright 2010 SBMAC ISSN Zeros of Jacobi-Sobolev orthogoal polyomials followig o-coheret pair of measures ELIANA X.L. DE ANDRADE 1, CLEONICE F. BRACCIALI 1, MIRELA V. DE MELLO 1 ad TERESA E. PÉREZ 2 1 DCCE, IBILCE, UNESP Uiversidade Estadual Paulista São José do Rio Preto, SP, Brazil 2 Departameto de Matemática Aplicada ad Istituto Carlos I de Física Teórica y Computacioal Uiversidad de Graada, Graada, Spai s: eliaa@ibilce.uesp.br / cleoice@ibilce.uesp.br / mirela_vaia@yahoo.com.br / tperez@ugr.es Abstract. Zeros of orthogoal polyomials associated with two differet Sobolev ier products ivolvig the Jacobi measure are studied. I particular, each of these Sobolev ier products ivolves a pair of closely related Jacobi measures. The measures of the ier products cosidered are beyod the cocept of coheret pairs of measures. Existece, real character, locatio ad iterlacig properties for the zeros of these Jacobi-Sobolev orthogoal polyomials are deduced. Mathematical subject classificatio: 33C45, 33C47, 26C10. Key words: Sobolev orthogoal polyomials, Jacobi orthogoal polyomials, Zeros of orthogoal polyomials. 1 Itroductio Cosider the ier product f, g S = k j=0 R f j) x) g j) x)dψ j x), k 1, 1) #CAM-117/09. Received: 04/VII/09. Accepted: 14/XII/09.

2 424 ZEROS OF JACOBI-SOBOLEV ORTHOGONAL POLYNOMIALS where dψ j, for j = 0, 1,..., k, are positive measures supported o R. This ier product is kow as Sobolev ier product ad the associated sequece of moic orthogoal polyomials, {S } =0, is kow as a sequece of moic Sobolev orthogoal polyomials. This kid of ier product is o-stadard i the sese that the shift operator, i.e., the multiplicatio operator by x, is ot self-adjoit x f, g S = f, x g S, where f ad g are polyomials with real coefficiets. Therefore, some of the usual properties of stadard orthogoal polyomials are ot true. I fact, the usual three term recurrece relatio ad the properties about the zeros real ad simple characters, iterlacig, etc.) are o loger valid. I 1991, A. Iserles et al. [9] studied Sobolev ier products as 1) for k = 1 whe the two measures dψ 0 ad dψ 1 are related. If we deote by {P ψ i } =0 i = 0, 1) the sequece of moic orthogoal polyomials with respect to the stadard ier product f, g ψi = f x) gx) dψ i x), i = 0, 1, the {dψ 0, dψ 1 } is a coheret pair of measures if R P ψ 1 x) = 1 [ P ψ x) + σ P ψ 0 ] x), for 1, where σ are o-zero costats. As a cosequece see [9]), the sequece of moic Sobolev orthogoal polyomials {S } =0 associated with the Sobolev ier product f, g S = f, g ψ0 + f, g ψ1 2) satisfies S +1 x) + a S x) = P ψ 0 +1 x) + σ P ψ 0 x), 1. 3) H.G. Meijer [13] has show that if {dψ 0, dψ 1 } is a coheret pair of measures, the both measures are closely related ad at least oe of them must be classical. Sobolev orthogoal polyomials associated with coheret pairs have bee exhaustively studied. Algebraic ad differetial properties, as well as properties

3 E.X.L. ANDRADE, C.F. BRACCIALI, M.V. MELLO ad T.E. PÉREZ 425 about zeros, have bee ivestigated. I particular, i [6], [8], ad [11], several results about existece, locatio ad iterlacig properties of the zeros of orthogoal polyomials with respect to Gegebauer-Sobolev ad Hermite-Sobolev ier products are show. Moreover, i [14], the authors cosidered special Jacobi-Sobolev ad Laguerre-Sobolev ier products, where the pair of measures forms a coheret pair ad they proved iterlacig properties of the zeros of Sobolev orthogoal polyomials. I [2] ad [3] it has bee itroduced a alterative approach to study Sobolev ier products such that the correspodig orthogoal polyomials still satisfy a relatio of the form 3). This kid of Sobolev ier products geeralizes Sobolev ier products defied from a coheret pair of measures ad it allows to exted the results about Sobolev orthogoal polyomials beyod the cocept of coheret pairs. I [7] the authors have cosidered the iverse problem: startig from the relatio 3) to obtai a pair of quasi-defiite momet fuctioals such that 3) holds. Their results show that the pair of measures ivolved beig coheret is ot ecessary for 3) to hold. I the preset paper, properties for the zeros of Sobolev orthogoal polyomials associated with ier products of the form 2) have bee ivestigated. The measures of the ier products ivolve Jacobi measures ad they are beyod the cocept of coheret pairs of measures. For α, β >, let dψ α,β) deotes the classical Jacobi measure o [, 1] give by dψ α,β) x) = 1 x) α 1 + x) β dx, {P α,β) } =0 be the sequece of classical moic Jacobi orthogoal polyomials ad ρ α,β) = P α,β), P α,β) ψ α,β). It is well kow that the zeros of P α,β), 1). We deote the zeros of P α,β) by p α,β) are all real, distict ad lie iside, i = 1, 2,...,, i icreasig order. For more details about these polyomials see, for istace, [5] ad [15]. I this paper we cosider two Sobolev ier products give i [3], amely Type I. For κ 1 1, κ 2 κ 3, dψ 0 = 1 + κ 1 x) dψ α,β) x), dψ 1 = κ 2 + κ 3 x) dψ α+1,β+1) x).

4 426 ZEROS OF JACOBI-SOBOLEV ORTHOGONAL POLYNOMIALS Type II. For κ 3 1, κ 2, κ 4 0, κ 1 κ 3 κ 2 /1 + κ 3 ), dψ 0 = dψ α,β) x), dψ 1 = κ 1 + κ ) 2 κ 3 dψ α+1,β+1) x) + κ 2 κ 4 δκ 3 ). κ 3 x Whe κ 1 = 0, κ 2 > 0 ad κ 3 = 0, the measures ivolved i type I form a coheret pair see [13]). I [14], the authors have proved that the zeros of the correspodig Sobolev orthogoal polyomials are real ad simple, ad they have iterlacig properties. Whe κ 3 = 0, the pair of measures o loger forms a coheret pair ad, i this case, we will show i Sectio 2 that, uder some coditios, the iterlacig properties established i [14] for κ 3 = 0 still hold eve whe κ 3 = 0. Whe κ 1 = 0 the pair of measures of type II is also a coheret pair of measures see [13]). I this case, properties about zeros for the correspodig Sobolev orthogoal appear i [14]. I [10] the authors have studied the special case whe κ 1 = κ 4 = 0, κ 2 > 0 ad κ 3 =. Whe κ 1 = 0, the pair of measures of type II o loger forms a coheret pair. Sectio 3 is devoted to show iterlacig properties for the zeros of Jacobi-Sobolev orthogoal polyomials of type II whe κ 1 = 0. Ideed, our results geeralize the iterlacig properties established i [14] for the particular case κ 1 = 0. I [1] it has bee studied properties of the zeros of orthogoal polyomials with respect to Gegebauer-Sobolev ier product where the associated pair of measures does ot form a symmetrically coheret pair. 2 Jacobi-Sobolev ier product of type I Let us cosider the followig modificatio of the Jacobi weight dψ α,β,κ 1) x) = 1 + κ 1 x)dψ α,β) x) = 1 + κ 1 x)1 x) α 1 + x) β dx, defied o [, 1], where κ 1 1. We deote by {P α,β,κ 1) } =0 of moic orthogoal polyomials associated with dψ α,β,κ1), ad ρ α,β,κ 1) = P α,β,κ 1 ) x) ) κ1 x)1 x) α 1 + x) β dx. the sequece

5 E.X.L. ANDRADE, C.F. BRACCIALI, M.V. MELLO ad T.E. PÉREZ 427 The followig result is kow see, for istace, [3]) where P α,β) x) = P α,β,κ 1) x) + d P α,β,κ 1) x), 1, Observe that sgd ) = sgκ 1 ) where, as usual, d = κ 1 ρ α,β), 1. 4) ρ α,β,κ 1) sga) = For 1, the zeros of P α,β,κ 1) { a /a, if a = 0, 0, if a = 0., 1). If we deote their zeros by p α,β,κ 1) are real, simple ad all lie i the iterval, i = 1, 2,...,, i icreasig order, the they iterlace with the zeros of the classical Jacobi polyomials ad their positio depeds o the sig of κ 1 see [4]): for κ 1 < 0 ad 1 i 1, < p α,β,κ 1) ad for 0 < κ 1 1 ad 1 i 1, < p α,β) < p α,β,κ 1),i < p α,β,κ 1) +1 < p α,β) +1 < 1, 5) < p α,β) < p α,β,κ 1) < p α,β,κ 1),i < p α,β) +1 < pα,β,κ 1) +1 < 1. 6) Cosider the followig Sobolev ier product, itroduced i [3], f, g J S1 = f x) gx)1 + κ 1 x)dψ α,β) x) 7) + f x) g x)κ 2 + κ 3 x)dψ α+1,β+1) x), with κ 1 1 adκ 2 κ 3. The, J S1 is positive defiite ad we will refer to it as Jacobi-Sobolev ier product of type I. Let { } =0 deote the sequece of moic orthogoal polyomials with respect to 7), we will refer to it as sequece of Jacobi-Sobolev orthogoal polyomials of type I. I additio, we deote ρ J 1 =, J S 1.

6 428 ZEROS OF JACOBI-SOBOLEV ORTHOGONAL POLYNOMIALS Furthermore, 0 x) = 1 ad for 0 where a J 1 +1 x) + a J 1 x) = Pα,β) +1 x) = Pα,β,κ 1) +1 x) + d P α,β,κ 1) x), 8) is give by the expressio [3]) a J 1 = κ α + β + 2)κ 3 )ρ α,β) +1 ρ J. 1 Observe that, if κ 1 = 0, the sga J 1 ) = sgκ 3). Otherwise, if sgκ 1 ) = sgκ 3 ), the sga J 1 ) = sgκ 1) = sgκ 3 ). 2.1 Zeros of For, i 0 we defie ad ad Pα,β) m J 1 i, = ˆm J 1 i, = x)[ x + sgκ 3 ) ] i dψ α,β) x), 9) S J 1 x) [ x + sgκ 3 ) ] i dψ α+1,β+1) x). 10) Sice, [x + sgκ 3)] i J S1 = 0 for 1 ad 0 i 1, we obtai κ 1 m J 1 i+1, + ) ) 1 sgκ 3 ) κ 1 m J 1 i, = i κ 3 ˆm J 1 i, + κ 2 κ 3 ) ˆm J 1 i,. 11) Usig itegratio by parts i 10) we have for 1 ad i 0, ˆm J 1 i, = i + α + β + 2) m J 1 i+1, 2sgκ 3)i + η) m J 1 i,, 12) where η = β + 1)[1 + sgκ 3 )] α + 1)[1 sgκ 3 )]. Sice { 2α + 1), if κ 3 < 0, η = 2β + 1), if κ 3 > 0, the sgη) = sgκ 3 ). Substitutig 12) i 11), the followig three term recurrece relatio holds m J 1 i+1, = 1 ) A i m J 1 i, κ 1 + i κ 3 i + α + β + 2) B i m J 1 i,, 13)

7 E.X.L. ANDRADE, C.F. BRACCIALI, M.V. MELLO ad T.E. PÉREZ 429 for 2 ad 1 i 1, where A i = κ 2 3 κ 3 )i 2 + [ κ 2 κ 3 )α + β + 1) κ 3 η) ] i + 1 sgκ 3 )κ 1 ), B i = κ 2 κ 3 ) [ 2sgκ 3 )i 1) + η ] i. Sice κ 2 κ 3 ad sgη) = sgκ 3 ), we observe that sgb i ) = sgκ 3 ) for i 1. Moreover, i order to assure the positivity of A i for i 1, we eed some additioal coditios. Lemma 2.1. Suppose α, β 0, κ 2 3 κ 3 ad sgκ 1 ) = sgκ 3 ) if κ 1 = 0. The A i > 0 for i 1. Proof. Observe that we ca write A i = [ κ 2 3 κ 3 ] i 2 + [ κ 2 3 κ 3 )α + β + 1) 2κ 3 β ] i + 1 κ 1 ) for κ 3 < 0, ad A i = [ κ 2 3 κ 3 ] i 2 + [ κ 2 3 κ 3 )α + β + 1) + 2κ 3 α ] i + 1 κ 1 ) for κ 3 > 0. The, the result holds. Lemma 2.2. Assume that the coditios of Lemma 2.1 hold. For 1, if κ 1 = 0, we get m J 1 0, = 0, sgm J 1 i, ) = )+i [ sgκ 3 ) ] +i, 1 i, ad if κ 1 = 0 sgm J 1 i, ) = )+i[ sgκ 3 ) ] +i, 0 i. Proof. Suppose κ 1 = 0. From 11), we get m J 1 0, = 0, 1. O the other had, usig the well kow property for the moic classical Jacobi polyomials see [15]) P α,β) x) = P α+1,β+1) x), 0,

8 430 ZEROS OF JACOBI-SOBOLEV ORTHOGONAL POLYNOMIALS i relatio 8), we have Therefore ˆm J 1 0, = a J 1 ˆm J 1 0, = ) a J 1 a J a J 1 2 a J 1 1 S J 1 x)dψ α+1,β+1) x). S J 1 1 x)dψ α+1,β+1) x). Sice sga J 1 ) = sgκ 3) ad S J 1 1 x)dψ α+1,β+1) x) > 0, we deduce sg ˆm J 1 0, ) = )+1 [ sgκ 3 ) ] +1, 1. By makig i = 0 i 12), we get ˆm J 1 0, = α + β + 2)m J 1 1,, ad the sgm J 1 1, ) = sg ˆm J 1 0, ) = )+1 [ sgκ 3 ) ] +1, 1. Now, suppose that κ 1 = 0. Usig 8), we obtai +1 x)dψ α,β) x) + a J 1 x)dψ α,β) x) = 0, 0, ad the m J 1 0,+1 = a J 1 m J 1 0,. Sice m J 1 0,0 = 0 x)dψ α,β) x) > 0 for 0, we get sgm J 1 0, ) = ) sga J 1 a J 1 2 a J 1 1 ) sgm J 1 0,0 ) = ) [sgκ 3 )]. The substitutio of i = 0 i 13) yields m J 1 1, = 1 sgκ 3)κ 1 )m J 1 0, /κ 1, the sgm J 1 1, ) = )+1 [sgκ 3 )] sgκ 1 ) = ) +1 [sgκ 3 )] +1. Therefore, usig mathematical iductio o i i 13), we get the result. Lemma 2.3. Uder the hypotheses of Lemma 2.1, let π r x) be a moic polyomial of degree r, with 1 r, such that all of its zeros are real ad lie i [, 1]. Let us defie I r, = The sgi r, ) = ) +r [ sgκ 3 ) ] +r. x)π rx)dψ α,β) x).

9 E.X.L. ANDRADE, C.F. BRACCIALI, M.V. MELLO ad T.E. PÉREZ 431 Proof. Let t r,1 t r,2 t r,r 1 be the zeros of π r x). The π r x) = r {x + sgκ 3 ) [t r, j + sgκ 3 )]} = j=1 r c i [x + sgκ 3 )] i, i=0 where c r = 1 ad, if c i = 0, the sgc i ) = ) r i [sgκ 3 )] r i, i = 0, 1,..., r. Usig 9), r I r, = c i m J 1 i,, ad the result holds from Lemma 2.2. i=0 Now we will show that, uder the hypotheses of Lemma 2.1, the -th Jacobi- Sobolev orthogoal polyomial of type I,, has real ad simple zeros ad they iterlace with the zeros of the classical Jacobi polyomial P α,β). Theorem 2.4. Suppose that the coditios of Lemma 2.1 hold. The, for for 1 i, i icreasig order, the they satisfy 2, has real ad simple zeros. If we deote s J 1 the zeros of for κ 3 < 0, for κ 3 > 0, s J 1 < p α,β) +1 < pα,β) +1, 1 i 1, 14) p α,β) < p α,β) +1 +1, 1 i 1. 15) Proof. Defie π j Pα,β) x) x) = x p α,β), j = i=1,i = j x p α,β) ), 1 j. The, deg π j = 1 ad, usig Lemma 2.3, we get sgi j, ) = )2 [sgκ 3 )] 2 = sgκ 3 ), where I j, = x) π j x)dψ α,β) x).

10 432 ZEROS OF JACOBI-SOBOLEV ORTHOGONAL POLYNOMIALS Applyig the Gaussia quadrature rule based o the zeros of P α,β), we obtai I j, = w i=1 pα,β) = w, j pα,β), j ) P α,β) ) π j pα,β) ) p α,β), j ) for j = 1, 2,...,. Suppose κ 3 < 0. I this case, for j = 1, 2,...,, we get pα,β), j )P α,β) p α,β), j ) > 0. Therefore, there is just oe zero of i each iterval p α,β), pα,β) i = 2, 3,...,. Sice P α,β) ad are moic, there is oe zero of, p α,β),1 ). The, 14) holds. For κ 3 > 0, we observe that pα,β), j )P α,β) p α,β), j ) < 0, for j = 1, 2,...,, ) for i ad a similar argumet as above shows 15). Moreover, it is possible to show iterlacig properties betwee the zeros of Jacobi-Sobolev orthogoal polyomials,, ad the zeros of the classical Jacobi polyomials P α,β+1), P α+1,β) ad P α+1,β+1). Theorem , we get Uder the hypotheses of Lemma 2.1, for 2 ad 1 i i) the zeros of iterlace with the zeros of Pα,β+1) ad P α+1,β) as follows s J 1 < p α,β+1) +1 < pα,β+1) +1 < 1, < p α+1,β) < p α+1,β) +1 +1, ii) the zeros of P α+1,β+1) separate the zeros of i the followig way s J 1 < p α+1,β+1),i +1.

11 E.X.L. ANDRADE, C.F. BRACCIALI, M.V. MELLO ad T.E. PÉREZ 433 Proof. Defiig π j Pα,β+1) x) x) = x + 1) x p α,β+1), j = x + 1) i=1,i = j x p α,β+1) ), ad ˆπ j Pα+1,β) x) x) = x 1) x p α+1,β), j = x 1) i=1,i = j a similar argumet used for Theorem 2.4 shows the result i). To get the result ii), take x p α+1,β) ), π j x) = x 2 1)P α+1,β+1) x) x p α+1,β+1), j = x 2 1) i=1,i = j x p α+1,β+1),i ) i Lemma 2.3. The oe obtais w, j pα+1,β+1), j ) P α+1,β+1) p α+1,β+1), j ) < 0, for j = 1, 2,..., 1. As a cosequece of this theorem, the followig result is established. Corollary 2.6. For 1, Jacobi-Sobolev orthogoal polyomial real ad simple zeros iside, 1). has Fially, iterlacig properties of the zeros of Jacobi-Sobolev orthogoal polyomials of two cosecutive degrees ca be show. Theorem 2.7. Uder coditios of Lemma 2.1, for 2, the 1 zeros of iterlace with the zeros of as follows s J 1,i +1, 1 i 1. Proof. From 8) we get a J 1 s J 1 ) = Pα,β) s J 1 ), for i = 1, 2,,.

12 434 ZEROS OF JACOBI-SOBOLEV ORTHOGONAL POLYNOMIALS O the other had, Theorem 2.4 provides ad the sg pα,β) ) P α,β) p α,β) )) = sgκ 3 ), sg s J 1 )Pα,β) s J 1 )) = sgκ 3). Sice sga J 1 ) = sgκ 3), we deduce s J 1 ) s J 1 ) > 0. Therefore, has a zero i each iterval s J 1, s J 1 ), for i = 2, 3,...,. Remark 2.8. If we have sgκ 1 ) = sgκ 3 ) istead of sgκ 1 ) = sgκ 3 ) i the hypotheses of Lemma 2.1, the sga J 1 0 ) = sgκ 1) ad there exists N N such that { sga J 1 ) = sgκ sgκ 1 ), if < N, 1++α+β+2)κ 3 ) = sgκ 3 ), if N, or a J 1 N = 0. Numerical experimets allow us to cojecture that, also i this case, the zeros of iterlace with the zeros of Pα,β). Moreover, for 1 i, if κ 1 < 0 ad κ 3 > 0, the s J 1 for > N, < p α,β) for N ad p α,β) if κ 1 > 0 ad κ 3 < 0, the p α,β) for N ad s J 1 for > N. < p α,β) Table 1 describes a example of this fact. Notice, from 8), that if a J 1 N = 0 the N+1 x) = Pα,β) N+1 x). 2.2 Zeros of ad Pα,β,κ 1) I this sectio we relate the zeros of Jacobi-Sobolev orthogoal polyomial of type I with the zeros of the polyomial P α,β,κ 1), κ 1 = 0, orthogoal with respect to the first measure i 7). For, i 0, we defie μ J 1 i, = x)1 + κ 1 x) i dψ α,β) x),

13 E.X.L. ANDRADE, C.F. BRACCIALI, M.V. MELLO ad T.E. PÉREZ 435 i s J 1 4,i p α,β) 4,i s J 1 4,i < p α,β) 4,i, i = 1, 2, 3, 4. Table 1 Zeros of polyomials κ 2 = 5, ad κ 3 = Here N = 4. ad P α,β) i s J 1 5,i p α,β) 5,i p α,β) 5,i 5,i, i = 1, 2, 3, 4, 5. for = 4, 5, α = 2, β = 5, κ 1 = 0.999, ad ˆμ J 1 i, = S J 1 x)1 + κ 1 x) i dψ α+1,β+1) x). Observe that, sice, 1 + κ 1 x) i J S1 = 0, for 1, μ J 1 i+1, = i κ 3 ˆμ J 1 i, + iκ 3 κ 1 κ 2 ) ˆμ J 1 i,, i = 0, 1,..., 1. 16) O the other had, usig itegratio by parts we get κ 1 ˆμ J 1 i, = i α + β)μj 1 i+1, [2 i + α + 1)1 κ 1 ) + β + 1)1 + κ 1 )] μ J 1 i, +i1 κ 2 1 )μj 1 i,, for 1 ad i 0. The, the followig recurrece relatio holds μ J 1 i+1, = κ 1 κ 1 + i κ 3 i α + β) A i μ J 1 i, B iμ J 1 i, + C iμ J 1 i 2, for 1 ad i = 1, 2,..., 1, where A i = κ 2 3 κ ) 3 i 2 + κ 2 α + β + 1) κ ) 3 [α2 κ 1 ) + β2 + κ 1 ) + 3] i, κ 1 κ 1 B i = 2κ 2 κ ) 3 3 κ 2 ) i 2 + κ 2 κ ) 3 α1 κ 1 ) + β1 + κ 1 )) i, κ 1 κ 1 C i = κ 2 κ ) 3 1 κ1 2 ) i i 1). κ 1 ),

14 436 ZEROS OF JACOBI-SOBOLEV ORTHOGONAL POLYNOMIALS I order to obtai A i, B i ad C i as o-egative coefficiets, we eed some additioal coditios. Observe that the coditios give i the ext lemma are sufficiet. Lemma 2.9. Suppose α, β 0, κ 2 3κ 3 /κ 1 0 ad sgκ 1 ) = sgκ 3 ). The A i, B i ad C i are o-egative for i 1. A similar argumet used i Lemma 2.2 allows us to obtai the sig of μ J 1 ˆμ J 1 i,, usig the hypotheses of Lemma 2.9. i, ad Lemma have Assume that the coditios of Lemma 2.9 hold. For 1, we i) μ J 1 1, = 0, sgμj 1 i, ) = )+i [sgκ 1 )], i = 2, 3,...,, ii) sg ˆμ J 1 i, ) = )+i+1 [sgκ 1 )] +1, i = 0, 1,..., 1. The ext lemma is aalogous to Lemma 2.3 ad it ca be proved usig the same techique. Agai, we assume that the hypotheses of Lemma 2.9 are valid. Lemma Uder coditios of Lemma 2.9, for 2 it follows i) Let π r be a moic polyomial of degree r, 1 r 1, such that all of its zeros are real ad lie i, 1). Defie I r, = 1 κ 1 the sgi r, ) = ) +r+1 [sgκ 1 )] +r+1. x)π rx)dψ α,β,κ 1) x), ii) Let π r be a moic polyomial of degree r, 1 r, with all real zeros i, 1). If we defie J r, = 1 κ 1 the sgj r, ) = ) +r+1 [sgκ 1 )] +r. S J 1 x)π r x)dψ α+1,β+1,κ1) x),

15 E.X.L. ANDRADE, C.F. BRACCIALI, M.V. MELLO ad T.E. PÉREZ 437 Proof. To prove i), let < t r,1 t r,2 t r,r < 1 be the zeros of π r. The r r π r x) = x t r, j ) = x + 1 )) r t r, j + 1κ1 = c i x + 1 ) i, κ 1 κ 1 j=1 j=1 with c r = 1. Whe κ 1 < 0 we have t r, j + 1 κ 1 < 0 ad whe 0 < κ 1 1 we have t r, j + 1 κ 1 > 0. Therefore i=0 sgc i ) = ) r i [sgκ 1 )] r i, i = 0, 1,..., r. Hece, I r, = r i=0 c 1 i κ i+1 x) 1 + κ 1 x) i+1 dψ α,β) x) = 1 r i=0 c i κ i+1 1 ad usig Lemma 2.10 the result holds. A similar argumet shows ii). μ J 1 i+1,, Now, we have the ecessary tools to get the aouced iterlacig property betwee the zeros of ad the zeros of Pα,β,κ 1). Theorem Uder the coditios of Lemma 2.9, for 2 ad 1 i 1, the zeros of satisfy i) If κ 1 < 0, ad, if 0 < κ 1 1, p α,β,κ 1) < p α,β,κ 1) +1 +1, s J 1 < p α,β,κ 1) +1 < pα,β,κ 1) +1. ii) The zeros of P α,β,κ 1) separate the zeros of. That is, s J 1 < p α,β,κ 1),i +1. Collectig all the iterlacig properties give i 5), 6) ad Theorems 2.4 ad 2.12, we have for κ 1 < 0, p α,β,κ 1) < p α,β) < p α,β,κ 1),i < p α,β,κ 1) +1, 1 i 1,

16 438 ZEROS OF JACOBI-SOBOLEV ORTHOGONAL POLYNOMIALS ad, for 0 < κ 1 1, p α,β) < p α,β,κ 1) < p α,β,κ 1),i < p α,β) +1, 1 i 1. To fiish this sectio, the extremal poits of ca be aalyzed. Deote the extremal poits of by ŝ J 1, i = 1, 2,..., 1, i icreasig order. Theorem Uder the hypotheses of Lemma 2.9, for 3 the polyomial has 1 extremal poits i the iterval, 1) ad they satisfy p α+1,β+1,κ 1) < ŝ J 1 < p α+1,β+1,κ 1) +1, 1 i 1. 3 Jacobi-Sobolev ier product of type II Let dψx) be the measure defied o [, 1] by meas of f, g ψ = κ 3 f x) gx) κ 3 x 1 x)α+1 1+x) β+1 dx +κ 4 f κ 3 ) gκ 3 ), 17) where κ 3 1 ad κ 4 0, ad let {P α,β,κ 3,κ 4 ) } =0 be the correspodig sequece of moic orthogoal polyomials. I Maroi [12] see also [3]), the author has obtaied the relatio where P α,β,κ 3,κ 4 ) d = ρα,β,κ 3,κ 4 ) x) = P α+1,β+1) x) + d P α+1,β+1) x), 1, κ 3 ρ α+1,β+1), ρ α,β,κ 3,κ 4 ) = P α,β,κ 3,κ 4 ), P α,β,κ 3,κ 4 ) ψ. Note that sgd ) = sgκ 3 ). I this sectio we cosider Jacobi-Sobolev ier product of type II, itroduced i [3], give by the expressio f, g J S2 = + f x) g x) f x) gx)dψ α,β) x) 18) κ 1 + κ ) 2 κ 3 κ 3 x dψ α+1,β+1) x) + κ 2 κ 4 f κ 3 ) g κ 3 ),

17 E.X.L. ANDRADE, C.F. BRACCIALI, M.V. MELLO ad T.E. PÉREZ 439 where κ 3 1, κ 2, κ 4 0 ad κ 1 κ 3 κ κ 3. We deote by {S J 2 } =0 the sequece of moic orthogoal polyomials associated with, J S2, ad we call it sequece of moic Jacobi-Sobolev orthogoal polyomials of type II. These polyomials satisfy S J 2 0 x) = 1, S J 2 1 x) = P α,β) 1 x) ad where S J 2 +1 x) + a J 2 a J 2 = ρ J 2 = S J 2, S J 2 J S 2 ad Sice ρ α+1,β+1) S J 2 b = + 1 ρ α,β) x) = Pα,β) +1 x) + b P α,β) x), 1, 19) ) + κ 1 2 ρ α+1,β+1) ρ J 2 d = + 1 b ρ α,β,κ 3,κ 4 ) κ 3 ρ α+1,β+1), 1, 20), 1. = + α + β + 1)ρ α,β) /, we ca also write a J 2 = 1 + κ1 + α + β + 1) ) b ρ α,β) Observe that sgb ) = sgd ) = sgκ 3 ) ad, if κ 1 > 0, the sga J 2 ) = sgb ) = sgκ 3 ). ρ J Zeros of S J 2 For, i 0, we defie ad Pα,β) ad ˆν J 2 i, = = ν J 2 i, = S J 2 x)x κ 3) i dψ α,β) x) 21) S J 2 x)x κ 3 ) i κ 3 x dψx) 22) κ 3 S J 2 x)x κ 3 ) i 1 x) α x) β+1 dx.

18 440 ZEROS OF JACOBI-SOBOLEV ORTHOGONAL POLYNOMIALS Because of the orthogoality property, we have S J 2, x κ 3) i J S2 1, ad the = 0, for ν J 2 i, = i κ 1 ˆν J 2 i, + κ 3 κ 2 ˆν J 2 i 2,), i 2. 23) Itegratio by parts i 22) for 1 ad i 1 provides ˆν J 2 i, = i α + β)ν J 2 i+1, + [ 2κ 3 i + α + 1)1 + κ 3 ) β + 1)1 κ 3 ) ] ν J 2 i, +iκ 2 3 1)ν J 2 i,, ad the, if we defie ν J 2, = 0, for 1 ad i = 2, 3,...,, the followig three term recurrece relatio ca be deduced ν J 2 i, = where i 1 + κ 1 i α + β)i A i ν J 2 i, B i ν J 2 i 2, + C i ν J 2 i 3, ), 24) A i = κ 3 κ2 2κ 1 ) i + κ1 [ κ3 α + β) + α β ] κ 3 κ 2 α + β), B i = [ 2κ 2 3 κ 2 κ 1 κ )] i + κ 2 [ κ 2 3 α + β 2) + κ 3 α β) ] + κ 1 κ ), C i = κ 3 κ 2 1 κ 2 3 ) i 2). The ext lemma establishes sufficiet coditios to determie the sig of the above coefficiets. Lemma 3.1. For κ 2 2κ 1 0, α + β > 2 ad { α β, if κ3, α β, if κ 3 1, the sga i ) = sgκ 3 ) ad B i > 0, for i 1. Moreover, if κ 3 = 1 the sgc i ) = sgκ 3 ) ad if κ 3 = 1 the C i = 0. We remark that Lemma 3.1 establishes sufficiet coditios i order to obtai the sig of A i, B i ad C i. Uder coditios of Lemma 3.1, aalogous techiques to those used i Lemmas 2.2 ad 2.3 allow us to prove the ext two lemmas.

19 E.X.L. ANDRADE, C.F. BRACCIALI, M.V. MELLO ad T.E. PÉREZ 441 Lemma 3.2. For 3, we have ν J 2 0, = 0 ad sg ν J 2 i,) = [ sgκ3 ) ] +i, 1 i 1. Lemma 3.3. For 3, let us cosider π r a moic polyomial of degree r, 1 r 1, such that all its zeros are real, simple ad lie i the iterval, 1). Defie I r, = The sgi r, ) = [ sgκ 3 ) ] +r. S J 2 x)π rx)dψ α,β) x). 25) Uder the same restrictios give i Lemma 3.1 for the parameters ad usig the above two lemmas, we ca show that the -th Jacobi-Sobolev orthogoal polyomial of type II, S J 2, has differet real zeros ad at least 1 zeros lie iside, 1). We deote the real zeros of S J 2, i icreasig order, by s J 2, i = 1, 2,...,. Theorem 3.4. Uder the coditios of Lemma 3.1, for 3,S J 2 has real zeros ad at least 1 of them lie iside the iterval, 1). Moreover, deotig the zeros of S J 2, by s J 2, i = 1, 2,...,, i icreasig order, the i) if κ 3, if κ 3 1, s J 2 < p α,β) < s J 2 +1 < pα,β) +1, 1 i 1, p α,β) < s J 2 < p α,β) +1 < s J 2 +1, 1 i 1. ii) For 1 i 1, the followig iterlacig property holds s J 2 < p α,β),i < s J We must poit out that oe zero of S J 2 ca be outside the iterval, 1). Figure 1 shows the graphs of Jacobi-Sobolev orthogoal polyomial of type II S J 2 6 ad the classical Jacobi orthogoal polyomial P α,β) 6. Accordig to Theorem 3.4, i Figure 1a), with κ 3 <, we ca see that the smallest zero is outside the iterval, 1). I Figure 1b), with κ 3 > 1, we ca see that the largest zero is outside, 1).

20 442 ZEROS OF JACOBI-SOBOLEV ORTHOGONAL POLYNOMIALS a) b) Figure 1 Graphs of S J 2 6 ad P α,β) 6 with κ 1 = 1, κ 2 = 4, κ 4 = 1. a) κ 3 =.1, α = 1, β = 2. b) κ 3 = 1.1, α = 2, β = Some coditios for all zeros of S J 2 to lie iside, 1) I this sectio we obtai some coditios for the parameters i the ier product 18) to assure that all zeros of S J 2 lie iside the iterval, 1).

21 E.X.L. ANDRADE, C.F. BRACCIALI, M.V. MELLO ad T.E. PÉREZ 443 We ow deote the polyomials S J 2 by S κ 1,κ 2,κ 3,κ 4 ) x) ad the coefficiets a J 2 by aκ 1,κ 2,κ 3,κ 4 ). Let κ 2 ted to i 1 κ 2 f, g J S2. The we fid that the moic polyomials, S κ 1,,κ 3,κ 4 ) x), must satisfy S κ 1,,κ 3,κ 4 ) m x) S κ 1,,κ 3,κ 4 ) x)dψx) = 0, for m =. Sice ρ J 2 > 2 κ 2 ρ α,β,κ 3,κ 4 ) ad, for a fixed κ 1, sga κ 1,κ 2,κ 3,κ 4 ) ) = sgκ 3 ), from 20) we verify that a κ 1,,κ 3,κ 4 ) = 0. The we coclude from 19) that S κ 1,,κ 3,κ 4 ) x) = P α,β) x) + b P α,β) x), 0, 26) with b = b 0 = 0. It is well kow that the sequece of moic Jacobi polyomials, {P α,β) } =0, satisfies P α,β) +1 x) = x λ α,β) ) +1 P α,β) x) γ α,β) +1 Pα,β) x), 1, with P α,β) 0 x) = 1, P α,β) 1 x) = x λ α,β) 1, γ α,β) +1 = λ α,β) +1 = β 2 α α + β)2 + α + β + 2), 0, 4 + α) + β) + α + β) 2 + α + β 1)2 + α + β) α + β + 1), 1. Now, we ca prove the followig result. Theorem 3.5. If the coditios of Lemma 3.1 are satisfied, 3 ad κ 2 large eough, the the zeros of S κ 1,κ 2,κ 3,κ 4 ) lie iside the iterval, 1) provided that i) for κ 3, α ad β are such that b < ii) for κ 3 1, α ad β are such that b > 2 + α + β + 1)2 + α + β) γ α,β) α), 2 + α + β + 1)2 + α + β) γ α,β) β).

22 444 ZEROS OF JACOBI-SOBOLEV ORTHOGONAL POLYNOMIALS Proof. i) For x = it is kow that P α,β) 2 + α + β + 1)2 + α + β) ) = γ α,β) α) Pα,β) ) ad sgp α,β) )) = ). The, from 26), b Choosig α ad β such that S κ 1,,κ 3,κ 4 ) ) = P α,β) ) 2 + α + β + 1)2 + α + β) γ α,β) α) ). b < 2 + α + β + 1)2 + α + β) γ α,β) α), we get sgs κ 1,,κ 3,κ 4 ) )) = ). From Theorem 3.4, for κ 3, at most s,1 lies outside, 1). Sice S κ 1,,κ 3,κ 4 ) is moic ad sgs κ 1,,κ 3,κ 4 ) )) = ) the s,1 > ad all zeros of S J 2 lie iside, 1). ii) For κ 3 1, the proof is aalogous usig x = 1. Ackowledgemets. This research was supported by grats from CAPES, CNPq ad FAPESP of Brazil ad by grats from Miisterio de Ciecia e Iovació Mici) of Spai ad Europea Regioal Developmet Fud MTM C02-02) ad by Juta de Adalucía G.I. FQM 0229). The authors would like to thak the referees for their valuable remarks, suggestios ad refereces. REFERENCES [1] E.X.L. Adrade, C.F. Bracciali ad A. Sri Raga, Zeros of Gegebauer-Sobolev orthogoal polyomials: beyod coheret pairs. Acta Appl. Math., ), [2] A.C. Berti ad A. Sri Raga, Compaio orthogoal polyomials: some applicatios. Appl. Numer. Math., ), [3] A.C. Berti, C.F. Bracciali ad A. Sri Raga, Orthogoal polyomials associated with related measures ad Sobolev orthogoal polyomials. Numer. Algorithms, ), [4] C.F. Bracciali, D.K. Dimitrov ad A. Sri Raga, Chai sequeces ad symmetric geeralized orthogoal polyomials. J. Comput. Appl. Math., ),

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