Asymptotic properties of generalized Laguerre orthogonal polynomials

Transcription

1 Asymptotic properties of geeralized Laguerre orthogoal polyomials Reato Álvarez odarse1,3 ad Jua J. Moreo Balcázar 2,3 1 Departameto de Aálisis Matemático, Uiversidad de Sevilla, Apdo. 1160, 1080, Sevilla, Spai 2 Departameto de Estadística y Matemática Aplicada, Uiversidad de Almería, 0120, Almería, Spai 3 Istituto Carlos I de Física Teórica y Computacioal, Uiversidad de Graada, 18071, Graada, Spai Abstract I the preset paper we deal with the polyomials L α,m,) x) orthogoal with respect to the Sobolev ier product p, q) = 1 Γα+1) 0 px)qx) x α e x dx + M p0)q0) + p 0)q 0),, M 0, α > 1, firstly itroduced by Koekoek ad Meijer i 1993 ad extesively studied i the last years. We preset some ew asymptotic properties of these polyomials ad also a it relatio betwee the zeros of these polyomials ad the zeros of Bessel fuctio J α x). The results are illustrated with umerical examples. Also, some geeral asymptotic formulas for geeralizatios of these polyomials are cojectured. MSC Subject Classificatio 2000: 33C5, 33C7, 2C05 Key words: Asymptotics, Laguerre polyomials, geeralized Laguerre polyomials. 1 Itroductio I this paper we will deal maily with the polyomials which are orthogoal with respect to the Sobolev-type ier product p, q) = 1 Γα + 1) 0 px)qx) x α e x dx + M p0)q0) + p 0)q 0), 1) where M, 0 ad α > 1. These polyomials were itroduced by Koekoek ad Meijer i [10] ad costitute a atural geeralizatio of the so called Koorwider s geeralized Laguerre Research partially supported by Juta de Adalucía, Grupo de Ivestigació FQM 0262, Direcció Geeral de Ivestigació Miisterio de Ciecia y Tecología) of Spai uder grat BFM C0-02 ad ITAS Project Research partially supported by Juta de Adalucía, Grupo de Ivestigació FQM 0229, Direcció Geeral de Ivestigació Miisterio de Ciecia y Tecología) of Spai uder grat BFM C02 02 ad ITAS Project

2 polyomials, earlier itroduced by Koorwider i [11], which are orthogoal with respect to 1) where = 0 for details see e.g. [8, 11]). I the followig we will deote L α,m,) x) ) the sequece of orthogoal polyomials with respect to 1). I [10] the authors established differet properties of the polyomials α,m,) L x) ) such as the differetial equatio that they satisfy, a five term recurrece relatio, a Christoffel Darboux type formula, a represetatio as a hypergeometric series 3 F 3 ad some properties of their zeros beig oe of them the fact that the zeros of these polyomials are all real ad simple). Later, although it was published earlier, Koekoek i [9] cosidered the more geeral ier product p, q) g = 1 Γα + 1) 0 px)qx) x α e x dx + s M i p i) 0)q i) 0), 2) where M i 0, i = 0,..., s ad α > 1 ad studied some properties of the correspodig orthogoal polyomials with respect to the ier product 2), usually called the discrete Sobolev-type Laguerre which costitutes a istace of a larger class of orthogoal polyomials: the discrete Sobolev-type orthogoal polyomials. For more detailed descriptio of this Sobolev-type orthogoal polyomials icludig the cotiuous oes) we refer the readers to the recet reviews [12, 1, 15]. Our mai aims here are two: 1. To fill a gap i the study of the polyomials L α,m,) x) orthogoal with respect to the ier product 1), that is, to obtai ew asymptotic properties such as strog asymptotics, Placherel Rotach type asymptotics ad Mehler Heie type formulas. This will be doe i Sectio 3, theorems 1 ad To establish it relatios whe betwee the zeros of the polyomial L α,m,) x) ad the zeros of Bessel fuctios J α, J α+2, J α+ or their combiatios accordig to the values of the masses ad M. This will be doe i Sectio, theorem 3. These kid of problems have bee also cosidered for the Jacobi Sobolev type orthogoal polyomials see [2]) ad for the cotiuous Sobolev orthogoal polyomials see, e.g., [, 13]). Also we will show that the techique used here for the L α,m,) x) polyomials ca be easily exteded to aother family of orthogoal polyomials correspodig to a o-diagoal case itroduced later o i [5, 6]. This will be doe i sectio.1. The structure of the paper is as follows: I sectio 2, some preiary results are quoted. I sectio 3, the asymptotics of the polyomials orthogoal with respect to the ier product 1) is deduced that allows us, i sectio, to obtai some iterestig properties of the zeros of these geeralized polyomials as well as to set out a cojecture about the asymptotic behavior of the polyomials orthogoal with respect to 2). I sectio.1 a example of a o-diagoal case will be discussed briefly ad, fially, i sectio.2 some umerical examples illustratig the above results are preseted. i=0 2

3 2 Preiaries 2.1 The classical Laguerre polyomials The classical Laguerre polyomials are defied by see e.g. [16]) x) = where α β) are the biomial coefficiets. k=0 ) + α x) k, k k! Propositio 1 Let x) ) be the sequece of Laguerre polyomials with leadig coefficiets 1) /!. They verifies the followig properties: a) For α R, [16, f )]) x) Lα) x) = Lα 1) x). 3) 1 b) Strog asymptotics Perro s formula) o C \ R [16, Th ]). Let α R. The x) = 2 1 π 1/2 e x/2 x) α/2 1/ α/2 1/ e 2 x)1/2 1 + O 1/2)). ) This relatio holds for x i the complex plae cut alog the positive real semiaxis; both x) α/2 1/ ad x) 1/2 must be take real ad positive if x < 0. The boud of the remaider holds uiformly i every closed domai which does ot overlap the positive real semiaxis. c) It holds [16, Sectio 8.22 ad formula )]) x) α/2 = e x/2 x α/2 J α 2 x) + O 3/ ), 5) uiformly o compact subsets of 0, + ) where J α is the Bessel fuctio. d) Mehler Heie type formula [16, Th ]) x/ + j)) α = x α/2 J α 2 x), 6) uiformly o compact subsets of C ad uiformly o j {0}. e) Scaled asymptotics o C \ [0, ]. It holds [7]) 2 2 π 1) x x) 2 + ) ) x 2 2x x exp x + = x 2 x 2 α 1/2 x α x 2 + ) 1/2 x 2 x x + α x x 2 x) 2 x ) 1/, 7) uiformly o compact subsets of C \ [0, ] takig ito accout that the square roots i 7) are egative if x is egative. Remark 1 Although the Mehler Heie type formula for Laguerre polyomials i Szegő s book is 6) with j = 0, it ca be show that this formula is true for j such as it appears i 6). 3

4 O the other had, formulas ) ad 7) allow us to obtai the ratio asymptotics for Laguerre ad scaled Laguerre orthogoal polyomials α > 1), respectively. I fact, from ) we deduce Lα+j) +k x) l j)/2 L α+l) +h x) = x)l j)/2, j, l R, h, k Z. 8) uiformly o compact subsets of C \ [0, ). We will use this result with j, l Z. Also, from 7) we get, for j {0}, 1 + j)x) + j)x) = 1 ϕ x 2)/2), 9) uiformly o compact subsets of C \ [0, ] where ϕ is the coformal mappig of C \ [ 1, 1] oto the exterior of the uit circle give by with x 2 1 > 0 whe x > Geeralizatio of Laguerre polyomials ϕx) = x + x 2 1, x C \ [ 1, 1], 10) I [10] Koekoek ad Meijer establish that the orthogoal polyomials, L α,m,) x), with respect to the ier product 1), M, 0, α > 1, ca be rewritte i terms of the Laguerre polyomials, x), L α,m,) x) = B 0 ) x) + B 1 )xl α+2) x) + B 2)x 2 L α+) x), 0, 11) 1 where it is assumed i x) = 0, for i = 1, 2 ad ) +α+1 B 0 ) = 1 B 2 ) = α+1 2 α+1)α+2)α+3), B 1 ) = M α+1 ) +α + 1 ) +α M α+1) 2 α+2)α+3) 2 α+2) α+1)α+3) ) +α, 12) 2 ) ) +α +α+1. 13) 1 otice that usig 11) ad the fact that the leadig coefficiets of the Laguerre polyomials are 1) /! we deduce that the leadig coefficiets of L α,m,) x) are 1)! B 0 ) B 1 ) + 1)B 2 )). Followig [10] we ca use the above formulas 12 13) to obtai the asymptotics of the coefficiets i 11): Case M > 0, = 0. B 0 ) = 1, B 1 ) α = M Γα + 2), B 2) = 0.

5 Case M = 0, > 0. B 0 ) α+3 = α + 1)Γα + ), Case M > 0, > 0. B 1 ) α + 2) = α+2 α + 1)Γα + ), B 2 ) α+1 = α + 1)Γα + ). 1) B 0 ) α+3 = α+1)γα+), B 2 ) 2α+2 = B 1 ) α+2) = α+2 α+1)γα+), M α+1)γα+3)γα+). 15) The polyomials L α,m,) x) verify several iterestig properties [10]: they satisfy a secod order differetial equatio with polyomial coefficiets of degree at most three, a five term recurrece relatio, ad Christoffel Darboux type formula ad they ca be represeted as a geeralized hypergeometric series 3 F 3. Of particular sigificace to our work is the followig: Theorem [Koekoek, Meijer [10]] The polyomial L α,m,) x) has real simple zeros. At least 1 of them lie i 0, + ). Furthermore, whe > 0 ad large eough they have exactly oe zero i, 0]. Before cocludig let us poit out that some of the above properties ad results have bee exteded to other more geeral polyomials. The reader iterested i these results ca cosult, e.g., [1, 3, 9]. 3 Asymptotic properties of geeralized Laguerre polyomials Alog this sectio B i ), i = 0, 1, 2, take the values give by 12 13), respectively. Theorem 1 The polyomials L α,m,) x) ), with α > 1, satisfy a) Exterior asymptotics. The followig its hold uiformly o compact subsets of C \ [0, + ), If M > 0 ad = 0, If M = 0 ad > 0, L α,m,0) x) α+1 x) = M Γα + 2). L α,0,) x) α+3 x) = α + 2) α + 1)Γα + ). If M > 0 ad > 0, L α,m,) x) 2α+ x) = M α + 1)Γα + 3)Γα + ). 5

6 b) Asymptotics o compact subsets of 0, + ). If M > 0 ad = 0, L α,m,0) x) 3α/2+1 = a) e x/2 x x α+2)/2 J α+2 2 ) 1)x + O mi{α+5/,3/}), 16) where a) = B 1) α If M = 0 ad > 0, 1 ) α/2+1 M Γα + 2) whe. L α,0,) x) 3α/2+3 = e x/2 b 0 ) x α/2 ) J α 2 x + b1 ) x x α+2)/2 J α+2 + b 2 ) x 2 x α+)/2 J α+ 2 )) 2)x 2 ) 1)x + O 3/), 17) where b 0 ) = B 0) α+3 α + 1)Γα + ) 1 b 1 ) = B 1) α+2 b 2 ) = B 2) α+1 ) α/2+1 ) 2 α/2+2 whe, α + 2) α + 1)Γα + ) α + 1)Γα + ) whe, whe. If M > 0 ad > 0, L α,m,) x) 5α/2+ = c)e x/2 x 2 x α+)/2 J α+ 2 ) 2)x +O mi{α+5/,3/}), 18) where c) = B 2) 2α+2 ) 2 α/2+2 α + 1)Γα + ) whe. Proof: a) We will prove here oly the case whe M, > 0. The proof of the other cases ca be doe i a similar way. First, we divide 11) by 2α+ x) L α,m,) x) 2α+ x) = B 0) 1 α+3 α+1 + B 1) 1 xl α+2) 1 x) α+2 α+1 + B 2) x 2 L α+) 2 x) x) 2α+2 2 x). ow, 8) ad 15) yield L α,m,) x) 2α+ x) = B 2 ) x 2 L α+) 2 x) 2α+2 2 x) = M α + 1)Γα + 3)Γα + ). b) We start cosiderig the case M = 0 ad > 0. I this case, if we divide 11) by 3α/2+3 L α,0,) x) 3α/2+3 = B 0) α+3 x) α/2 + x B 1) α+2 L α+2) 1 x) α+2)/2 + x 2 B 2) α+1 L α+) 2 x) α+)/2, 6

7 ad use the formulas 5) ad 1), the expressio 17) follows. To prove the case whe M, > 0 we divide 11) by 5α/2+ to get L α,0,) x) 5α/2+ = 1 B 0 ) α+1 α+3 x) α/2 + 1 α+1 x B 1) α+2 L α+2) 1 x) α+2)/2 + x 2 B 2) L α+) 2 x) 2α+2 α+)/2. Therefore, takig ito accout the asymptotic formula see [16, p.15]) J α 2 ) 1 1/2 x) = π cos 2 x α π x 2 π ) + O 3/ ), whe, valid for ay x o compact subsets of C \ [0, ), ad usig 5) as well as 15) we fid L α,0,) ) x) 1 5α/2+ = O α+5/ + c)e x/2 x 2 x α+)/2 J α+ 2 ) ) 1 2)x + O 3/, with c) = B ) 2) 2 α/2+2 2α+2. Thus, the applicatio of 15) leads to 18). The case 16) is similar to this oe ad we omit it here. Remark 2 We ca deduce the strog exterior asymptotic of the polyomials L α,m,) x), for all M, 0, directly from a) i the above theorem usig Perro s formula ). For example, for case M, > 0, i the same coditios of Propositio 1 b) we have L α,m,) x) 5α/2+15/ e = M ) 2 x)1/2 2 π α + 1)Γα + 3)Γα + ) ex/2 x) α/2 1/ 1 + O 1/2 ). Theorem 2 The polyomials L α,m,) x) ), with α > 1, satisfy a) Exterior Placherel Rotach type asymptotics. The followig its hold uiformly o compact subsets of C \ [0, ], If M > 0 ad = 0, L α,m,0) x) α+1 x) = M Γα + 2) x ϕ x 2)/2) ϕ x 2)/2) + 1) 2. 19) If M = 0 ad > 0, L α,0,) x) α+3 x) = α+1)γα+) x 2 ϕ 2 x 2)/2)+α+2)x ϕ x 2)/2) ϕ 2 x 2)/2)+1 ) 2 ϕ 2 x 2)/2)+1 ) ϕ x 2)/2)+1). If M > 0 ad > 0, L α,m,) x) 2α+ x) = M α + 1)Γα + 3)Γα + ) I the three cases ϕx) is give by 10). x 2 ϕ 2 x 2)/2) ϕ x 2)/2) + 1). 20) 7

8 b) Mehler Heie type formulas. The followig its hold uiformly o compact subsets of C, If M > 0 ad = 0, If M = 0 ad > 0, L α,0,) x/) 2α+3 = L α,m,0) x/) 2α+1 = M Γα + 2) x x α+2)/2 J α+2 2 x). 21) α + 1)Γα + ) x 2 x α+)/2 J α+ 2 x) α + 2)x x α+2)/2 J α+2 2 x) x α/2 J α 2 ) x). 22) If M > 0 ad > 0, L α,m,) x/) 3α+ = M α + 1)Γα + 3)Γα + ) x2 x α+)/2 J α+ 2 x). 23) Proof: a) To prove 19-20) we will use a similar idea as i the proof Theorem 1 a). We first use the idetity 3) to rewrite the quotiet x) L α+2) 1 x) = L α+2) x) 2L α+2) 1 x) + Lα+2) 2 x) L α+2) 1 x), ad the use the ratio asymptotics for the scaled Laguerre polyomials 9) to obtai x) L α+2) 1 x 2)/2) + 1) = ϕ x) ϕ x 2)/2) 2, x) L α+) 2 ϕ x 2)/2) + 1) = x) ϕ 2 x 2)/2) Thus, dividig 11) by x), scalig x as x ad usig the above it relatios the result follows. b) Sice the proof of the three cases are completely aalogous we will prove oly the secod case ad will omit the other two cases. To prove the case M = 0, > 0, we use agai the relatio 11). Scalig the variable x as x/, i 11) yields L α,0,) x/) = B 0 ) x/) + B 1 ) x Lα+2) x/) + B 2) x2 The, dividig the above expressio by 2α+3 we get L α,0,) x/) 2α+3 = B 0) x/) α+3 α + B 1) α+2 1 x L α+2) 1 x/) α+2 + B 2) α+1 2 Lα+) 2 x/). x 2 L α+) 2 x/) α+. Fially, we take the it ad use 6) ad 1) that lead to the result 22).. 8

9 Remark 3 The exterior Placherel Rotach type asymptotics of the polyomials L α,m,) x) ca be obtaied i a straightforward way usig a) of the above theorem ad 7). For example, i the simplest case M > 0, = 0 Koorwider polyomials) i the same coditios of Propositio 1 e) we have 1) 2 +1 L α,m,0) x) α+1/2 ϕ x 2)/2) exp 2x x+ x 2 x ) = M 2 α π Γα + 2) x 1 α x + x 2 x) α x 2 x ) 1/ ϕ 3/2 x 2)/2) ϕ x 2)/2) + 1) 2. Zeros of the geeralized Laguerre polyomials I this sectio we will obtai the asymptotic properties of the zeros of L α,m,) x) that follow from the Mehler Heie type formulas give i Theorem 2 b). More cocretely, we will establish a it relatio betwee the zeros of the geeralized Laguerre polyomials ad the zeros of Bessel fuctio J α, J α+2 or J α+ or their combiatios. I fact, we will prove the followig: Theorem 3 Deote by j α,i the i-th positive zero of the Bessel fuctio J α x). Let ) x,i i=1 be the zeros i icreasig order of the polyomial L α,m,) x) orthogoal with respect to the ier product 1) with α > 1. The, a) If M > 0 ad = 0, we have b) If M = 0 ad > 0, we have x,1 = 0 ad x,i = j2 α+2,i 1, i 2. 2) x,i = h α,i, where h α,i deotes the i-th real zero of fuctio hx) defied as hx) = x 2 x α+)/2 J α+ 2 x) α + 2)x x α+2)/2 J α+2 2 x) α + 1)Γα + ) x α/2 J α 2 ) x). Moreover, hx) has oly oe egative real zero. c) If M, > 0, we have x,i = 0, i = 1, 2 ad x,i = j2 α+,i 2, i 3. 25) Proof: The results i the three cases are a cosequece of Theorem 2 b) ad Hurwitz s theorem [16, Thm ]. Let prove ow that whe M = 0 ad > 0, the it fuctio hx) has oly oe egative real zero. Usig the defiitio of Bessel fuctio J α x) we have: x α/2 J α 2 x) = i=0 x) i i!γi + α + 1). 9

10 It is well kow [16, 8.1] that for α > 1 the above fuctio oly has positive real zeros. Therefore, hx) = = α + 1)Γα + ) α + 1)Γα + ) i=0 1) i x i+2 i!γi + α + 5) α + 2) 1)i x i+1 i!γi + α + 3) 1 Γα + 1) + i=0 x) i+2 i!γi + α + 3) 1) i x i ) i!γi + α + 1) ) α + 2 i + 2)i + α + ). Thus, Moreover, hx) = +, h0) = x Γα + 2)Γα + ) < 0. 26) h x) = α + 1)Γα + ) i=0 i + 2) x) i+1 i!γi + α + 3) α + 2 < 0, for all x < 0. i + 2)i + α + ) The, hx) is a cotiuous decreasig fuctio o, 0) ad gatherig with 26) we obtai that hx) has oe ad oly oe zero o, 0). The last theorem has several importat cosequeces. First of all, from 2) follows that i the case whe M > 0, = 0 the first scaled zero, x,1, of the orthogoal polyomials Koorwider polyomials) L α,m,0) x) goes to 0 whe. Secod, i the case M > 0, > 0 the two first scaled zeros x,1, x,2 ), beig oe of them a egative zero [10], of the correspodig orthogoal polyomials L α,m,) x) also ted to 0 whe., i.e., i these cases the origi attracts oe or two zeros of the correspodig orthogoal polyomials. This situatio agrees with the results i [1]. I fact, i [1] the authors proved, i a more geeral framework, that if M, > 0 the there are two zeros ot scaled oes of L α,m,) x) that are attracted by 0 ad if M > 0, = 0 there is oly oe zero that is attracted by 0. We will call this the simple or regular situatio. otice that if x,i 0 the x,i 0 but ot vice versa. Let also poit out that to apply the geeral results of [1] we eed to obtai the ratio asymptotics of the sequece L α,m,) x) ). This relatio ca be deduced from Theorem 1 ad the relatio 8), i fact we have that L α,m,) x) = 1, uiformly o +1 x)/l α,m,) compact subsets of C \ [0, ) with M, > 0. The situatio for the case M = 0, > 0 is differet because i this case we ca ot apply the geeral results of [1]. Furthermore, i this case, for large eough, the first scaled zero x,1 is always egative ad does ot teds to zero whe, i.e., x,1 = h α,1 is a egative real umber. Let us ow cosider the polyomials L α,m 0,...,M s ) x) orthogoal with respect to the ier product 2). If all masses M 0,..., M s are positive the, usig the results of [1] it ca be show that s + 1 zeros of L α,m 0,...,M s ) x) go to zero as teds to ifiity. But ow, usig the fact proved here that asymptotic behavior of the smaller zeros of L α,m,) x) is determied by the Mehler Heie type formulas, it is reasoable to expect for this geeral case a simple Mehler Heie type formulas similar to the 21-23). More exactly, we pose the followig: 10

11 Cojecture 1 Let L α,m 0,...,M s ) x) be the polyomials orthogoal with respect to the ier product 2). If M i > 0, i = 1,..., s, the, for some real umbers β ad K L α,m 0,...,M s ) x/) β = K x s+1 x α+2s+2)/2 J α+2s+2 2 x), uiformly o compact subsets of C where β R ad K is a o-zero costat. Moreover, i the geeral case these simple Mehler Heie type formulas should ot be expected whe some of the costats vaish sice eve i the case of the polyomials L α,m,) x) discussed here it does ot appear. What happes i this case is still a ope questio..1 Aother geeralizatio of the Laguerre polyomials I [5, 6] a differet geeralizatio of the Laguerre polyomials has bee itroduced. I fact i [5, 6] the authors cosidered the moic polyomials, R α,m 0,M 1 x), orthogoal with respect to the liear fuctioal U o the liear space of polyomials with real coefficiets defied as U, P = 0 P x) x α e x dx + M 0 P 0) + M 1 P 0), M 0, M 1 R, α > 1. Although the fuctioal U is ot positive defiite, for large eough there exists the orthogoal polyomial R α,m 0,M 1 x) for all the values of the masses M 0 ad M 1. Furthermore, we have the followig expressio for these geeralized moic Laguerre polyomials, i terms of the moic Laguerre polyomials ˆL α x) beig R α,m 0,M 1 x) = ˆL α x) + A 1 ˆL α ) x) + A 2 ˆL α ) x), A 1 = 2α + 2) > 0, 2 A 2 = α + 2)α + 3) > 0 I [6] it was established for large eough ad M 0, M 1 0, that all zeros of these polyomials are real, simple ad oe of them is egative. Thus, usig the same ideas preseted i this paper we ca obtai the Mehler Heie type formula, that is, for ay M 1 > 0, ad M 0 0, R α,m 0,M 1 x/) α = x 2 x α+)/2 J α+ 2 x), from where a similar formula to 25) follows, i.e., if M 1 > 0 ad M 0 0 we have x,i = 0, i = 1, 2 ad x,i = j2 α+,i 2, i 3, where x,i deotes, as before, the zeros of the polyomial R α,m 0,M 1 x) ordered i icreasig order. I other words, for this o diagoal case the origi always attracts the two first scaled zeros of R α,m 0,M 1 x). I this case we expect to be true the followig: Cojecture 2 Let R α,m 0,...,M s x) be the polyomials orthogoal with respect to liear fuctioal U defied by U, P = 0 P x) x α e x dx + 11 s M k P 0)) k), α > 1. k=0

12 If M s > 0, the, for some real umbers β ad K L α,m 0,...,M s ) x/) β = K x s+1 x α+2s+2)/2 J α+2s+2 2 x), uiformly o compact subsets of C where β R ad K is a o-zero costat, i.e., i this case we have a simple Mehler Heie type formula for ay choice of the masses M 0,..., M s umerical examples Fially, we illustrate with umerical examples the asymptotic behavior of the first three scaled zeros of L α,m,) x) i the three cases. a) Koorwider polyomials) α = 0.5, M = ad = 0. x 1 x,2 x,3 = = = Limit value 0 j 2 α+2,1 = j 2 α+2,2 = b) α = 0.5, M = 0 ad =. x 1 x,2 x,3 = = = Limit value h α,1 = h α,2 = h α,3 = c) Whe M, > 0 we preset three examples with the objective to compare our umerical results with the lower boud for x,1 obtaied i [10]. c1) α = 0.5, M = 2 ad = 30. x 1 x,2 x,3 = = = Limit value 0 0 j 2 α+,2 =

13 c2) α = 0.5, M = 2 ad =. x 1 x,2 x,3 = = = Limit value 0 0 j 2 α+,2 = c3) α = 0.5, M = 60 ad =. x 1 x,2 x,3 = = = Limit value 0 0 j 2 α+,2 = The lower boud obtaied i [10] for the smallest zero of L α,m,) x) whe is large eough ad M, > 0 is b, M) := 1 2 M x,1 0 ad furthermore x,1 = 0. We compare this lower boud with the data of the umerical examples: α = 0.5, If M = 2, = 30, the b2, 30) , x 100, , x 300, , x 500, If M = 2, =, the b2, ) x 100, , x 300, , x 500, If M = 60, =, the b60, ) x 100, , x 300, , x 500, otice that the smallest zero, at least i these umerical examples, is very close to 0 eve for values of ot excessively large it is also true for values of such as 10, 20, etc.) Ackowledgemets: The authors are very grateful to M. Alfaro ad M.L. Rezola for their helpful discussio ad remarks that allow us improve the paper. This research has supported by the Juta de Adalucía uder the grat Acció Coordiada etre los grupos FQM-207 y FQM

14 Refereces [1] M. Alfaro, G. López, M.L. Rezola, Some properties of zeros of Sobolev-type orthogoal polyomials, J. Comp. Appl. Math ), [2] M. Alfaro, F. Marcellá, M.L. Rezola, Estimates for Jacobi Sobolev type orthogoal polyomials. Appl. Aal ), [3] M. Alfaro, F. Marcellá, M.L. Rezola, A. Roveaux, O orthogoal polyomials of Sobolev type: Algebraic properties ad zeros, SIAM J. Math. Aal. 233) 1992), [] M. Alfaro, J.J. Moreo-Balcázar, M.L. Rezola, Laguerre Sobolev orthogoal polyomials: asymptotics for coheret pairs of type II. J. Approx. Theory, ), [5] R. Alvarez-odarse, F. Marcellá, A geeralizatio of the classical Laguerre polyomials. Red. Circ. Mat. Palermo 2) 2)1995), [6] R. Alvarez-odarse, F. Marcellá, A geeralizatio of the class Laguerre polyomials: asymptotic properties ad zeros. Appl. Aal. 623-)1996), [7] J.S. Geroimo, W. Va Assche, Relative asymptotics for orthogoal polyomials with ubouded recurrece coefficiets, J. Approx. Theory ), [8] R. Koekoek, Koorwider s Laguerre polyomials, Delft Progress Report ), [9] R. Koekoek, Geeralizatios of Laguerre polyomials, J. Math. Aal. Appl. 1532)1990), [10] R. Koekoek, H.G. Meijer, A geeralizatio of Laguerre polyomials, SIAM J. Math. Aal. 23)1993), [11] T.H. Koorwider, Orthogoal polyomials with weight fuctio 1 x) α 1+x) β +Mδx+ 1) + δx 1). Caad. Math. Bull. 272) 198), [12] F. Marcellá, M. Alfaro, M.L. Rezola, Orthogoal polyomials o Sobolev spaces: old ad ew directios. J. Comput. Appl. Math. 81-2)1993), [13] F. Marcellá, J.J. Moreo-Balcázar, Strog ad Placherel-Rotach asymptotics of odiagoal Laguerre Sobolev orthogoal polyomials, J. Approx. Theory ), [1] A. Martíez-Fikelshtei, Asymptotic properties of Sobolev orthogoal polyomials. J. Comput. Appl. Math )1998), [15] A. Martíez-Fikelshtei, Aalytic aspects of Sobolev orthogoal polyomials revisited. J. Comput. Appl. Math )2001), [16] G. Szegő, Orthogoal Polyomials, Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., Providece, RI, fourth editio, [17] W. Va Assche, Asymptotics for Orthogoal Polyomials, Lecture otes i Math., 1265, Spriger, Berli,