KU Leuven Department of Computer Science

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1 O orthogoal polyomials related to arithmetic ad harmoic sequeces Adhemar Bultheel ad Adreas Lasarow Report TW 687, February 208 KU Leuve Departmet of Computer Sciece Celestijelaa 200A B-300 Heverlee (Belgium)

2 O orthogoal polyomials related to arithmetic ad harmoic sequeces Adhemar Bultheel ad Adreas Lasarow Report TW 687, February 208 Departmet of Computer Sciece, KU Leuve Abstract I this paper we study special systems of orthogoal polyomials o the uit circle. More precisely, with a view to the recurrece relatios fulfilled by these orthogoal systems, we aalyze a lik of o-egative arithmetic to harmoic sequeces as a mai subject. Here, arithmetic sequeces appear as coefficiets of orthogoal polyomials ad harmoic sequeces as correspodig Szegő parameters. Keywords : Orthogoal polyomials o the uit circle, recurrece relatios of Szegő-type, arithmetic sequeces, harmoic sequeces MSC 200 : Primary : 42C05; Secodary : 30C5.

3 TW Report Preprit, 208 Research Article Adhemar Bultheel ad Adreas Lasarow O orthogoal polyomials related to arithmetic ad harmoic sequeces Abstract: I this paper we study special systems of orthogoal polyomials o the uit circle. More precisely, with a view to the recurrece relatios fulfilled by these orthogoal systems, we aalyze a lik of o-egative arithmetic to harmoic sequeces as a mai subject. Here, arithmetic sequeces appear as coefficiets of orthogoal polyomials ad harmoic sequeces as correspodig Szegő parameters. Keywords: Orthogoal polyomials o the uit circle, recurrece relatios of Szegőtype, arithmetic sequeces, harmoic sequeces MSC 200: Primary 42C05; secodary 30C5 Itroductio Throughout the paper, let be a positive iteger. Suppose that p is a (complexvalued) polyomial of degree which admits the represetatio p ( x ) a x + a x + + a x + a 0 with o-egative real coefficiets. I this paper, especially, we are iterested i the case that the differece of cosecutive coefficiets is positive ad costat, i.e. a k a k d, k,...,, for some (arbitrary, but fixed) positive real umber d. We will deote by P ; ar the set of all polyomials of this type. The coefficiets are related to arithmetic sequeces (hece the " ar " i the otatio) ad, sice d > 0, we have the mootoicity a > a > > a > a 0 0. Thereby, the cosideratios below ca be see as a cotiuatio of those i [ 2 ] ad [5] o special polyomials appearig i orthogoal systems o the uit circle T : { z C : z } of the complex plae C. Adhemar Bultheel, Departemet Computerweteschappe, KU Leuve, Belgium; adhemar.bultheel@cs.kuleuve.be Adreas Lasarow, Fakultät Iformatik, Mathematik ud Naturwisseschafte, HTWK Leipzig, Germay; adreas.lasarow@htwk-leipzig.de

4 2 A. Bultheel ad A. Lasarow TW Report Suppose that μ is a measure belogig to M, where M stads for the set of all fiite measures defied o the σ -algebra B T of all Borel subsets of T. We will call a (fiite or ifiite) sequece ( φ k ) τ k 0, where each φ k is a polyomial of degree ot greater tha k, a orthoormal polyomial system for μ whe φ s ( z )φ t ( z ) μ (d z ) δ st, T where δ st : if s t ad δ st : 0 if s t (for each choice of idices). Here ad heceforth, τ is a o-egative iteger or τ (arbitrarily chose, but fixed). If there exists a orthoormal polyomial system for some μ M, the we fid a special oe which is uiquely determied by the extra coditio that the leadig coefficiet of each φ k is a positive real umber. We will call this ( φ k ) τ k 0 the ( up to τ ) ormalized orthoormal polyomial system for μ. As a aside, we remark that there are explicit descriptios of ormalized orthoormal polyomial systems by certai determiat formulas or (equivalet) by usig etries of the iverse of Toeplitz matrices give by the Fourier coefficiets of the measure μ (see, e.g., [ 7 ], [ 6 ], [ 3 ], ad [ 4, Sectio 3.6]). Sice we are more iterested i the recurrece relatios for such systems, we omit here the details. Orthoormal polyomial systems for some μ M fulfill specific recurrece relatios, where the elemet φ ca be calculated based o φ ad vice versa. By the degree of freedom of the choice of such orthoormal system oe ca successively choose the elemets so that the related recursios oly deped i each step o some parameter from the uit disk D : { w C : w < } (cf. [ 7 ], [ 6, Sectios.5 ad.7], but below we follow more the approach of [3], [4, Sectio 3.6]). Cocerig the recurrece relatios i questio, the followig term will be take ceter stage. We will call ( φ k ) τ k 0 a sequece of Szegő polyomials whe φ 0 is a costat fuctio o C with a value p 0 C { 0 } ad the other polyomials i the sequece with higher idex are ( i each -th step ) coected via φ ( x ) ( ) xφ e 2 ( x ) + e φ [ ] ( x ) with some Szegő parameter e D (cf. [ 3 ] ad [ 4, Defiitio 3.6.7]). Herei ad furthermore (with some o-egative iteger m ), the otatio p [ m ] stads for the polyomial which is uiquely determied via ( ) p [ m ] ( x ) x m p x, x C { 0 }, (2) for a polyomial p of degree ot greater tha m. I additio, a sequece ( φ k ) τ k 0 of Szegő polyomials is called caoical whe φ 0 is a costat fuctio o C with some positive real umber p 0 as value (cf. [4, Defiitio 3.6.4]). ()

5 TW Report Orthogoal polyomials related to arithmetic sequeces 3 I Sectio 2 we will give some iformatio about sequeces of Szegő polyomials which are kow, but we recall them because that is useful for our mai results. Sectio 3 forms the body of this paper, where we study the set P ; ar with a view to the recurrece relatios fulfilled by sequeces of Szegő polyomials. This approach is similar to the ivestigatios i [ 2 ] ad [ 5 ] betwee properties of coefficiets of Szegő polyomials ad correspodig properties of Szegő parameters. However, as a mai result i Sectio 3, it will be revealed that a arithmetic sequece of coefficiets is related to a harmoic sequece of Szegő parameters. Fially, we remark that the restrictio of the case that the polyomials have oegative real coefficiets is ot essetial. However, the calculatios are somewhat more labor-itesive for the more geeral situatio. We will give the details later. 2 Hits o sequeces of Szegő polyomials We give i this sectio some otes o a sequece ( φ k ) τ k 0 of Szegő polyomials which are useful for our mai results. Eve if we mostly fix the statemets below oly as a remark, there is usually more tha oe lie eeded to prove them precisely (depedig o the kowledge). Sice the k -th elemet φ k of a sequece of Szegő polyomials is of degree ot greater tha k (i fact, of exact degree k ; cf. [ 2, Lemma 2.4]), there are some coefficiets a k ;0, a k ;,..., a k ; k C so that φ k ( x ) k a k ; j x j. (3) j 0 Especially, we are iterested i this paper i the case that all coefficiets of the correspodig polyomials are o-egative real. Remark 2.. Suppose that φ is a polyomial of degree with o-egative real coefficiets. If φ is the polyomial give by () with some real umber e, where 0 e <, the φ ( x ) j 0 a ; j + e a ; j x j e 2 usig the otatio of the coefficiets as i (3) for φ ad settig a ; : 0. I particular, we ca see, that all coefficiets of the polyomial φ are o-egative real i this situatio as well. The followig example ca be see as the iitial poit of the cosideratios i this paper. Note that the structure of the parameters which appear i this example

6 4 A. Bultheel ad A. Lasarow TW Report are closely related to that i [ 6, Examples.6.3 ad.6.4] cocerig (ifiite) sequeces of orthogoal polyomials o the uit circle. Example 2.. Let φ 0 be the costat fuctio with value ad let e l :, l, 2, 3, 4. l + The the (caoical) sequece ( φ k ) 4 k 0 of Szegő polyomials with φ 0 ad the sequece of Szegő parameters ( e l ) 4 l is give by φ 0 ( x ), φ ( x ) (2 x +), 3 φ 2 ( x ) φ 3 ( x ) φ 4 ( x ) 6 (3 x 2 +2 x +), 0 (4 x 3 +3 x 2 +2 x +), 5 (5 x 4 +4 x 3 +3 x 2 +2 x +). Example 2. suggests a lik betwee a harmoic sequece of Szegő parameters ad correspodig arithmetic sequeces of coefficiets of Szegő polyomials. This lik will be studied i Sectio 3 i somewhat more detail. Note that () is equivalet to xφ ( x ) ( ) φ ( x ) e φ [ ] e 2 ( x ) (4) (cf. [ 2, Lemma 2.4]). Thus, i additio to Remark 2., it follows from () that the polyomial φ ca also be expressed i terms of the coefficiets of the polyomial φ. Usig the otatio of the coefficiets as i (3) for φ, we get the followig. Remark 2.2. Suppose that φ is a polyomial of degree ad that φ is the polyomial give by () with o-egative real coefficiets ad with some e D. The we get a ; > a ;0 0 ad 0 e <, where φ ( x ) j 0 a ; a ; j + a ;0 a ; j x j a 2 ; a 2 ;0 ad e a ;0 a ; (cf. [ 2, Lemma 2.6]). I particular, if we kow the coefficiets of the polyomial φ, the the parameter e ad the polyomial φ are uiquely determied. The ext example illustrates that the case cocerig () ad sequeces of Szegő polyomials is uspectacular (but a exceptio).

7 TW Report Orthogoal polyomials related to arithmetic sequeces 5 Example 2.2. Suppose that p is a polyomial admittig p ( x ) a x + a 0 with some o-egative real umbers a ad a 0. With a view to Remark 2.2 oe ca see that there is a ( φ k ) k 0 of Szegő polyomials with φ p if ad oly if a > a 0, where the Szegő polyomials with φ p is the uiquely determied ad caoical (sice φ 0 is the costat fuctio with value a 2 a 2 0 ). With the followig example we will emphasize that, uder the exclusive terms of Remark 2.2, it is possible that a coefficiet of the polyomial φ is egative real. Example 2.3. If φ 2 ad φ 3 are the polyomials give by φ 2 ( x ) 8 x 2 x + 7 ad φ 3 ( x ) 0 x x x + 6, the () is fulfilled with 3 ad e 3 5. I view of the defiitio of sequeces of Szegő polyomials, a fiite sequece of this type ca be exteded to a ifiite oe (by choosig the missig Szegő parameters arbitrary, but belogig to D ). The followig ote poits out a simple extesio regardig (ifiite) orthoormal polyomial systems by fixig the measure μ. Remark 2.3. Suppose that ( φ k ) k 0 is a sequece of Szegő polyomials. The ( φ k ) k 0 is a orthoormal polyomial system for μ give by μ ( B ) : 2 π B φ ( z ) 2 λ (d z ), B B T, (5) where λ stads for the liear Lebesgue-Borel measure o T. I fact ( cf. [ 6, Theorems.7.5 ad.7.8] or [2, Propositio 2.5 ad Remark 5.2] ), if we choose φ + l( x ) : x l φ ( x ), l, 2,..., the ( φ k ) k 0 is a sequece of Szegő polyomials with parameter e + l 0 for all itegers l, where ( φ k ) k 0 is a orthoormal polyomial system for μ. If ( φ k ) k 0 is a sequece of Szegő polyomials, the Remark 2.3 clarifies particularly that there is a μ M so that ( φ k ) k 0 is a orthoormal polyomial system for μ. I additio to that (cf. [ 6, Theorem.7.] or [ 4, Theorem 3.6.2]), if we cosider a ifiite sequece ( φ k ) k 0 of Szegő polyomials, the there is exactly oe measure μ M so that ( φ k ) k 0 is a orthoormal polyomial system for μ. Note that, coversely, if we have a orthoormal polyomial system ( φ k ) τ k 0 for μ M, the i the set of all such systems are icluded sequeces of Szegő polyomials ad we fid a special oe which is caoical (cf. [ 6, Chapter ] or [ 4, Sectio 3.6]). Fially, we recall the followig maipulatio by multiplicatio of sequeces of Szegő polyomials (cf. [2, Remark 2.0]).

8 6 A. Bultheel ad A. Lasarow TW Report Remark 2.4. Suppose that ( φ k ) τ k 0 is a sequece of Szegő polyomials with related sequece ( e l ) τ l of Szegő parameters ad let a be a positive real umber. The ( aφ k ) τ k 0 is a sequece of Szegő polyomials with sequece ( e l ) τ l of Szegő parameters. Furthermore, if ( φ k ) τ k 0 is a orthoormal polyomial system for the measure μ, the ( aφ k ) τ k 0 is a orthoormal polyomial system for μ. a 2 3 O Szegő polyomials belogig to P ; ar Now, with a view to the set P ; ar, we study special sequeces of Szegő polyomials. Thereby, the followig result is the lychpi. Lemma 3.. Suppose that φ is a polyomial of degree ad that φ is the polyomial give by () with some e D ad 2. If φ P ; ar, the φ P ; ar ad, usig the otatio of the coefficiets as i (3) for φ, the φ ( x ) j 0 ( j +) d ( d + 2 a ;0) x j, (6) a 2 ; a 2 ;0 with d a ; a ;0 > 0. e a ;0 d + a ;0 (7) Proof. Let φ P ; ar. With a view to (3) for φ, we get a ; > a ;0 0, φ ( x ) j 0 a ; a ; j + a ;0 a ; j x j, (8) a 2 ; a 2 ;0 ad e a ;0 a ; (cf. Remark 2.2). Furthermore, sice φ P ; ar, there is a d > 0 so that (9) Thus, it follows that i.e. (7), ad a ; k a ; k d, k,...,. e a ;0 a ; a ;0, d + a ;0 a ; a ; j + a ;0 a ; j ( d + a ;0) ( ( j +) d + a ;0 ) ( ) a ;0 ( j ) d + a ;0 ( j +) d 2 + da ;0+( j +) da ;0 ( j ) da ;0 ( j +) d ( d +2 a ;0)

9 TW Report Orthogoal polyomials related to arithmetic sequeces 7 for j 0,,...,, i.e. we get (6). I particular, we see φ P ; ar. Theorem 3.. Suppose that p P ; ar for some 2. The there is a uiquely determied sequece ( φ k ) k 0 of Szegő polyomials, where φ p. This sequece ( φ k ) k 0 of Szegő polyomials is caoical, where the associated Szegő parameter e is give via (7) ad (3). Furthermore, the polyomial φ l belogs to P l ; ar ad the associated Szegő parameter e l is give by e l, l,...,. l + Proof. We set φ p ad use the otatio (3) with k replaced by. Because p P ; ar, we have a ; > a ;0 0 so that the parameter e ad the polyomial φ accordig to (9) ad (8), respectively, are well-defied. Sice the relatio (4) is equivalet to (), from (8) ad (9) we get (). Furthermore, Lemma 3. yields φ P ; ar ad the represetatio (7) ad (6) for e ad φ, respectively. I particular, if 2, we ca proceed with the approach ad get φ 2 P 2; ar, where Remark 2.2 ad (6) imply e a ;0 a ; d ( d +2 a ;0) a 2 ; a 2 ;0 d ( d +2 a ;0) a 2 ; a 2 ;0 d ( d + 2 a ;0) d ( d + 2 a ;0). For l,...,, by the priciple of iductio, we get φ l P l ; ar ad e l l +. Fially, with a view to Example 2.2, oe ca see that the costructio used above leads to a uiquely determied sequece ( φ k ) k 0 of Szegő polyomials, where φ p, ad that this is caoical. Corollary 3.. If p P ; ar with, the all zeros of p belog to D. Proof. For, the statemet follows immediately from the defiitio of P ; ar. If 2, the Theorem 3. implies that there is a sequece ( φ k ) k 0 of Szegő polyomials with φ p. Thus, i this case, the statemet follows from a geeral result o sequeces of Szegő polyomials (see, e.g., [2, Propositio 2.5 (a)]). As a aside, we ote that the statemet of Corollary 3. follows from a classical theorem due to Eeström Kakeya as well (see, e.g., [, Theorem A]). The result, that is revealed by Theorem 3., comprises a very special structure of Szegő polyomials. This will be emphasized by the followig. Propositio 3.. Suppose that p P ; ar for some 2 ad let ( φ k ) k 0 be the uiquely determied sequece of Szegő polyomials, where φ p. (a) The sequece ( φ k ) k 0 is the ( up to ) ormalized orthoormal polyomial system correspodig to the measure μ give by (5).

10 8 A. Bultheel ad A. Lasarow TW Report (b) Let d m be the differece of cosecutive coefficiets of φ m for m, 2,..., ad let d 0 : φ 0 (0). The d 0 > d > > d d, where d l φ l (0) 2 2 φ ad d l d l + l for l,..., ad where d d + (0) d. ( Furthermore, d m m φ [ m ] m (0) φ m (0) ) for m, 2,...,. (c) Deotig the coefficiets as i (3), the ( + ( a k ; j j + k 2 p +3 [ ] (0) ) 2 ( p (0) ) ) 2, j 0,,..., k, for all idices k 0,,...,. (d) The followig statemets are equivalet: (i) p (0) 0. (ii) e 0. (iii) d d. (e) The followig statemets are equivalet: (iv) p (0) d. +. (v) e (vi) d d +2. Proof. As is well-kow (cf. Remark 2.3), the sequece ( φ k ) k 0 is a orthoormal polyomial system for the measure μ give by (5). Sice φ p particularly implies φ P ; ar ad sice Theorem 3. yields φ l P l ; ar for l,..., as well as that φ 0 (0) is a positive real umber, the leadig coefficiet of φ k is a positive real umber for each k 0,,...,. Thus, the sequece ( φ k ) k 0 is the (up to ) ormalized orthoormal polyomial system for μ. Hece, (a) is prove. Because φ m P m ; ar, the umber d m is well-defied accordig to (b) ad φ [ m ] m (0) is the leadig coefficiet of φ m for m, 2,...,. Cosequetly, based o (3), for m, 2,..., we have φ [ m ] m (0) a m ; m md m + a m ;0 md m + φ m (0), ( i.e. d m m φ [ m ] m (0) φ m (0) ). Moreover, by Theorem 3. ad Lemma 3. follows the represetatio (6) for φ, where d d. This implies d d ( d + 2 a ;0) a 2 ; a 2 ;0 d ( d + 2 a ;0) ( d + a ;0) 2 a 2 ;0 d 2 (+ 2 a ;0 d ) d + 2 a ;0 d d + 2 a ;0 d d d ( d + 2 a ;0) ( d ) d a ;0 + 2 φ (0). d I particular, takig φ (0) 0 ad d > 0 ito accout, we see d d. Let l,...,. Theorem 3. yields e l l + o the oe had ad o the other

11 TW Report Orthogoal polyomials related to arithmetic sequeces 9 had e l a l ;0 ld l + a l ;0 by Lemma 3. ad Remark 2.2. Hece, we get a l ;0 0 ad d l a l ;0 φ l (0). Therefore, similar as above, i view of Theorem 3. ad Lemma 3. it follows d l d l + 2, l,...,, l ad particularly d l > d l. Hece, (b) is prove. By (b) ad p φ, we have (ote p [ ] (0) > p (0)) d + 2 φ (0) d ( p [ ] (0) p (0) ) + ( p [ ] (0) p (0) ) 2 p (0) p [ ] (0) p (0) p [ ] (0) + p (0) p [ ] (0) p (0) Furthermore, by the priciple of iductio, oe ca show that ( p [ ] (0) ) 2 ( p (0) ) 2. ( + 2 ) j j l + 2 l + l 2, l,...,. Thus, usig the coefficiets as i (3), from (b) we get a k ;0 φ k (0) d k + k + + ( k + ) 2 + k φ (0) d ( ( p [ ] (0) ) 2 l k + ( ( p [ ] (0) ) 2 ( p (0) ) ) 2 d l ( p (0) ) 2 ) for k 0,...,. Recallig φ l P l ; ar ad d l φ l (0) for l,...,, the assertio of (c) follows. Takig p φ ad (7) ito accout the assertios of (d) ad (e) are a cosequece of (b). Corollary 3.2. Suppose that ( φ k ) τ k 0 is a sequece of Szegő polyomials. (a) If φ l P l ; ar for some idex l with τ > l, where the differece of cosecutive coefficiets of the polyomial φ l is ot equal to φ l (0), the φ k P k ; ar for all idices k with τ k > l. (b) If φ l P l ; ar for some idex l with τ > l, where the associated Szegő parameter e l is zero, the φ k P k ; ar for all idices k with τ k > l.

12 0 A. Bultheel ad A. Lasarow TW Report Proof. Let φ l P l ; ar for a idex l with τ > l, where the differece d l of cosecutive coefficiets of φ l is differet from φ l (0). Furthermore, we assume temporarily that there is a idex k with τ k > l, where φ k P k ; ar. Thus, Theorem 3. ad part (b) of Propositio 3. with p φ k yield d l φ l (0). But this coflicts with the coditio of φ l. Therefore, φ k P k ; ar for all idices k with τ k > l, ad (a) is prove. Sice the coditio e l 0 leads to φ l (0) 0 (cf. Remark 2.2), the statemet of (b) is a cosequece of (a). I view of the iterdepedecy of sequeces of Szegő polyomials ad orthoormal polyomial systems for measures μ M (see, e.g., the otes from Remark 2.3 to the ed of Sectio 2), we ca simply reformulate the statemets above i terms of orthoormal systems. I particular, we get the followig result which turs out to be somewhat more surprisig (if you do ot have the recurrece relatio i mid). Theorem 3.2. Suppose that ( φ k ) τ k 0 is a orthoormal polyomial system for some measure μ M. (a) If φ P ; ar for some idex with 2, the a k ;0 0 ad u a k ; j k + ( a j + k 2 2 ; a 2 ) +3 ;0, j 0,,..., k, with some u k T for all idices k 0,,..., based o (3). (b) If φ l P l ; ar for some idex l with τ > l, where the differece of cosecutive coefficiets of the polyomial φ l is ot equal to φ l (0), the φ k P k ; ar for all idices k with τ k > l. (c) If φ l P l ; ar for some idex l with τ > l, where φ l (0) is zero, the φ k P k ; ar for all idices k with τ k > l. Based o Theorem 3., we ca also see that the case of a ifiite sequece of Szegő polyomials, where the coefficiets of all polyomials are related to o-egative real arithmetic sequeces, is a very special oe. Theorem 3.3. Suppose that ( φ k ) k 0 is a sequece of Szegő polyomials ad let ( e l ) l be the associated sequece of Szegő parameters. The the followig statemets are equivalet: (i) There is a positive real umber p 0 so that the sequece ( φ k ) k 0 is give by φ k ( x ) p 0 k 2 +3 k ( j +) x j, k 0,, 2, 3,.... j 0 (ii) For each idex l with l the polyomial φ l belogs to P l ; ar ad φ 0 (0) is a positive real umber. (iii) There is some l 0 so that for each idex l with l l 0 the polyomial φ l belogs to P l ; ar.

13 TW Report Orthogoal polyomials related to arithmetic sequeces (iv) The sequece ( e l ) l is give by e l, l, 2, 3,..., l + ad φ 0 (0) is a positive real umber. I particular, if (i) is fulfilled, the ( φ k ) k 0 is the ormalized orthoormal polyomial system for the (uiquely determied) measure μ give by μ ( B ) : p 2 0 π ( R e z ) λ (d z ), B B T, (0) B where λ stads for the liear Lebesgue-Borel measure o T. Proof. The implicatios (i) (ii) ad (ii) (iii) are a cosequece of the settigs. Furthermore, the implicatios (ii) (iv) ad (iii) (ii) follow from Theorem 3.. It remais to prove that (iv) (i). Therefore, we suppose that (iv) holds. Sice ( φ k ) k 0 is a sequece of Szegő polyomials, for each positive iteger, the relatio () is fulfilled with the special Szegő parameter e give by (iv). Takig ito accout that φ 0 (0) is a positive real umber, i.e. φ 0 is the costat fuctio with (positive) value φ 0 (0), we have φ 0 ( x ) φ 0 (0) p ( j +) x j with the positive real umber p 0 : 2 φ 0 (0) ad () for implies φ ( x ) ( 2 ) 2 ( xp 0 + ) p 0 2 j 0 p ( j +) x j. j 0 Now, by the priciple of iductio, we suppose that φ k is give by φ k ( x ) p 0 k 2 +3 for some positive iteger k. The we have where ( φ [ k ] k ( x ) p 0 k 2 +3 ) 2 k 2 +3 k ( j +) x j, j 0 k ( j +) x k j, j 0 ( ( ) 2 ) ( k 2 +3 ) ( k +)( k +3)( k +)( ) k +3 ( k +),

14 2 A. Bultheel ad A. Lasarow TW Report ad k j + j + j j + k j + ( k +)( j +), ( )( k +3) k 2 +5 k +6 ( k +) 2 +3( k +)+2. Thus, i view of (iv) ad () for k +, we get φ k +( x ) ( p ( ) 0 k 2 k 2 x ( j +) x j + +3 j 0 p 0 k ( +2 k + k ( x 0 + j + k + k +3 j k p 0 + ( j +) x j ( )( k +3) j 0 k ) ( j +) x k j j 0 ) ) k j + x j + ( k +) x k + k p 0 + ( j +) x j, ( k +) 2 +3( k +)+2 j 0 so that we have prove, by the priciple of iductio, that (i) follows from (iv). Suppose that (i) holds. If p 0 2, the the cosideratios i [ 6, Example.6.4] imply that ( φ k ) k 0 is the ormalized orthoormal polyomial system for the (uiquely determied) measure μ give by (0). Usig this special case i combiatio with Remark 2.4, we see that this holds for ay positive real umber p 0. Fially, i additio to Lemma 3. ad Corollary 3.2, we preset the followig result cocerig the oe-step extesio give by (). Propositio 3.2. Suppose that φ P ; ar for some 2 ad let d be the differece of cosecutive coefficiets of the polyomial φ. Furthermore, let φ be the polyomial give by () ad some parameter e D, where φ ( x ) a ; x + a ; x + + a ; x + a ;0 with some coefficiets a ;0, a ;,..., a ; C as i (3). (a) The followig statemets are equivalet: (i) The parameter e is a real umber with 0 e < ad φ (0) d. (ii) The polyomial φ belogs to P ; ar. I particular, if (i) is satisfied, the the coefficiets a ;0, a ;,..., a ; of the polyomial φ are real ad a ; > a ; > > a ; > a ;0 0. () (b) Suppose that φ (0) 0, but also that φ (0) d, ad let { } φ m : mi (0) d,. d φ (0) The the followig statemets are equivalet:

15 TW Report Orthogoal polyomials related to arithmetic sequeces 3 (iii) The parameter e is a real umber with 0 e m. (iv) The coefficiets a ;0, a ;,..., a ; of the polyomial φ are real ad a ; a ; a ; a ;0 0. (2) I particular, if (iii) is satisfied, the φ P ; ar, although all zeros of φ belog to D ad 0 e < m is actually equivalet to (). (c) Suppose that φ (0) 0. The e 0 holds if ad oly if the coefficiets a ;0, a ;,..., a ; of the polyomial φ are real ad (2) holds. I particular, if e 0 is satisfied, the a ; a ;0 0 ad all zeros of φ belog to D. Proof. I view of Lemma 3. we see that (ii) implies (i). Now, we suppose that (i) holds. By φ P ; ar ad φ (0) d, we have the represetatio φ ( x ) ( j +) d x j. j 0 Hece (cf. Remark 2.), it follows that φ ( x ) j 0 j d + e ( j ) d x j, e 2 where ( ) ( j +) d + e ( j ) d j d + e ( j ) d ( e ) d for j 0,,...,, i.e. we get (ii). Therefore, (i) ad (ii) are equivalet. I particular, if (i) is satisfied, the (ii) as well so that the coefficiets a ;0, a ;,..., a ; of the polyomial φ are real ad () holds. Thus, (a) is prove. We ow suppose that φ (0) 0 ad φ (0) d hold. I particular, due to (a), we have φ P ; ar. Recallig the choice of the umber m accordig to (b) ad φ P ; ar, the equivalece of (iii) ad (iv) is a cosequece of [ 2, Propositio 5.4 (a)]. Furthermore, from [ 2, Propositio 5.4 (e)] ad φ (0) 0 we see that 0 e < m is actually equivalet to (). Takig φ P ; ar ad Corollary 3. ito accout, by [ 2, Propositio 2.5 (a)] ad the choice of φ it follows that all zeros of φ belog to D. Thus, (b) is prove. Fially, we suppose that φ (0) 0 holds. Note that e 0 ad () imply φ ( x ) xφ ( x ). Hece, if e 0, the the coefficiets a ;0, a ;,..., a ; of the polyomial φ are real ad (2) holds, where a ; a ;0 0 (sice φ P ; ar ad φ (0) 0 ). Furthermore, from Corollary 3. we kow that all zeros of φ

16 4 REFERENCES TW Report belog to D. Therefore, all zeros of φ belog to D whe e 0. Usig [ 2, Propositio 5.4 (a)] oe ca also see that, if the coefficiets a ;0, a ;,..., a ; of the polyomial φ are real ad (2) holds, the e 0. As a aside to () ad (2), we ote that a ;0 0 is equivalet to e 0 (cf. Remark 2.2). Furthermore, if φ (0) 0, the (c) of Propositio 3.2 shows that there is oly the choice e 0 which realize (2) ad o choice which realize (). Ackowledgmet: The secod author has to thak Felix Klemd for support durig the preparatio of these results. Refereces [] N. Aderso, E.B. Saff, ad R.S. Varga, A extesio of the Eeström Kakeya theorem ad its sharpess, SIAM J. Math. Aal. 2 (98), [2] A.E. Choque Rivero ad A. Lasarow, The Eeström Kakeya theorem ecouters the theory of orthogoal polyomials o the uit circle, Liear Algebra Appl. 439 (203), [3] Ph. Delsarte, Y.V. Gei, ad Y.G. Kamp, Orthogoal polyomial matrices o the uit circle, IEEE Tras. Circuits ad Systems 25 (978), [4] V.K. Dubovoj, B. Fritzsche, ad B. Kirstei, Matricial Versio of the Classical Schur Problem, Teuber-Texte zur Math. 29, Teuber, Leipzig 992. [5] A. Lasarow, O some special polyomials appearig i orthogoal systems o the uit circle, Aalysis 34 (204), [6] B. Simo, Orthogoal Polyomials o the Uit Circle, Part : Classical theory, Amer. Math. Soc. Coll. Publ. 54, America Mathematical Society, Providece, [7] G. Szegő, Orthogoal Polyomials, Amer. Math. Soc. Colloq. Publ. 23, America Mathematical Society, Providece, 939.