Infinite log-concavity: developments and conjectures

Transcription

1 Ifiite log-cocavity: developmets ad cojectures Peter R. W. McNamara Departmet of Mathematics, Bucell Uiversity, Lewisburg, PA 17837, USA, ad Bruce E. Saga Departmet of Mathematics, Michiga State Uiversity, East Lasig, MI , USA, April 13, 2009 Key Words: biomial coefficiets, computer proof, Gaussia polyomial, ifiite log-cocavity, real roots, symmetric fuctios, Toeplitz matrices AMS subject classificatio (2000): Primary 05A10; Secodary 05A20, 05E05, 39B12. Abstract Give a sequece (a ) = a 0, a 1, a 2,... of real umbers, defie a ew sequece L(a ) = (b ) where b = a 2 a 1a +1. So (a ) is log-cocave if ad oly if (b ) is a oegative sequece. Call (a ) ifiitely log-cocave if L i (a ) is oegative for all i 1. Boros ad Moll [4] cojectured that the rows of Pascal s triagle are ifiitely log-cocave. Usig a computer ad a stroger versio of log-cocavity, we prove their cojecture for the th row for all We also use our methods to give a simple proof of a recet result of Umisy ad Yeats [30] about regios of ifiite log-cocavity. We ivestigate related questios about the colums of Pascal s triagle, q-aalogues, symmetric fuctios, real-rooted polyomials, ad Toeplitz matrices. I additio, we offer several cojectures. 1 Itroductio Let (a ) = (a ) 0 = a 0, a 1, a 2,... 1

2 be a sequece of real umbers. It will be coveiet to exted the sequece to egative idices by lettig a = 0 for < 0. Also, if (a ) = a 0, a 1,..., a is a fiite sequece the we let a = 0 for >. Defie the L-operator o sequeces to be L(a ) = (b ) where b = a 2 a 1a +1. Call a sequece i-fold log-cocave if L i (a ) is a oegative sequece. So logcocavity i the ordiary sese is 1-fold log-cocavity. Log-cocave sequeces arise i may areas of algebra, combiatorics, ad geometry. See the survey articles of Staley [25] ad Breti [8] for more iformatio. Boros ad Moll [4, page 157] defied (a ) to be ifiitely log-cocave if it is i-fold log-cocave for all i 1. They itroduced this defiitio i cojuctio with the study of a specializatio of the Jacobi polyomials whose coefficiet sequece they cojectured to be ifiitely log-cocave. Kauers ad Paule [16] used a computer algebra pacage to prove this cojecture for ordiary log-cocavity. Sice the coefficiets of these polyomials ca be expressed i terms of biomial coefficiets, Boros ad Moll also made the statemet: Prove that the biomial coefficiets are -logcocave. We will tae this to be a cojecture that the rows of Pascal s triagle are ifiitely log-cocave, although we will later discuss the colums ad other lies. Whe give a fuctio of more tha oe variable, we will subscript ( the L-operator by the parameter which is varyig to form the sequece. So L ) would refer to the operator actig o the sequece ( ). Note that we drop the sequece paretheses 0 for sequeces of biomial coefficiets to improve readability. We ow restate the Boros-Moll cojecture formally. Cojecture 1.1. The sequece ( ) is ifiitely log-cocave for all 0. 0 I the ext sectio, we use a stregtheed versio of log-cocavity ad computer calculatios to verify Cojecture 1.1 for all Umisy ad Yeats [30] set up a correspodece betwee certai symmetric sequeces ad poits i R m. They the described a ifiite regio R R m bouded by hypersurfaces ad such that each sequece correspodig to a poit of R is ifiitely log-cocave. I Sectio 3, we show how our methods ca be used to give a simple derivatio of oe of their mai theorems. We ivestigate ifiite log-cocavity of the colums ad other lies of Pascal s triagle i Sectio 4. Sectio 5 is devoted to two q-aalogues of the biomial coefficiets. For the Gaussia polyomials, we show that certai aalogues of some ifiite log-cocavity cojectures are false while others appear to be true. I cotrast, our secod q-aalogue seems to retai all the log-cocavity properties of the biomial coefficiets. I Sectio 6, after showig why the sequece (h ) 0 of complete homogeeous symmetric is a appropriate aalogue of sequeces of 2

3 biomial coefficiets, we explore its log-cocavity properties. We ed with a sectio of related results ad questios about real-rooted polyomials ad Toeplitz matrices. While oe purpose of this article is to preset our results, we have writte it with two more targets i mid. The first is to covice our audiece that ifiite log-cocavity is a fudametal cocept. We hope that its defiitio as a atural extesio of traditioal log-cocavity helps to mae this case. Our secod aspiratio is to attract others to wor o the subject; to that ed, we have preseted several ope problems. These cojectures each represet fudametal questios i the area, so eve solutios of special cases may be iterestig. 2 Rows of Pascal s triagle Oe of the difficulties with provig the Boros-Moll cojecture is that log-cocavity is ot preserved by the L-operator. For example, the sequece 4, 5, 4 is log-cocave but L(4, 5, 4) = 16, 9, 16 is ot. So we will see a coditio stroger tha log-cocavity which is preserved by L. Give r R, we say that a sequece (a ) is r-factor log-cocave if a 2 ra 1 a +1 (1) for all. Clearly this implies log-cocavity if r 1. We see a r > 1 such that (a ) beig r-factor log-cocave implies that (b ) = L(a ) is as well. Assume the origial sequece is oegative. The expadig rb 1 b +1 b 2 i terms of the a ad rearragig the summads, we see that this is equivalet to provig (r 1)a 2 1a a 1 a 2 a +1 a 4 + ra 2 a (a 2 +1 a a +2 ) + ra 2 1a a +2. By our assumptios, the two expressios with factors of r o the right are oegative, so it suffices to prove the iequality obtaied whe these are dropped. Applyig (1) to the left-had side gives (r 1)a 2 1a a 1 a 2 a +1 r 1 r 2 a r a4. So we will be doe if r r 2 r = 1. Fidig the root r 0 > 1 of the correspodig quadratic equatio fiishes the proof of the first assertio of the followig lemma, while the secod assertio follows easily from the first. 3

4 Lemma 2.1. Let (a ) be a oegative sequece ad let r 0 = (3 + 5)/2. The (a ) beig r 0 -factor log-cocave implies that L(a ) is too. So i this case (a ) is ifiitely log-cocave. Now to prove that ay row of Pascal s triagle is ifiitely log-cocave, oe merely lets a computer fid L( i ) for i up to some boud I. If these sequeces are all oegative ad L( I ) is r0 -factor log-cocave, the the previous lemma shows that this row is ifiitely log-cocave. Usig this techique, we have obtaied the followig theorem. Theorem 2.2. The sequece ( ) 0 is ifiitely log-cocave for all We ote that the ecessary value of I icreases slowly with icreasig. As a example, whe = 100, our techique wors with I = 5, while for = 1000, we eed I = 8. Of course, the method developed i this sectio ca be applied to ay sequece such that L i (a ) is r 0 -factor log-cocave for some i. I particular, it is iterestig to try it o the origial sequece which motivated Boros ad Moll [4] to defie ifiite log-cocavity. They were studyig the polyomial P m (x) = m d l (m)x l (2) l=0 where d l (m) = m j=l 2 j 2m ( 2m 2j m j )( m + j m )( ) j. l Kauers [private commuicatio] has used our method to verify ifiite log-cocavity of the sequece (d l (m)) l 0 for m 129. For such values of m, L 5 l applied to the sequece is r 0 -factor log-cocave. 3 A regio of ifiite log-cocavity Umisy ad Yeats [30] too a differet approach to the Boros-Moll Cojecture as described i the Itroductio. Sice they were motivated by the rows of Pascal s triagle, they oly cosidered real sequeces a 0, a 1,..., a which are symmetric (i that a = a for all ) ad satisfy a 0 = a = 1. Each such sequece correspods to a poit (a 1,..., a m ) R m where m = /2. Their regio, R, whose poits all correspod to ifiitely log-cocave sequeces, is bouded by m parametrically defied hypersurfaces. The parameters are x ad 4

5 d 1, d 2,..., d m ad it will be coveiet to have the otatio s = d i. i=1 We will also eed r 1 = (1 + 5)/2. Note that r 2 1 = r 0. The th hypersurface, 1 < m, is defied as while where H = {(x s 1,..., x s 1, r1 x s, x s +1 +d d +1,..., x s m+d d +1) : x 1, 1 = d 1 > > d > d +2 > > d m > 0}, H m = {(x s 1,..., x s m 1, cx s m 1 ) : x 1, 1 = d 1 > > d m 1 > 0}, c = { r1 if = 2m, 2 if = 2m + 1. Let us say that the correct side of H for 1 m cosists of those poits i R m that ca be obtaied from a poit o H by icreasig the th coordiate. The let R be the regio of all poits i R m havig icreasig coordiates ad lyig o the correct side of H for all. We will show how our method of the previous sectio ca be used to give a simple proof of oe of Umisy ad Yeats mai theorems. But first we eed a modified versio of Lemma 2.1 to tae care of the case whe = 2m + 1. Lemma 3.1. Let a 0, a 1,..., a 2m+1 be a symmetric, oegative sequece such that (i) a 2 r 0a 1 a +1 for < m, ad (ii) a m 2a m 1. The L(a ) has the same properties, which implies that (a ) is ifiitely log-cocave. Proof. Clearly L(a ) is still symmetric. To show that the other two properties persist, ote that i demostratig Lemma 2.1 we actually proved more. I particular, we showed that if equatio (1) holds at idex of the sequece (a ) (with r = r 0 ), the it also holds at idex of the sequece L(a ) provided that the origial sequece is log-cocave. Note that the assumptios of the curret lemma imply log-cocavity of (a ): This is clear at idices m, m + 1 because of coditio (i). Also, usig symmetry ad multiplyig coditio (ii) by a m gives a 2 m 2a m 1 a m = 2a m 1 a m+1 (ad symmetrically for = m + 1). 5

6 So ow we ow that coditio (i) is also true for L(a ). As for coditio (ii), usig symmetry we see that we eed to prove a 2 m a m 1 a m 2 ( a 2 m 1 a m 2 a m ). Rearragig terms ad droppig oe of them shows that it suffices to demostrate 2a 2 m 1 + a m 1 a m a 2 m. But this is true because of (ii), ad we are doe. Theorem 3.2 ([30]). Ay sequece correspodig to a poit of R is ifiitely logcocave. Proof. It suffices to show that the sequece satisfies the hypotheses of Lemma 2.1 whe = 2m, or Lemma 3.1 whe = 2m + 1. Suppose first that < m. Beig o the correct side of H implies that there are values of the parameters such that a 2 (r 1 x s ) 2 = r 2 1x (s 1+d )+(s +1 d +1 ) = r 0 a 1 a +1. Thus we have the ecessary iequalities for this rage of. If = m the we ca use a argumet as i the previous paragraph if = 2m. If = 2m + 1, the beig o the correct side of H m implies that a m 2x s m 1 = 2a m 1. This is precisely coditio (ii) of Lemma 3.1, which fiishes the proof. 4 Colums ad other lies of Pascal s triagle While we have treated Boros ad Moll s statemet about the ifiite log-cocavity of the biomial coefficiets to be a statemet about the rows of Pascal s triagle, their wordig also suggests a examiatio of the colums. Cojecture 4.1. The sequece ( ) is ifiitely log-cocave for all fixed 0. We will give two pieces of evidece for this cojecture. Oe is a demostratio that various colums correspodig to small values of are ifiitely log-cocave. Aother is a proof that L( i ) is oegative for certai values of i ad all. Propositio 4.2. The sequece ( ) is ifiitely log-cocave for

7 Proof. Whe = 0 we have, for all i 1, ( ) L i = (1, 0, 0, 0,...). 0 For = 1 we obtai ( ) L = (1, 1, 1,... ) 1 so ifiite log-cocavity follows from the = 0 case. The sequece whe = 2 is a fixed poit of the L-operator, agai implyig ifiite log-cocavity. I what follows, we use the otatio L(a ) for the th elemet of the sequece L(a ), ad similarly for L ad L. Propositio 4.3. The sequece L i ( ) is oegative for all ad for 0 i 4. Proof. By the previous propositio, we oly eed to chec 3. Usig the expressio for a biomial coefficiet i terms of factorials, it is easy to derive the followig expressios: ( ) L = 1 ( )( ) 1 ad L 2 ( ) = 2 2 ( 1) ( ) 2 ( )( ). 1 2 With a little more wor, oe ca show that L 3 ( ) ca be expressed as a product of oegative factors times the polyomial (4 6) 2 ( ) 2. To show that this is oegative, we write = + m for m 0 to get (4 6)m 2 + ( )m + (3 2 6). But the coefficiets of the powers of m are all positive for 3, so we are doe with the case i = 3. Whe i = 4, we follow the same procedure, oly ow the polyomial i m has coefficiets which are polyomials i up to degree 7. For example, the coefficiet of m 3 is , , , , To mae sure this is oegative for itegral 3, oe rewrites the polyomial as ( ,248) 5 + (25, ,296) 2 + (16, ), fids the smallest such that each of the factors i paretheses is oegative from this value o, ad the checs ay remaiig by direct substitutio. 7

8 Kauers ad Paule [16] proved that the rows of Pascal s triagle are i-fold logcocave for i 5. Kauers [private commuicatio] has used their techiques to cofirm Propositio 4.3 ad to also chec the case i = 5 for the colums. For the latter case, Kauers used a computer to determie (L 5 ( ) ) ( ) 2 5 (3) explicitly, which is just a ratioal fuctio i ad. He the showed that (3) is oegative by meas of cylidrical algebraic decompositio. We refer the iterested reader to [16] ad the refereces therei for more iformatio o such techiques. More geerally, we ca loo at a arbitrary lie i Pascal s triagle, i.e., cosider the sequece ( +mu +mv ) m 0. The uimodality ad (1-fold) log-cocavity of such sequeces has bee ivestigated i [3, 27, 28, 29]. We do ot require that u ad v be coprime, so such sequeces eed ot cotai all of the biomial coefficiets i which a geometric lie would itersect Pascal s triagle, e.g., a sequece such as ( 0 ) (, ) ( 2, ) 4,... would be icluded. By lettig u < 0, oe ca get a fiite trucatio of a colum. For example, if = 5, = 3, u = 1, ad v = 0 the we get the sequece ( ) 5, 3 ( ) 4, 3 ( ) 3 3 which is ot eve 2-fold log-cocave. So we will oly cosider u 0. Also ( ) ( ) + mu + mu = + mv + m(u v) so we ca also assume v 0. We offer the followig cojecture, which icludes Cojecture 1.1 as a special case. Cojecture ) 4.4. Suppose that u ad v are distict oegative itegers. The is ifiitely log-cocave for all 0 if ad oly if u < v or v = 0. ( +mu mv m 0 We first give a quic proof of the oly if directio. Supposig that u > v 1, we cosider the sequece ( ) ( ) ( ) 0 u 2u,,,... 0 v 2v obtaied whe = 0. We claim that this sequece is ot eve log-cocave ad that log-cocavity fails at the secod term. Ideed, the fact that ( ) u 2 ( v < 2u 2v) follows immediately from the idetity ( )( ) ( )( ) ( )( ) ( )( ) ( ) u u u u u u u u 2u =, 0 2v 1 2v 1 v v 2v 0 2v 8

9 which is a special case of Vadermode s Covolutio. The proof just give shows that subsequeces of the colums of Pascal s triagle are the oly ifiite sequeces of the form ( ) +mu that ca possibly be ifiitely mv m 0 log-cocave. We also ote that the previous cojecture says othig about what happes o the diagoal u = v. Of course, the case u = v = 1 is Cojecture 4.1. For other diagoal values, the evidece is coflictig. Oe ca show by computer that ( ) +mu is ot 4-fold log-cocave for = 2 ad ay 2 u 500. However, mu m 0 this is the oly ow value of for which ( ) +mu is ot a ifiitely log-cocave mu m 0 sequece for some u 1. We coclude this sectio by offerig cosiderable computatioal evidece i favor of the if directio of Cojecture 4.4. Theorem 2.2 provides such evidece whe u = 0 ad v = 1. Sice all other sequeces with u < v have a fiite umber of ozero etries, we ca use the r 0 -factor log-cocavity techique for these sequeces as well. For all 500, 2 v 20 ad 0 u < v, we have checed that ( ) +mu mv m 0 is ifiitely log-cocave. 5 q-aalogues This sectio will be devoted to discussig two q-aalogues of biomial coefficiets. For the Gaussia polyomials, we will see that the correspodig geeralizatio of Cojecture 1.1 is false, ad we show oe exact reaso why it fails. I cotrast, the correspodig geeralizatio of Cojecture 4.1 appears to be true. This shows how delicate these cojectures are ad may i part explai why they seem to be difficult to prove. After itroducig our secod q-aalogue, we cojecture that the correspodig geeralizatios of Cojectures 1.1, 4.1 ad 4.4 are all true. This secod q-aalogue arises i the study of quatum groups; see, for example, the boos of Jatze [15] ad Majid [21]. Let q be a variable ad cosider a polyomial f(q) R[q]. Call f(q) q-oegative if all the coefficiets of f(q) are oegative. Apply the L-operator to sequeces of polyomials (f (q)) i the obvious way. Call such a sequece q-logcocave if L(f (q)) is a sequece of q-oegative polyomials, with i-fold q-logcocavity ad ifiite q-log-cocavity defied similarly. We will be particularly iterested i the Gaussia polyomials. The stadard q-aalogue of the oegative iteger is [] = [] q = 1 q 1 q = 1 + q + q2 + + q 1. The, for 0, the Gaussia polyomials or q-biomial coefficiets are defied 9

10 as [ ] = [ ] = q [] q! [] q![ ] q! where [] q! = [1] q [2] q [] q. For more iformatio, icludig proofs of the assertios made i the ext paragraph, see the boo of Adrews [2]. Clearly substitutig q = 1 gives [ ]1 = ( ). Also, it is well ow that the Gaussia polyomials are ideed q-oegative polyomials. I fact, they have various combiatorial iterpretatios, oe of which we will eed. A (iteger) partitio of is a wealy decreasig positive iteger sequece λ = (λ 1, λ 2,..., λ l ) such that λ def = i λ i =. The λ i are called parts. For otatioal coveiece, if a part is repeated r times i a partitio λ the we will deote this by writig r i the sequece for λ. We say that λ fits iside a s t box if λ 1 t ad l s. Deote the set of all such partitios by P (s, t). It is well ow, ad easy to prove by iductio o, that [ ] = q λ. (4) λ P (,) We are almost ready to prove that the sequece ([ ]) is ot ifiitely q-logcocave. I fact, we will show it is ot eve 2-fold q-log-cocave. First we eed a 0 lemma. I it, we use mit f(q) to deote the ozero term of least degree i f(q). ([ Lemma 5.1. Let L ]) = B (q). The for /2, { q if < /2, mit B (q) = 2q if = /2. Proof. Sice B (q) = [ ] 2 [ ][ 1 +1] it suffices to prove, i view of (4), the followig two statemets. If i ad (λ, µ) P ( + 1, 1) P ( 1, + 1) with λ + µ = i, the (λ, µ) P (, ) 2. Furthermore, the umber of elemets i P (, ) 2 P ( + 1, 1) P ( 1, + 1) is 0 or 1 or 2 depedig o whether i < or i = < /2 or i = = /2, respectively. The first statemet is a easy cosequece of λ + µ = i. A similar argumet wors for the i < case of the secod statemet. If i = the the pair ((), ) is i the differece ad if i = = /2 the the pair (, (1 )) is as well. Propositio 5.2. Let L 2 ([ ]) = C (q). The for 2 ad = /2 we have Cosequetly, ([ ]) 0 mit C (q) = q 2. is ot 2-fold q-log-cocave. 10

11 Proof. The proofs for eve ad odd are similar, so we will oly do the former. So suppose = 2 ad cosider C (q) = B (q) 2 B 1 (q)b +1 (q) = B (q) 2 B 1 (q) 2. By the previous lemma mit B (q) 2 = 4q 2 ad mit B 1 (q) 2 = q 2 2. Thus mit C (q) = q 2 2 = q 2 as desired. After what we have just proved, it may seem surprisig that the followig cojecture, which is a q-aalogue of Cojecture 4.1, does seem to hold. Cojecture 5.3. The sequece ([ ]) is ifiitely q-log-cocave for all fixed 0. As evidece, we will prove a q-aalogue of Propositio 4.2 ad commet o Propositio 4.3 i this settig. Propositio 5.4. The sequece ([ ]) is ifiitely q-log-cocave for 0 2. Proof. Whe = 0 oe has the same sequece as whe q = 1. Whe = 1 we claim that ([ ]) L = (1, q, q 2, q 3,...). 1 Ideed, [] 2 [ 1][ + 1] = (1 q ) 2 (1 q 1 )(1 q +1 ) (1 q) 2 = q 1 2q + q +1 (1 q) 2 = q 1 (ad recall that the sequece starts at = 1). It follows that ([ ]) L i = (1, 0, 0, 0,...) 1 for i 2. For = 2, the maipulatios are much lie those i the previous paragraph. Usig iductio o i, we obtai ([ ]) [ ] L i = q (2i 1)( 2) 2 2 for i 0. This completes the proof of the last case of the propositio. 11

12 If we ow cosider arbitrary it is ot hard to show, usig algebraic maipulatios lie those i the proof just give, that ([ ]) [ ][ ] L = q. (5) [] 1 These are, up to a power of q, the q-narayaa umbers. They were itroduced by Fürliger ad Hofbauer [13] ad are cotaied i a specializatio of a result of MacMaho [20, page 1429] which was stated without proof. They were further studied by Brädé [5]. As show i the refereces just cited, these polyomials are the geeratig fuctios for a umber of differet families of combiatorial objects. Thus they are q-oegative. More computatios show that L 2 ([ ]) [ ] 2 [ ][ ] = q3 3 [2]. (6) [] 2 [ 1] 1 2 It is ot clear that these polyomials are q-oegative, although they must be if Cojecture 5.3 is true. Furthermore, whe q = 1, the triagle made as ad vary is ot i Sloae s Ecyclopedia [24] (although it has ow bee submitted). We expect that these itegers ad polyomials have iterestig, yet to be discovered, properties. We coclude our discussio of the Gaussia polyomials by cosiderig the sequece ([ ]) + mu (7) mv for oegative itegers u ad v, as we did i Sectio 4 for the biomial coefficiets. Whe u > v the sequece has a ifiite umber of ozero etries. We ca use (4) to show that the highest degree term i [ ] +u 2 [ v +2u ] 2v has coefficiet 1, so the sequece (7) is ot eve q-log-cocave. Whe u < v, it seems to be the case that the sequece is ot 2-fold q-log-cocave, as show for the rows i Propositio 5.2. Whe u = v, the evidece is coflictig, reflectig the behavior of the biomial coefficiets. Sice settig q = 1 i [ ] ( +mu mu yields +mu ) ([ mu, we ow that 2+mu ]) mu m 0 is ot always 4-fold q-log-cocave. It also traspires that the case = 3 is ot always 5-fold q-log-cocave. We have ot ecoutered other values of that fail to yield a q-log-cocave sequece whe u = v. While the variety of behavior of the Gaussia polyomials is iterestig, it would be desirable to have a q-aalogue that better reflects the behavior of the biomial coefficiets. A q-aalogue that arises i the study of quatum groups serves this purpose. Let us replace the previous q-aalogue of the oegative iteger with m 0 12

13 the expressio = q q q q 1 = q1 + q 3 + q q 1. From this, we obtai a q-aalogue of the biomial coefficiets by proceedig as for the Gaussia polyomials: for 0, we defie! =!! where! = 1 2. Lettig q 1 i ( gives ), ad a straightforward calculatio shows that [ ] = 1 q 2 q 2. (8) So is, i geeral, a Lauret polyomial i q with oegative coefficiets. Our defiitios of q-oegativity ad q-log-cocavity for polyomials i q exted to Lauret polyomials i the obvious way. We offer the followig geeralizatios of Cojectures 1.1, 4.1 ad 4.4. Cojecture 5.5. (a) The row sequece ( ) is ifiitely q-log-cocave for all 0. 0 (b) The colum sequece ( ) is ifiitely q-log-cocave for all fixed 0. (c) For all itegers 0 u < v, the sequece ( ) +mu is ifiitely q-log-cocave mv m 0 for all 0. Several remars are i order. Suppose that for f(g), g(q) R[q, q 1 ], we say f(q) g(q) if g(q) f(q) is q-oegative. The the proofs of Lemmas 2.1 ad 3.1 wor equally well if the a i s are Lauret polyomials ad we replace the term logcocave by q-log-cocave. Usig these lemmas, we have verified Cojecture 5.5(a) for all 53. Eve though (a) is a special case of (c), we state it separately sice (a) is the q-geeralizatio of the Boros-Moll cojecture, the primary motivatio for this paper. As evidece for Cojecture 5.5(b), it is ot hard to prove the appropriate aalogue of Propositios 4.2 ad 5.4, i.e. that the sequece is ifiitely q-logcocave for all 0 2. To obtai the expressios for L ( ) ad L 2 ( ), tae equatios (5) ad (6), replace all square bracets by agle bracets ad replace each the terms q ad q 3 3 by the umber 1. 13

14 Cojecture 5.5(c) has bee verified for all 24 with v 10. Whe u > v, we ca use (8) to show that the lowest degree term i +u 2 v +2u 2v has coefficiet 1, so the sequece is ot eve q-log-cocave. Whe u = v, the quatum groups aalogue has exactly the same behavior as we observed above for the Gaussia polyomials. 6 Symmetric fuctios We ow tur our attetio to symmetric fuctios. We will demostrate that the complete homogeeous symmetric fuctios (h ) 0 are a atural aalogue of the rows ad colums of Pascal s triagle. We show that the sequece (h ) 0 is i- fold log-cocave i the appropriate sese for i 3, but ot 4-fold log-cocave. Lie the results of Sectio 5, this result uderlies the difficulties ad subtleties of Cojectures 1.1 ad 4.1. I particular, it shows that ay proof of Cojecture 1.1 or Cojecture 4.1 would eed to use techiques that do ot carry over to the sequece (h ) 0. For a more detailed expositio of the bacgroud material below, we refer the reader to the texts of Fulto [12], Macdoald [19], Saga [23] or Staley [26]. Let x = {x 1, x 2,...} be a coutably ifiite set of variables. For each 0, the elemets of the symmetric group S act o formal power series f(x) R[[x]] by permutatio of variables (where x i is left fixed if i > ). The algebra of symmetric fuctios, Λ(x), is the set of all series left fixed by all symmetric groups ad of bouded (total) degree. The vector space of symmetric fuctios homogeeous of degree has dimesio equal to the umber of partitios λ = (λ 1,..., λ l ) of. We will be iterested i three bases for this vector space. The moomial symmetric fuctio correspodig to λ, m λ = m λ (x), is obtaied by symmetrizig the moomial x λ 1 1 x λ l l. The th complete homogeeous symmetric fuctio, h, is the sum of all moomials of degree. For partitios, we the defie h λ = h λ1 h λl. Fially, the Schur fuctio correspodig to λ is s λ = det(h λi i+j) 1 i,j l. We remar that this determiat is a mior of the Toeplitz matrix for the sequece (h ). We will have more to say about Toeplitz matrices i the ext sectio. Our iterest will be i the sequece just metioed (h ) 0. Let h (1 ) deote the iteger obtaied by substitutig x 1 = = x = 1 ad x i = 0 for i > ito h = h (x). The h (1 ) = ( ) + 1 (the umber of ways of choosig thigs from 14

15 thigs with repetitio) ad so the above sequece becomes a colum of Pascal s triagle. By the same toe h (1 ) = ( ) 1 ad so the sequece becomes a row. We will ow collect the results from the theory of symmetric fuctios which we will eed. Partially order partitios by domiace where λ µ if ad oly if for every i 1 we have λ λ i µ µ i. Also, if {b λ } is ay basis of Λ(x) ad f Λ(x) the we let [b λ ]f deote the coefficiet of the basis elemet b λ i the expasio of f i this basis. First we have a simple cosequece of Youg s Rule. Theorem 6.1. For ay partitios λ, µ we have [m µ ]s λ is a oegative iteger. I particular, { 1 if µ = λ, [m µ ]s λ = 0 if µ λ. Let λ + µ deote the compoetwise sum (λ 1 + µ 1, λ 2 + µ 2,...). The ext result follows from the Littlewood-Richardso Rule ad iductio. is a o- Theorem 6.2. For ay partitios λ 1,..., λ r ad µ we have [s µ ]s λ 1 s λ r egative iteger. I particular, { 1 if µ = λ [s µ ]s λ 1 s λ r = λ r, 0 if µ λ λ r. Because of this result we call λ λ r the domiat partitio for s λ 1 s λ r. Fially, we eed a result of Kirillov [17] about the product of Schur fuctios, which was proved bijectively by Kleber [18] ad Fulme ad Kleber [11]. This result ca be obtaied by applyig the Desaot-Jacobi Idetity also ow as Dodgso s codesatio formula to the Jacobi-Trudi matrix for s r+1. Note that, to improve readability, we drop the sequece paretheses whe a sequece appears as a subscript. Theorem 6.3 ([11, 17, 18]). For positive itegers, r we have (s r) 2 s ( 1) rs (+1) r = s r 1s r+1. To state our results, we eed a few more defiitios. If b λ is a basis for Λ(x) ad f Λ(x) the we say f is b λ -oegative if [b λ ]f 0 for all partitios λ. Note that m λ -oegativity is the atural geeralizatio to may variables of the q-oegativity defiitio for R[q]. Also ote that s λ -oegativity implies m λ - oegativity by Theorem 6.1. Theorem 6.4. The sequece L i (h ) is s λ -oegative for 0 i 3. sequece L 4 (h ) is ot m λ -oegative. But the 15

16 Proof. From the defiitio of the Schur fuctio we have L 0 (h ) = h = s ad L 1 (h ) = (h ) 2 h 1 h +1 = s 2. Now Theorem 6.3 immediately gives L 2 (h ) = (s 2) 2 s ( 1) 2s (+1) 2 = s s 3 which is s λ -oegative by the first part of Theorem 6.2. Usig Theorem 6.3 twice gives L 3 (h ) = (s ) 2 (s 3) 2 s 1 s ( 1) 3s +1 s (+1) 3 = (s ) 2 (s 3) 2 (s ) 2 s ( 1) 3s (+1) 3 + (s ) 2 s ( 1) 3s (+1) 3 s 1 s ( 1) 3s +1 s (+1) 3 = (s ) 2 s 2s 4 + s ( 1) 3s 2s (+1) 3 which is agai s λ -oegative. This fiishes the cases 0 i 3. We ow assume 2. Computig L 4 (h ) from the expressio for L 3 (h ) gives the sum of the terms i the left colum below. The right colum gives the domiat partitio for each term, as determied by Theorem (s ) 4 (s 2) 2 (s 4) 2 (8, 4, 2, 2) +2(s ) 2 (s 2) 2 s 4s ( 1) 3s (+1) 3 (7, 5, 3, ) +(s ( 1) 3) 2 (s 2) 2 (s (+1) 3) 2 (6, 6, 4) (s 1 ) 2 s ( 1) 2s ( 1) 4(s +1 ) 2 s (+1) 2s (+1) 4 (8, 4, 2, 2) (s 1 ) 2 s ( 1) 2s ( 1) 4s 3s (+1) 2s (+2) 3 (7 1, 5 + 1, 3 + 1, 1) s ( 2) 3s ( 1) 2s 3(s +1 ) 2 s (+1) 2s (+1) 4 (7 + 1, 5 1, 3 1, + 1) s ( 2) 3s ( 1) 2(s 3) 2 s (+1) 2s (+2) 3 (6, 6, 4) Now cosider λ = (7 + 1, 5 1, 3 1, + 1), the domiat partitio for the peultimate term above. Observe that if µ is the domiat partitio for ay other term, the λ µ. So, by the secod part of Theorem 6.2, s λ appears i the Schurbasis expasio for L 4 (h ) with coefficiet 1. It the follows from the secod part of Theorem 6.1, that the coefficiet of m λ is 1 as well. 7 Real roots ad Toeplitz matrices We ow cosider two other (almost equivalet) settigs where, i cotrast to the results of the previous sectio, Cojecture 1.1 does seem to geeralize. I fact, this may be the right level of geerality to fid a proof. 16

17 Let (a ) = a 0, a 1,..., a be a fiite sequece of oegative real umbers. It was show by Isaac Newto that if all the roots of the polyomial p[a ] def = a 0 + a 1 x + a x are real, the the sequece (a ) is log-cocave. For example, sice the polyomial (1 + x) has oly real roots, the th row of Pascal s triagle is log-cocave. It is atural to as if the real-rootedess property is preserved by the L-operator. The literature icludes a umber of results about operatios o polyomials which preserve real-rootedess; for example, see [6, 7, 8, 22, 31, 32]. Cojecture 7.1. Let (a ) be a fiite sequece of oegative real umbers. If p[a ] has oly real roots the the same is true of p[l(a )]. This cojecture is due idepedetly to Richard Staley [private commuicatio]. It is also oe of a umber of related cojectures made by Steve Fis [10]. If true, Cojecture 7.1 would immediately imply the origial Boros-Moll Cojecture. As evidece for the cojecture, we have verified it by computer for a large umber of radomly chose real-rooted polyomials. We have also checed that p[l( i ) ] has oly real roots for all i 10 ad 40. It is iterestig to ote that Boros ad Moll s polyomial P m (x) i equatio (2) does ot have real roots eve for m = 2. So if the correspodig sequece is ifiitely log-cocave the it must be so for some other reaso. Alog with the rows of Pascal s triagle, it appears that applyig L to the other fiite lies we were cosiderig i Sectio 4 also yields sequeces with real-rooted geeratig fuctios. So we mae the followig cojecture which implies the if directio of Cojecture 4.4. Cojecture 7.2. For 0 u < v, the polyomial p[l i m( ( ) +mu mv )] has oly real roots for all i 0. We have verified this assertio for all 24 with i 10 ad v 10. I fact, it follows from a theorem of Yu [33] that the cojecture holds for i = 0 ad all 0 u < v. So it will suffice to prove Cojecture 7.1 to obtai this result for all i. We ca obtai a matrix-theoretic perspective o problems of real-rootedess via the followig reowed result of Aisse, Schoeberg ad Whitey [1]. A matrix A is said to be totally oegative if every mior of A is oegative. We ca associate with ay sequece (a ) a correspodig (ifiite) Toeplitz matrix A = (a j i ) i,j 0. I comparig the ext theorem to Newto s result, ote that for a realrooted polyomial p[a ] the roots beig opositive is equivalet to the sequece (a ) beig oegative. Theorem 7.3 ([1]). Let (a ) be a fiite sequece of real umbers. The every root of p[a ] is a opositive real umber if ad oly if the Toeplitz matrix (a j i ) i,j 0 is totally oegative. 17

18 To mae a coectio with the L-operator, ote that a 2 a 1 a +1 = a a +1 a 1 a, which is a mior of the Toeplitz matrix A = (a j i ) i,j 0. Call such a mior adjacet sice its etries are adjacet i A. Now, for a arbitrary ifiite matrix A = (a i,j ) i,j 0, let us defie the ifiite matrix L(A) by ) ( a L(A) = i,j a i,j+1 a i+1,j a i+1,j+1. i,j 0 Note that if A is the Toeplitz matrix of (a ) the L(A) is the Toeplitz matrix of L(a ). Usig Theorem 7.3, Cojecture 7.1 ca ow be stregtheed as follows. Cojecture 7.4. For a sequece (a ) of real umbers, if A = (a j i ) i,j 0 is totally oegative the L(A) is also totally oegative. Note that if (a ) is fiite, the Cojecture 7.4 is equivalet to Cojecture 7.1. As regards evidece for Cojecture 7.4, cosider a arbitrary -by- matrix A = (a i,j ) i,j=1. For fiite matrices, L(A) is defied i the obvious way to be the ( 1)- by-( 1) matrix cosistig of the 2-by-2 adjacet miors of A. I [9, Theorem 6.5], Fallat, Herma, Gehtma, ad Johso show that for 4, L(A) is totally oegative wheever A is. However, for = 5, a example from their paper ca be modified to show that if A = 1 t t t t t 2 0 t 2 t 3 + 2t 1 + 4t 2 2t 3 + t 0 0 t 2 2t 3 + 2t t 4 + 2t t 0 0 t 2 t 3 + t t 2 the A is totally oegative for t 0, but L(A) is ot totally oegative for sufficietly large t (t 2 will suffice). We coclude that the Toeplitz structure would be importat to ay affirmative aswer to Cojecture 7.4. We fiish our discussio of the matrix-theoretic perspective with a positive result similar i flavor to Cojecture 7.4. Propositio 7.5. If A is a fiite square matrix that is positive semidefiite, the L(A) is also positive semidefiite. Proof. The ey idea is to costruct the secod compoud matrix C 2 (A) of A, which is the array of all 2-by-2 miors of A, arraged lexicographically accordig to the row ad colum idices of the miors [14]. 18

19 We claim that if A is positive semidefiite, the so is C 2 (A). Ideed, sice the compoud operatio preserves multiplicatio ad iverses, the eigevalues of C 2 (A) are equal to the eigevalues of C 2 (J), where J is the Jorda form of A. If J is upper-triagular ad has diagoal etries λ 1, λ 2,..., λ, the we see that C 2 (J) is upper-triagular with diagoal etries λ i λ j for all i < j. Sice the λ i s are all oegative, so too are the eigevalues of C 2 (J), implyig that C 2 (A) is positive semidefiite. Fially, sice L(A) is a pricipal submatrix of C 2 (A), L(A) is itself positive semidefiite. Acowledgemets. We tha Bodo Lass for suggestig that we approach Cojecture 1.1 from the poit-of-view of real roots of polyomials. Sectio 7 also beefited from iterestig discussios with Charles R. Johso. Refereces [1] Aisse, M., Schoeberg, I. J., ad Whitey, A. M. O the geeratig fuctios of totally positive sequeces. I. J. Aalyse Math. 2 (1952), [2] Adrews, G. E. The theory of partitios. Cambridge Mathematical Library. Cambridge Uiversity Press, Cambridge, Reprit of the 1976 origial. [3] Belbachir, H., Becherif, F., ad Szalay, L. Uimodality of certai sequeces coected with biomial coefficiets. J. Iteger Seq. 10, 2 (2007), Article , 9 pp. (electroic). [4] Boros, G., ad Moll, V. Irresistible itegrals. Cambridge Uiversity Press, Cambridge, Symbolics, aalysis ad experimets i the evaluatio of itegrals. [5] Brädé, P. q-narayaa umbers ad the flag h-vector of J(2 ). Discrete Math. 281, 1-3 (2004), [6] Brädé, P. O liear trasformatios preservig the Pólya frequecy property. Tras. Amer. Math. Soc. 358, 8 (2006), [7] Breti, F. Uimodal, log-cocave ad Pólya frequecy sequeces i combiatorics. Mem. Amer. Math. Soc. 81, 413 (1989), viii+106. [8] Breti, F. Log-cocave ad uimodal sequeces i algebra, combiatorics, ad geometry: a update. I Jerusalem combiatorics 93, vol. 178 of Cotemp. Math. Amer. Math. Soc., Providece, RI, 1994, pp

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