ON MULTIPLE AND INFINITE LOG-CONCAVITY

Transcription

1 ON MULTIPLE AND INFINITE LOG-CONCAVITY LUIS A. MEDINA AND ARMIN STRAUB Abtract. Following Boro Moll, a equence (a n) i m-log-concave if L j (a n) 0 for all j = 0,,..., m. Here, L i the operator defined by L(a n) = a 2 n a n a n+. By a criterion of Craven Corda and McNamara Sagan it i known that a equence i -log-concave if it atifie the tronger inequality a 2 k ra k a k+ for large enough r. On the other hand, a recent reult of Brändén how that -log-concave equence include equence whoe generating polynomial ha only negative real root. In thi paper, we invetigate equence which are fixed by a power of the operator L and are therefore - log-concave for a very different reaon. Surpriingly, we find that equence fixed by the non-linear operator L and L 2 are, in fact, characterized by a linear 4-term recurrence. In a final conjectural part, we oberve that poitive equence appear to become -log-concave if convoluted with themelve a finite number of time.. Introduction A equence (a n ) i aid to be log-concave if a 2 n a n a n+ for all n. If all term of the equence are poitive, then log-concavity implie unimodality. For a very nice introduction and many example of both unimodal and log-concave equence we refer to [7]. Following Boro and Moll [2], we denote with L the operator which end a equence (a n ) to the equence (a 2 n a n a n+ ). Then (a n ) i log-concave if and only if L(a n ) 0. Similarly, the equence (a n ) i aid to be m-log-concave if L j (a n ) 0 for all j = 0,,..., m. If (a n ) i m-log-concave for all m > 0, then it i aid to be -log-concave. Often, we will conider the generating function f(x) = n 0 a nx n. In that cae, we write L[f](x) = n 0 (a2 n a n a n+ )x n with the undertanding that a = 0. Example.. The equence ( n 2) i fixed by the operator L becaue ( ) 2 ( )( ) ( ) n n n + n = Since the equence i nonnegative, it i therefore -log-concave. In Section 3, we will characterize all equence fixed by L. More generally, a a warm-up problem, it wa conjectured in [2] that the binomial coefficient (either row, that i ( ( n 0), n ( ),..., n ( n), or column, that i n ) ( n, n+ ) ( n, n+2 ) n,..., of Pacal triangle) are -log-concave. That the row of Pacal triangle are indeed -log-concave wa recently proven by Brändén [3] who etablihed, much more generally, that the coefficient of polynomial, all of whoe root are negative Date: May 7, Mathematic Subject Claification. Primary 05A20, 39B2. Key word and phrae. log-concavity, linear recurrence, convolution.

2 2 LUIS A. MEDINA AND ARMIN STRAUB and real, are alway -log-concave. Thi connection of the location of root and log-concavity will be briefly reviewed in Section 2. We remark that, on the other hand, the cae of column of Pacal triangle i till wide open; baed on extenive computation by Kauer and Paule [2] it i only known that they are 5-log-concave. A will alo be reviewed in Section 2, a equence i -log-concave if it atifie the tronger condition a 2 n ra n a n+ for r (3 + 5)/ Thi criterion, due to Craven and Corda [5] a well a McNamara and Sagan [4], generalize to a powerful approach of howing that a pecific equence i -log-concave. On the other hand, there are equence, like the one in Example., that are fixed by the operator L (or one of it power) and are -log-concave for thi reaon. Thi lead u to invetigate the equence fixed by L in Section 3 a well a the equence fixed by L 2 in Section 4. In both cae, we find that, urpriingly, thee equence fixed by non-linear operator are characterized by linear 4-term recurrence with contant coefficient. Thi phenomenon doe not appear to extend to the equence fixed by L m for m > 2. A an application, we ak if thee criteria for -log-concavity can be combined to yield an algorithm that decide, in finite time, whether or not a given finite equence i -log-concave. In the final ection, Section 5, we tart with poitive equence and repeatedly convolute them with themelve. That i to ay, we conider the coefficient of power of a given polynomial. The central limit theorem ugget that, a the exponent increae, the hape of the reulting equence approache the hape of a normal ditribution. One therefore expect the equence to become more and more log-concave. Indeed, we oberve that each equence appear to become - log-concave in a finite number of tep. 2. Review of multiple log-concavity Newton famou and claical theorem on real root tate that if the polynomial p(x) = a 0 + a x a d x d ha only negative real root, then it coefficient (a n ) are log-concave. We refer to [5] for hitoric information and related reult. Remark 2.. In fact, omewhat tronger, Newton theorem implie that the number a n / ( d n) form a log-concave equence. Thi i indeed tronger ince the binomial coefficient are log-concave and the Hadamard product of two log-concave equence i again log-concave. If only log-concavity i deired, the aumption of Newton theorem can be weakened: it i hown in [] that the condition that all root are negative (and therefore real) can be replaced with the condition that all root lie in the ector defined by arg( z) π/3 (equivalently, all root z = a + bi atify both a 0 and b 2 3a 2 ). Brändén [3] recently etablihed the following more general theorem which wa previouly, and independently, conjectured by Stanley, McNamara Sagan [4] and Fik [7]. Theorem 2.2. ([3]) Let p(x) = d n=0 a nx n. If all root of p are negative and real then o are the root of d L[p](x) = (a 2 n a n a n+ )x n. n=0 Here, it i undertood that a = 0 a well a a d+ = 0.

3 ON MULTIPLE AND INFINITE LOG-CONCAVITY 3 Corollary 2.3. If the polynomial p(x) = a 0 + a x a d x d ha only negative real root, then it coefficient (a n ) are -log-concave. Example 2.4. If a n = ( d n), then p(x) = d n=0 a nx n = (x + ) d clearly ha only negative real root. It follow that (row of) the binomial coefficient are -logconcave, confirming the motivating conjecture in [2]. Note that the convere of Corollary 2.3 i not true, a i illutrated by the polynomial P 2 (x) = 3 2 x x + 2 8, which ha non-real root but coefficient which are -log-concave; thi i eaily proved uing the notion of r-factor log-concavity which i reviewed next. We remark that P 2 (x) i one of the Boro Moll polynomial, which occurred in the evaluation of a quartic integral [2], and have been invetigated by many author hence; we only refer to the recent article [4] a well a the reference therein. Neverthele, a illutrated by Theorem 2.8 below, a partial convere to Brändén reult i poible if -log-concavity i replaced by an even tronger property. An important tool for etablihing infinite log-concavity of a pecific equence i the notion of r-factor log-concavity, which ha been introduced and ued by McNamara and Sagan [4]. Let r. A equence (a n ) i r-factor log-concave if a 2 n ra n a n+. In fact, uing lightly different terminology, thi notion ha already been conidered by Craven and Corda in [5]. Let M r be the r-factor log-concave equence. It i hown in [5, Theorem 4.] that () L(M r ) M r+ if and only if r Generalization, for intance to the cae of decreaing equence, appear in [8]. Of particular importance for our purpoe i the following conequence of the containment (). Lemma 2.5. If a equence (a n ) i r-factor log-concave for ome r , then L(a n ) i r-factor log-concave a well. In particular, the equence (a n ) i - log-concave. A illutrated in the next example, thi give rie to a more general approach, ued in [4], to how that a pecific equence i -log-concave. Example 2.6. The polynomial p(x) = + 4x + 6x 2 + 4x 3 ha coefficient which are 2.25-factor log-concave. The polynomial L[p](x) = + 0x + 20x 2 + 6x 3 ha coefficient which are 2.5-factor log-concave, while the coefficient of L 2 [p](x) = + 80x + 240x x 3 are factor log-concave. Lemma 2.5 therefore how that, in fact, the coefficient of p(x) are -log-concave. The concept of r-factor log-concavity of the coefficient of a polynomial i intimately related to the location of zero dicued earlier in thi ection. For intance, the following reult on Hurwitz tability i due to Katkova and Vihnyakova. We refer to [] for further detail, including the cae of lower degree, and reference to related and earlier reult. Here, trong log-concavity, and likewie r-factor trong log-concavity, mean that the defining inequalitie are trict (for intance, a equence (a n ) i r-factor trongly log-concave if a 2 n > ra n a n+ ).

4 4 LUIS A. MEDINA AND ARMIN STRAUB Theorem 2.7. ([]) Let p(x) be a polynomial with poitive coefficient and degree larger than 5. If the coefficient of p(x) are r 0 -factor trongly log-concave, where r i the unique real root of r 3 r 2, then all the root of p(x) have negative real part. A a econd example, we cite the following reult of Kurtz [3], a tated in [6], which may be viewed a a partial convere to Newton theorem on real root. Theorem 2.8. ([3]) Let p(x) be a polynomial with poitive coefficient. If the coefficient of p(x) are 4-factor log-concave, then all the root of p(x) are real (and hence negative). It i alo hown in [3] that 4-factor log-concavity cannot be replaced (4 ε)- factor log-concavity for any ε > Sequence fixed by L A oberved in Example., the equence ( n 2) i fixed by L and therefore -logconcave. In thi ection, we characterize all equence (a n ) n 0 that are fixed by the operator L, that i L(a n ) = (a n ), or, equivalently, (2) a 2 n a n a n+ = a n for all indice n 0 (with the undertanding that a = 0). Remark 3.. Let u note that the, apparently more general, characterization of equence (a n ) n 0 uch that L(a n ) = λ(a n ) for ome number λ reduce to the cae λ =. Indeed, if L(a n ) = λ(a n ) then L(b n ) = (b n ) with b n = a n /λ. Suppoe that (a n ) n 0 i a equence fixed by L. Note that if a m 0 and a m = 0 for ome m > 0, then a m+ = 0 a well. In particular, the equence (a n+m+2 ) n 0 i again fixed by L. In characterizing all equence that are fixed by L, it i therefore no lo of generality to aume that the equence (a n ) ha no internal zero, meaning that if a m = 0 for ome m 0 then a n = 0 for all n m. If (a n ) ha no internal zero, then a 0 = unle (a n ) i the zero equence. Auming that (a n ) i not the zero equence, the value of a then determine (a n ). Example 3.2. With the value k given, let (a n ) n 0 be the unique equence with a = k which i fixed by L and ha no internal zero. When k {0,, 2} thi equence i finitely upported with value (), (, ) and (, 2, 2, ), repectively. On the other hand, if k 3 i an integer, then the correponding equence i infinite and all of it term are poitive integer. Thee claim are not a priori obviou but will be direct conequence of the characterization in Theorem 3.3. The firt few cae, with k = 3, 4, 5, are:, 3, 6, 0, 5, 2, 28, 36,..., 4, 2, 33, 88, 232, 609, 596,..., 5, 20, 76, 285, 065, 3976, 4840,... In the firt cae, correponding to k = 3, the equence i given by a n = ( ) n+2 2, the introductory Example., and in the cae k = 4 we identify the equence a a n = F 2n+3 with F n denoting the Fibonacci number. The alert reader may have noticed that the infinite equence given in Example 3.2 each have a rational generating function. The next reult and it corollary how that thi i alway the cae.

5 ON MULTIPLE AND INFINITE LOG-CONCAVITY 5 Theorem 3.3. Let k be an arbitrary number. Then the equence (a n ) n 0 defined by a n x n = kx + kx 2 x 3 = ( x)( (k )x + x 2 ) n 0 i fixed by L. Note that a 0 =, a = k. Proof. Let S be the (invere) hift operator defined by Sa n = a n, and conider the operator (3) L := ks + ks 2 S 3 = ( S) ( (k ) S + S 2). We note that the rational generating function for (a n ) i equivalent to the fact that the recurrence La n = 0 hold, for all n, with initial condition a 0 = and a = a 2 = 0. The factorization of L, a on the right-hand ide of (3), implie that the equence b n = ( (k ) S + S 2) a n = a n (k ) a n + a n 2 i contant. Since b 0 = a 0 = it follow that b n = for all n 0. In other word, (a n ) atifie the nonhomogeneou recurrence (4) a n (k ) a n + a n 2 = for all n 0. To prove that (a n ) i fixed under L, we need to how that (5) a n (a n ) = a n+ a n for all n 0. Equation (5) clearly hold for n = 0. For the purpoe of induction, aume that (5) hold for ome n. Then a n+ (a n+ ) (4) = a n+ ((k ) a n a n ) = (k ) a n+ a n a n+ a n (5) = (k ) a n+ a n a n (a n ) = a n ((k ) a n+ (a n )) (4) = a n+2 a n, which how that (5) alo hold for n + in place of n. The reult therefore follow by induction. Remark 3.4. Note that the equence (a n ), defined in Theorem 3.3, i C-finite [8]. Moreover, ince C-finite equence form an algebra, the equence (a 2 n) and (a n a n+ ), a well a any linear combination of thee, are again C-finite. In order to obtain an alternative and automatic proof of Theorem 3.3, we can therefore ue the C-finite anatz, recently advertied in [9], to how that a 2 n a n a n+ = a n, thu proving that (a n ) i indeed fixed by L. Theorem 3.3 ha the, poibly urpriing, conequence that the olution of the non-linear three-term recurrence (2) in fact atify a linear four-term recurrence. Corollary 3.5. Let N N { }. Suppoe that the equence (a n ) N n=0 i fixed by L and that a n 0 for all 0 n N. Then, with k = a, (6) a n k(a n a n 2 ) a n 3 = 0

6 6 LUIS A. MEDINA AND ARMIN STRAUB for all n =, 2,..., N, with initial condition a 0 = and a = a 2 = 0. Proof. Recall that if (a n ) i fixed by L and a 0 0, then a 0 = and the value of a determine the initial egment of nonzero term of (a n ). On the other hand, for any value k = a, Theorem 3.3 provide a equence that i fixed by L. It follow that the two equence have to agree for all initial nonzero term. A illutrated by Example 3.2, the equence fixed by L uually have infinite upport. We now determine all equence with finite upport that are fixed by L. Propoition 3.6. Let 3 and r < be integer. If 2r, then (7) p,r (x) = x ( x)( 2 co(2πr/)x + x 2 ) i a degree 3 polynomial, and the coefficient of p,r are fixed by L. Moreover, every finitely upported equence which i fixed by L and ha no internal zero arie in thi way. Proof. Aume that (a n ) i one of the equence of Theorem 3.3 with the property that there i N 0 uch that a N+ = 0. Chooe the minimal uch N. Then (a n ) n 0 = (, a,..., a N,, 0, 0,, a,...). Converely, conider a finite equence (a n ) N n=0 of nonzero term which i fixed by L (o that, in particular, a 0 = a N = ). Then p(x) = a 0 + a x a N x N necearily atifie p(x) x N+3 = ( x)( (a )x + x 2 ). It follow that (a )x + x 2 = (x ζ )(x ζ 2 ) where ζ, ζ 2 are nontrivial -th root of unity, with = N + 3. In fact, one clearly ha ζ 2 = ζ = ζ. Writing ζ = exp(2πir/) for ome r, we thu find ( ) 2πr a = ζ + ζ = 2 co. The claim follow from here. We oberve that the polynomial p,r (x) are palindromic, that i, their coefficient form ymmetric equence. Thi property follow from x 3 p,r (/x) = p,r (x) which i readily verified from (7). While the coefficient of thee polynomial are not in general poitive, we now how they are poitive in the cae r =. In thi cae, the coefficient therefore form an intereting family of equence which are -log-concave; ee Example 3.8 which lit the firt few equence explicitly. Theorem 3.7. Let 3 be an integer. The polynomial (8) p (x) = p, (x) = have the following propertie: x ( x)( 2 co(2π/)x + x 2 ) (a) The polynomial p (x) i palindromic of degree 3. (b) The coefficient of p (x) are poitive. (c) The coefficient of p (x) are fixed by L. (d) The coefficient of p (x) are -log-concave.

7 ON MULTIPLE AND INFINITE LOG-CONCAVITY 7 (e) The coefficient of p (x) are r-factor log-concave if and only if { co(2π/) r, if i even, [ 2 co(π/)], if i odd. 2 Proof. Part (a) i a pecial cae of the palindromicity of the polynomial p,r (x) which wa oberved above. Note that claim (c) wa proved in Propoition 3.6. Together with the poitivity claimed in part (b), thi implie the -log-concavity of part (d). It therefore only remain to how part (b) and (e). Let u prove (b). Since the polynomial p (x) = a 0 + a x a 3 x 3 i palindromic of degree 3, it i ufficient to how that a j > 0 for j = 0,,..., k where k = ( 3)/2. We recall the claical generating function 2zx + x 2 = U n (z)x n of the Chebyhev polynomial U n (z) of the econd kind. Since U n (co(θ)) =, we have in((n+)θ) in(θ) and hence n=0 2 co(θ)x + x 2 = in(θ) (9) p (x) = + x x in(2π/) in((n + )θ)x n n=0 in(2π(n + )/)x n. Note that in(2π/) > 0. The coefficient a j therefore i a poitive linear combination of in(2π(n + )/) for n = 0,,..., j. On the other hand, for n = 0,,..., k, 2π(n + ) 2π(k + ) n=0 π( ) and hence in(2π(n + )/) > 0. Therefore, the coefficient of p (x) are poitive a claimed in (b). To how part (e), recall that (a n ) i r-factor log-concave if and only if a 2 n ra n+ a n 0 for all n =, 2,..., 4. Uing the fact that a 2 n a n+ a n = a n, thi inequality i clearly equivalent to r a 2 n = a n a n+ a n a n. We have already hown that the coefficient a n are poitive and fixed by L. Hence they are log-concave and, a a conequence, unimodal. Becaue the equence (a n ) i ymmetric and unimodal, it maximum i a N, with N = ( 3)/2, and it follow a N a N. that (a n ) i r-factor log-concave if and only if r The value a N can be obtained from the expanion (9). Indeed, writing ζ = e 2πi/, we find that, for n = 0,,..., 3, a n = in(2π/) Im n j= < π [ ζ j+ = in(2π/) Im ζ 2 ζ n ]. ζ

8 8 LUIS A. MEDINA AND ARMIN STRAUB A imple calculation how that [ Im ζ 2 ζ n ] [ ζ + ζ 2 + ζ n+ ζ n+2 ] = Im ζ ζ 2 = in( 2π ) + in( 4π 2π(n+) ) + in( ) in( 2π(n+2) ). 2 2 co(2π/) In the cae when n i N = ( 3)/2, the ine can all be expreed with argument in term of only, and one obtain a N = in(2π/) { in(4π/) 2 2 co(2π/), if i even, 2 in(π/) in(2π/)+in(4π/) 2 2 co(2π/), if i odd. Uing a N a N = + a N, it i now traightforward to verify the claim uing tandard trigonometric identitie. Indeed, ince co(2t) = 2 co(t) 2, we have, by definition of the Chebyhev polynomial U n, { U 3 (t), if i even, a N = 4( t 2 )U (t) 2U 0 (t) U (t) + U 3 (t), if i odd, with t = co(π/), from which the claim i immediate. Example 3.8. The firt few polynomial of Theorem 3.7 are p 3 (x) =, p 4 (x) = + x, a well a p 5 (x) = x + x 2, p 6 (x) = + 2x + 2x 2 + x 3, p 7 (x) = + ( + α)x + α( + α)x 2 + ( + α)x 3 + x 4, α = 2 in ( 3π 4 ), p 8 (x) = + ( + 2)x + (2 + 2)x 2 + (2 + 2)x 3 + ( + 2)x 4 + x 5. In each cae, poitivity of the coefficient, together with the fact that they are fixed under L, how that they are -log-concave. None of thee polynomial ha all their root on the negative real axi. Hence, it i not poible to deduce the -logconcavity of their coefficient uing Brändén reult in form of Corollary 2.3. Remark 3.9. Note that / co(2π/) a. Part (e) of Theorem 3.7 therefore how that, given any ε > 0, there i a finite equence (a n ) which i -log-concave but not ( + ε)-factor log-concave. In particular, let (a n ) be the coefficient of the polynomial p (x) with 6. The above how that (a n ) i fixed by L but i not r-factor log-concave for r > 2. In particular, there i no m > 0 uch that L m (a n ) i r-factor log-concave for r > 2. It i therefore not poible to apply Lemma 2.5 a in Example 2.6 to how that the equence (a n ) i -log-concave. Apart from the pecial cae dicued in Remark 3.9, we have been able to uccefully apply the approach of Example 2.6 to etablih -log-concavity in all the example we have encountered. Thi motivate the next quetion which, in particular, ak whether -log-concavity of a finite equence i decidable. Quetion 3.0. Given a finite poitive equence (a n ), compute L m (a n ) for m =, 2,... until either (a) L m (a n ) ha negative term, or (b) L m (a n ) = λ(a n ) for ome λ > 0, or

9 ON MULTIPLE AND INFINITE LOG-CONCAVITY 9 (c) L m (a n ) i r-factor log-concave for ome r In the cae (a) the equence i (m )-log-concave but not m-log-concave, and in the cae (b) and (c) the equence i -log-concave. Doe thi imple algorithm alway terminate? Or, if thi i not the cae, i there ome other algorithm which determine, in finite time, whether a given finite equence i -log-concave? 4. Sequence fixed by L 2 We now conider an analog of Theorem 3.3 which characterize equence fixed by L 2. No uch characterization appear to exit for equence fixed by L n with n > 2; ee Example 4.3. Note that the rational generating function in Theorem 4. i equivalent to the equence (a n ) atifying the recurrence a n βa n + (β 2 γ)a n 2 a n 3 = 0, with initial condition a 0 = and a = a 2 = a 3 = 0. Theorem 4.. Let β, γ be arbitrary number. Then the equence (a n ) n 0 defined by a n x n = βx + (β 2 γ)x 2 x 3 n 0 i fixed by L 2. Note that a 0 =, a = β, a 2 = γ. Proof. Let α, α 2, α 3 be uch that βx + (β 2 γ)x 2 x 3 = (α x)(α 2 x)(α 3 x). Necearily, α α 2 α 3 =. In the equel, we aume that α, α 2, α 3 are ditinct; the general cae may then be obtained by a limiting argument. Uing the general partial fraction expanion d d α j x = d, α k x α j α k j= k= valid for ditinct α j, we obtain the expanion d α j x = j= In the cae d = 3, we thu have (0) a n = n=0 x n d k= (α α 2 )(α 2 α 3 )(α 3 α ) j= j k α n+ k d j= j k [ α3 α 2 α n+ α j α k. + α α 3 α n+ 2 + α ] 2 α α3 n+. By a traight-forward direct computation, uing α α 2 α 3 =, we find that [ a 2 α3 α 2 n a n a n+ = (α α 2 )(α 2 α 3 )(α 3 α ) α n 2 + α α 3 α2 n 2 + α 2 α α n 2 3 Comparing thi expreion for L(a n ) with (0) for (a n ), it become clear that iterating thi computation to compute L 2 (a n ) will reult in the original equence. In other word, we have hown that L 2 (a n ) = (a n ), a claimed. ].

10 0 LUIS A. MEDINA AND ARMIN STRAUB An alternative and automatic, but arguably le illuminating and rather more computational, proof can be obtained by applying the C-finite anatz [9] a indicated in Remark 3.4. Example 4.2. If β = 2 and γ = 3, then we obtain the equence (a n ) whoe firt few term are given by, 2, 3, 5, 9, 6, 28, 49, 86, 5, 265,... Indeed, thi equence [6, A00534] may be written a a n = (n )/3 k=0 ( ) n k, 2k + and a n count the number of compoition of n into part congruent to or 2 modulo 4. Theorem 4. how that thi equence i fixed by L 2. However, L(a n ) = (,,, 2,, 4,...) i not poitive, o (a n ) i not even log-concave. Example 4.3. After the ucce of Theorem 3.3 and 4., one may be tempted to hope that imilar characterization exit for fixed equence of L m when m > 2. Thi doe not eem to be the cae, a i illutrated by the following example for m = 3. The equence (a n ), characterized by a 0 =, a = 2, a 2 = 5, a 3 = 9 together with the fact that it i fixed by L 3, i given by, 2, 5, 9, 96 5, , ,... Computing everal more term, one can check that (a n ) cannot atify a linear recurrence of mall degree and mall order. For intance, we have found that it doe not atify a recurrence with contant coefficient of order up to 0 (recall that in the cae m = and m = 2 any uch equence atifie a recurrence with contant coefficient of order 3). We alo found that (a n ) doe not atify a recurrence with linear coefficient of order up to 6, or a recurrence with quadratic coefficient of order up to 4. In all the example we have conidered of equence that are fixed by L 3 with initial integral value a in Example 4.3, the reulting equence involved fraction of rapidly increaing ize. We are therefore lead to wonder whether there are example of integer equence fixed by L 3 (but not by L). Quetion 4.4. For n > 2, are there (poitive) integer equence that are fixed by L n, but not by L m for any m < n? 5. Convolution of equence It i a well-known reult of Hoggar [9] that the product of polynomial with poitive and log-concave coefficient again ha log-concave coefficient. Thi i generalized in [0] by Johnon and Goldchmidt who apply their reult to how, for intance, log-concavity of the Stirling number of the econd kind a a equence in the econd parameter. Thi naturally lead one to wonder whether anything intereting can be aid about the log-concavity of the coefficient of the product p(x) q(x) if p(x) and q(x) have m- and n-log-concave coefficient, repectively. The next example, contructed a an application of the previou ection, dampen exceive expectation

11 ON MULTIPLE AND INFINITE LOG-CONCAVITY by demontrating that the product of two polynomial with -log-concave coefficient may fail to be 5-log-concave. Example 5.. Conider the polynomial () p(x) = + ( + 2)x + (2 + 2)x 2 + (2 + 2)x 3 + ( + 2)x 4 + x 5 from Example 3.8. Since it i fixed under L, it coefficient are -log-concave. However, the coefficient of p(x) 2 are 4-log-concave but not 5-log-concave. On the other hand, the coefficient of p(x) n for n = 3, 4,..., 50 are again -log-concave. We remark that light perturbation of the polynomial p(x) in () can be ued to contruct many further example of polynomial with imilar propertie but which are not fixed under L. For intance, the degree 6 polynomial q(x) = p(x) + (x/4) 6 ha the property that it coefficient are 5-log-concave, while the coefficient of q(x) 2 are only 4-log-concave but not 5-log-concave. Note that Example 5. demontrate that, in general, if the coefficient of p(x) λ are m-log-concave for ome m > thi doe not imply that the coefficient of p(x) λ+ are m-log-concave a well. On the other hand, note that each polynomial p(x) = a 0 +a x+ +a d x d with poitive coefficient a j > 0, j = 0,,..., d, induce a probability ditribution on the et {0,,..., d} with probability weight pecified by the coefficient of p(x). Namely, a random variable X i ditributed according to p(x), if, for j = 0,,..., d, Prob(X = j) = a j p(). If X and Y are random variable ditributed according to p(x) and q(x), repectively, then X + Y i ditributed according to the product p(x)q(x). In particular, let X,..., X λ be independent random variable ditributed according to p(x). Then their um X + + X λ i ditributed according to p(x) λ. The central limit theorem therefore ugget that the coefficient of p(x) λ hould eventually become more log-concave a λ increae. Indeed, in all the example we have conidered, we have oberved that the coefficient of p(x) λ become -log-concave for finite λ. Conjecture 5.2. Let p(x) be a polynomial with poitive coefficient, that i p(x) = a 0 + a x + + a d x d with a j > 0 for all j = 0,..., d. Then there exit N uch that, for all λ N, p(x) λ ha coefficient that are -log-concave. Example 5.3. Conider p(x) = +x+x 2. Computer experiment ugget that the coefficient of p λ are -log-concave for λ 0. For 0 λ 500 we have proved thi uing Lemma 2.5 a in Example 2.6. For intance, in the cae λ = 0, one oberve that the coefficient of L 5 [p 0 ] are 9.0-factor log-concave. More generally, it appear that p λ i log-concave for λ (and trongly log-concave for λ 2), 2-log-concave for λ 4, 3-log-concave for λ 7, 4-log-concave for λ 8, 5-logconcave for λ 9, and, a mentioned above, -log-concave for λ 0. Note that thi information i included in Table. Example 5.4. Conider the polynomial p(x) = + x + 2x 2 and note that the coefficient (,, 2) are not log-concave. However, p 3 ha log-concave coefficient, p 0 ha 2-log-concave and p 6 ha 3-log-concave coefficient. Thi information i further extended in Table. In particular, it turn out that p 23 ha -log-concave coefficient. Indeed, the coefficient of L 5 [p 23 ] are 4.23-factor log-concave.

12 2 LUIS A. MEDINA AND ARMIN STRAUB Example 5.5. Let u conider general quadratic polynomial p(x) = a 0 + a x + a 2 x 2. Note that caling a polynomial p(x) λp(x) doe not affect quetion of log-concavity, and neither doe the tranformation p(x) p(λx). Without lo of generality, we may therefore aume a 0 = and a =. In the cae of the polynomial p(x) = + x + ax 2, a {, 2,..., 6}, and value m {, 2,..., 0, }, Table lit the minimal exponent λ uch that p λ ha m-log-concave coefficient x + x x + 2x x + 3x x + 4x x + 5x x + 6x Table. Minimal exponent λ uch that p λ ha m-log-concave coefficient. We invite the reader to oberve the variou looe pattern uggeted by the data contained in Table. Acknowledgement. We wih to thank Danylo Radchenko for intereting and helpful comment. The econd author would like to thank the Max-Planck-Intitute for Mathematic in Bonn, where part of thi work wa completed, for providing wonderful working condition. Reference [] P. Bahl, R. Devitt-Ryder, and T. Nguyen. The location of root of logarithmically concave polynomial. Preprint, 20. [2] G. Boro and V. H. Moll. Irreitible Integral: Symbolic, Analyi and Experiment in the Evaluation of Integral. Cambridge Univerity Pre, [3] P. Brändén. Iterated equence and the geometry of zero. J. Reine Angew. Math., 20(658):5 3, 20. [4] W. Y. C. Chen, A. L. B. Yang, and E. L. F. Zhou. Ratio monotonicity of polynomial derived from nondecreaing equence. Electronic Journal of Combinatoric, 7:N37, 200. [5] T. Craven and G. Corda. Iterated Laguerre and Turán inequalitie. Journal of Inequalitie in Pure and Applied Mathematic, 3(3), [6] G. Corda. Iterated Turán inequalitie and a conjecture of P. Brändén. In P. Brändén, M. Paare, and M. Putinar, editor, Notion of Poitivity and the Geometry of Polynomial, Trend in Mathematic, page Springer Bael, Jan. 20. [7] S. Fik. Quetion about determinant and polynomial. Preprint, Aug Available at: [8] L. Grabarek. Non-Linear Coefficient-Wie Stability and Hyperbolicity Preerving Tranformation. PhD thei, Univerity of Hawai i at Mānoa, 202. [9] S. G. Hoggar. Chromatic polynomial and logarithmic concavity. Journal of Combinatorial Theory, Serie B, 6(3): , June 974. [0] O. Johnon and C. Goldchmidt. Preervation of log-concavity on ummation. ESAIM: Probability and Statitic, 0:206 25, [] O. M. Katkova and A. M. Vihnyakova. A ufficient condition for a polynomial to be table. Journal of Mathematical Analyi and Application, 347():8 89, Nov [2] M. Kauer and P. Paule. A computer proof of Moll log-concavity conjecture. Proc. Amer. Math. Soc., 35: , [3] D. C. Kurtz. A ufficient condition for all the root of a polynomial to be real. Amer. Math. Monthly, 99(3): , Mar. 992.

13 ON MULTIPLE AND INFINITE LOG-CONCAVITY 3 [4] P. R. W. McNamara and B. E. Sagan. Infinite log-concavity: development and conjecture. Dicrete Mathematic & Theoretical Computer Science, Proceeding of FPSAC 2009, page , [5] C. P. Niculecu. A new look at Newton inequalitie. Journal of Inequalitie in Pure and Applied Mathematic, (2), [6] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequence, 203. Publihed electronically at [7] R. Stanley. Log-concave and unimodal equence in Algebra, Combinatoric and Geometry. Graph theory and it application: Eat and Wet (Jinan, 986). Ann. New York Acad. Sci., 576: , 989. [8] D. Zeilberger. A holonomic ytem approach to pecial function identitie. Journal of Computational and Applied Mathematic, 32(3):32 368, 990. [9] D. Zeilberger. The C-finite anatz. The Ramanujan Journal, 3(-2):23 32, June 203. Department of Mathematic, Univerity of Puerto Rico, Box 70377, San Juan, PR , Puerto Rico addre: lui.medina7@upr.edu Department of Mathematic, Univerity of Illinoi at Urbana-Champaign, 409 W. Green St, Urbana, IL 680, United State Current addre: Max-Planck-Intitut für Mathematik, Vivatgae 7, 53 Bonn, Germany addre: atraub@illinoi.edu