iterated sequences and the geometry of zeros

J. reine angew. Math. 658 (2011), 115—131

DOI 10.1515/CRELLE.2011.063

Journal fur die reine undangewandte Mathematik( Walter de Gruyter

Berlin � New York 2011

Iterated sequences and the geometry of zeros

Dedicated to the memory of Julius Borcea

By Petter Branden at Stockholm

Abstract. We study the e¤ect on the zeros of generating functions of sequencesunder certain non-linear transformations. Characterizations of Polya–Schur type are givenof the transformations that preserve the property of having only real and non-positivezeros. In particular, if a polynomial a0 þ a1z þ � � � þ anzn has only real and non-positivezeros, then so does the polynomial a2

0 þ ða21 � a0a2Þz þ � � � þ ða2

n�1 � an�2anÞzn�1 þ a2nzn.

This confirms a conjecture of Fisk, McNamara–Sagan and Stanley, respectively. A con-sequence is that if a polynomial has only real and non-positive zeros, then its Taylor coef-ficients form an infinitely log-concave sequence. We extend the results to transcendentalentire functions in the Laguerre–Polya class, and discuss the consequences to problemson iterated Turan inequalities, studied by Craven and Csordas. Finally, we propose a newapproach to a conjecture of Boros and Moll.

1. Introduction

Let F be a transformation of sequences of real numbers, and let fakg be a real se-quence. We are interested in when the iterates F iðfakgÞ, for i A N, are non-negative. Suchquestions appear in the theory of entire functions [4], [5], and recently in the theory of spe-cial functions [2], [7], [8] and combinatorics [10], [17]. It has been made evident that the zeroset of the generating function of fakg plays a prominent role in such questions. One pur-pose of this paper is to make this correspondence explicit.

Let fakg ¼ fakgnk¼0, where n A NW fyg, be a sequence of real numbers. The se-

quence is log-concave if a2k � ak�1akþ1 f 0, for all 1e k e n � 1. Define an operator on

sequences by LðfakgÞ ¼ fbkgnk¼0, where bk ¼ a2

k � ak�1akþ1 for all 0e k e n, anda�1 ¼ anþ1 ¼ 0. Hence, fakg is log-concave if and only if LðfakgÞ is non-negative. Thesequence is i-fold log-concave if the ith iterate, L iðfakgÞ, is non-negative, and infinitely

log-concave if it is i-fold log-concave for all i A N. Boros and Moll [2] conjectured that the

Supported by the Goran Gustafsson Foundation.

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sequence of binomial numbers,n

k

� �� n

k¼0

, is infinitely log-concave for each n A N. If the

polynomialPnk¼0

akzk has only real and non-positive zeros, then it follows that the sequence

fakg is log-concave. Motivated by this fact and Boros and Moll’s conjecture on binomialnumbers, Stanley [17], McNamara–Sagan [10] and Fisk [7], independently made the fol-lowing conjecture.

Conjecture 1.1. Suppose that the polynomialPnk¼0

akzk has only real and negative zeros.

Then so does the polynomial

Pnk¼0

ða2k � ak�1akþ1Þzk; where a�1 ¼ anþ1 ¼ 0:

In particular, the sequence fakgnk¼0 is infinitely log-concave.

It should be mentioned that similar questions were raised already in [4], [5], seeSection 8. In Section 3, we prove Conjecture 1.1. However, we take a general approachand study a large class of transformations of sequences. Let a ¼ fakgyk¼0 be a fixed se-quence of complex numbers and define two sequences LE

a ðfakgnk¼0Þ ¼ fbkðaÞgn

k¼0 andLO

a ðfakgnk¼0Þ ¼ fckðaÞgn

k¼0, where

bkðaÞ ¼Pyj¼0

ajak�jakþj and ckðaÞ ¼Pyj¼0

ajak�jakþ1þj;

and aj ¼ 0 if j B f0; . . . ; ng. In Theorems 5.7 and 5.8 we characterize the sequences a, forwhich LE (or LO) preserves the property of having generating polynomial with only realand non-positive zeros. The characterization is of Polya–Schur type; that is, LE

a (or LOa )

has the desired properties if and only if the generating function of LEa ðf1=k!gÞ (or

LOa ðf1=k!gÞ) is an entire function that can be approximated, uniformly on compact subsets

of C, by polynomials with only negative zeros. Similar characterizations of classes of trans-formations were given in [12] and [1]. The fundamental di‰culty in our setting is that in [1],[12], the transformations in question are linear, whereas the transformations that weconsider are not. This potential problem is overcome by a symmetric function identity(Theorem 2.1) that linearizes the problem.

In Section 8, we propose a new approach to the original conjecture (see Con-jecture 8.4) of Boros and Moll [2]. We state a conjecture that would imply 3-fold log-concavity of the sequences in question.

2. Symmetric function identities

Let fekðzÞgnk¼0 denote the elementary symmetric functions in the variables

z ¼ ðz1; . . . ; znÞ, and set ekðzÞ ¼ 0 for k B f0; . . . ; ng. If m ¼ fmkgyk¼0 is a sequence of com-

plex numbers, we define a symmetric function by

Wm;nðzÞ ¼Pie j

mj�ieiðzÞejðzÞ:

116 Branden, Iterated sequences and the geometry of zeros



Theorem 2.1. Let m ¼ fmkgyk¼0 be a sequence of complex numbers, and let

gk ¼Pbk=2c

j¼0

k

j

� �mk�2j; for k A N:ð1Þ

Then

Wm;nðzÞ ¼ enðzÞPnk¼0

gken�k zþ 1

z

� �;ð2Þ

where 1=z ¼ ð1=z1; . . . ; 1=znÞ.

Proof. By linearity it is enough to prove the theorem for the case when there is anumber m A N such that mm ¼ 1, and mk ¼ 0 for each k 3m.

Following [3], [11], for k; r; n A N, define a symmetric function srkðzÞ by

srkðzÞ ¼

Pa¼ða1;...;anÞ

za1

1 � � � zann ;

where the summation is over all a A f0; 1; 2gn such that a1 þ � � � þ an ¼ k, andjfi : ai ¼ 2gj ¼ r. By a simple counting argument, see [3], [11],

eiðzÞejðzÞ ¼P

r

i � r þ j � r

i � r

� �sr

iþjðzÞ;ð3Þ

and thus

eiðzÞeiþmðzÞ ¼P

j

2j þ m

j

� �s

i�jmþ2iðzÞ:ð4Þ

From the definition of s i�jmþ2iðzÞ, we see that

si�jmþ2iðzÞ ¼

PjSj¼mþ2j

zSei�jðz2t : t B SÞ;

where zS ¼Q

s AS

zs. Summing over all i in the equation above yields

PjSj¼mþ2j

zSQt BS

ð1 þ z2t Þ ¼ enðzÞen�m�2j z1 þ

1

z1; . . . ; zn þ

1

zn

� �:

Equation (2), for our choice of m, now follows from (4) when summing over all j. r

We pause here to sketch an alternative combinatorial proof of the important caseof (2) when m ¼ f1; 0;�1; 0; 0; . . .g. For undefined symmetric function terminology we referto [16], Chapter 7. For our particular choice of m, we want to prove the identity

Pnk¼0

�ekðzÞ2 � ek�1ðzÞekþ1ðzÞ

�¼ enðzÞ

Pbn=2c

k¼0

Cken�2k zþ 1

z

� �;ð5Þ

117Branden, Iterated sequences and the geometry of zeros



where Ck ¼ 2k

k

� �=ðk þ 1Þ is a Catalan number, see [16], Exercise 6.19. We may rewrite (5)

as

Pnk¼0

�ekðzÞ2 � ek�1ðzÞekþ1ðzÞ

�¼Pbn=2c

k¼0

Ck

PjSj¼2k

zSQj BS

ð1 þ z2j Þ:ð6Þ

The polynomial ekðzÞ2 � ek�1ðzÞekþ1ðzÞ is the Schur-function s2kðzÞ, where

2k ¼ ð2; 2; . . . ; 2Þ.

By the combinatorial definition of the Schur-function, the left-hand side of (6) is the gener-ating polynomial of all semi-standard Young tableaux with entries in f1; . . . ; ng, that areof shape 2k for some k A N. Call this set An. Given T A An, let S be the set of entrieswhich occur only once in T . By deleting the remaining entries, we obtain a standard Youngtableau of shape 2k, where 2k ¼ jSj. There are exactly Ck standard Young tableaux ofshape 2k with set of entries S, see e.g. [16], Exercise 6.19.ww. The original semi-standardYoung tableau is then determined by the set of duplicates. This explains the right-handside of (6).

3. Grace–Walsh–Szego type theorems and a proof of Conjecture 1.1

The Grace–Walsh–Szego theorem is undoubtably one of the most useful theoremsgoverning the location of zeros of polynomials, see [13]. A circular region is a proper subsetof the complex plane that is bounded by either a circle or a straight line, and is either openor closed. A polynomial is multi-a‰ne provided that each variable occurs at most to thefirst power.

Theorem 3.1 (Grace–Walsh–Szego). Let f A C½z1; . . . ; zn� be a multi-a‰ne and sym-

metric polynomial, and let K be a circular region. Assume that either K is convex or that the

degree of f is n. For any z1; . . . ; zn A K , there is a z A K such that f ðz1; . . . ; znÞ ¼ f ðz; . . . ; zÞ:

We are now in a position to prove Conjecture 1.1.

Proof of Conjecture 1.1. Let PðzÞ ¼Pnk¼0

akzk ¼Qnk¼0

ð1 þ rkzÞ, where rk > 0 for all1e k e n, and let

QðzÞ ¼Pnk¼0

ða2k � ak�1akþ1Þzk:

Suppose that there is a z A C, with z B fx A R : xe 0g, for which QðzÞ ¼ 0. We may write zas z ¼ x2, where ReðxÞ > 0. By (5),

0 ¼ QðzÞ ¼ anxn Pbn=2c

k¼0

Cken�2k r1xþ1

r1x; . . . ; rnxþ

1

rnx

� �;




where Ck ¼ 2k

k

� �=ðk þ 1Þ. Since Re

�rjxþ 1=ðrjxÞ

�> 0 for all 1e j e n, the Grace–

Walsh–Szego theorem provides an h A C, with ReðhÞ > 0, such that

0 ¼Pbn=2c

k¼0

Cken�2kðh; . . . ; hÞ ¼Pbn=2c

k¼0

Ck

n

2k

� �hn�2k ¼: hnpn

1

h2

� �:

Since ReðhÞ > 0, we have 1=h2 A Cnfx A R : xe 0g. Hence, the desired contradiction fol-lows if we can prove that all the zeros of pnðzÞ are real and negative. This follows fromthe identity

Pbn=2c

k¼0

Ck

n

2k

� �zkð1 þ zÞn�2k ¼

Pnk¼0

1

n þ 1

n þ 1

k

� �n þ 1

k þ 1

� �zk

¼ 1

n þ 1ð1 � zÞn

Pð1;1Þn

1 þ z

1 � z

� �;

where fPð1;1Þn ðzÞgn are Jacobi polynomials, see [14], p. 254. The zeros of the Jacobi poly-

nomials fPð1;1Þn ðzÞgn are located in the interval ð�1; 1Þ. Note that the first identity in the

equation above follows immediately from (5). r

Now that Conjecture 1.1 is established, we shall see how the ideas in the proof can beextended considerably.

If m is a sequence of complex numbers, define a (non-linear) operator,Tm : C½z� ! C½z�, by

Tm

�Pnk¼0

akzk

�¼Pie j

mj�iaiajziþj:ð7Þ

Define polynomials, Pm;nðzÞ, for n A N, by

Pm;nðzÞ ¼Pnk¼0

gk

n

k

� �zn�k ¼

Pj;k

n

k

� �k

j

� �mk�2jz

n�k:

A complex polynomial Fðz1; . . . ; znÞ is weakly Hurwitz stable if Fðz1; . . . ; znÞ3 0whenever ReðzjÞ > 0 for all 1e j e n. The following theorem can be seen as a Grace–Walsh–Szego theorem for certain non-multi-a‰ne polynomials:

Theorem 3.2. Let m be a sequence of complex numbers. The following are equivalent:

(i) Wm;nðzÞ is weakly Hurwitz stable.

(ii) For all polynomials PðzÞ of degree at most n, with only real and non-positive zeros,the polynomial Tm

�PðzÞ

�is either identically zero or weakly Hurwitz stable.

(iii) Tm

�ð1 þ zÞn� ¼ Wm;nðz; . . . ; zÞ is weakly Hurwitz stable.

(iv) The polynomial Pm;nðzÞ is weakly Hurwitz stable.




Proof. Suppose that Wm;nðzÞ is weakly Hurwitz stable, and that PðzÞ is a real poly-nomial of degree at most n with only real and non-positive zeros. By Hurwitz’ theorem on

the continuity of zeros, see e.g. [13], Theorem 1.3.8, we may assume that PðzÞ ¼Qnj¼1

ð1þ rjzÞ,

where rj > 0 for all 1e j e n. Suppose that ReðzÞ > 0. Then ReðrjzÞ > 0, for all 1e j e n.Hence

Wm;nðr1z; . . . ; rnzÞ ¼ Tm

�PðzÞ

�3 0;

which proves (i) ) (ii).

The implication (ii) ) (iii) is obvious. Clearly, by (2),

Wm;nðz; . . . ; zÞ ¼ znPm;n z þ 1

z

� �:

Hence, the equivalence of (iii) and (iv) follows from the set identity

fz þ 1=z : z A C and ReðzÞ > 0g ¼ fz A C : ReðzÞ > 0g:

Now, suppose that Wm;nðz1; . . . ; znÞ ¼ 0, where ReðzjÞ > 0 for all 1e j e n. Then, by (2),

Pnk¼0

gken�k z1 þ1

z1

; . . . ; zn þ1

zn

� �¼ 0:

Since Reðzj þ 1=zjÞ > 0, for all 1e j e n, the Grace–Walsh–Szego theorem provides anumber x A C, with ReðxÞ > 0, such that

0 ¼Pnk¼0

gken�kðx; . . . ; xÞ ¼ Pm;nðxÞ:

This verifies (iv) ) (i). r

4. Algebraic Polya–Schur characterizations of transformations

Let us turn to the cases when all the non-zero mi’s have the same parity. Leta ¼ fakgyk¼0 be a fixed sequence of complex numbers and define two sequencesLE

a ðfakgnk¼0Þ ¼ fbkðaÞgn

k¼0 and LOa ðfakgn

k¼0Þ ¼ fckðaÞgnk¼0, where

bkðaÞ ¼Pyj¼0

ajak�jakþj and ckðaÞ ¼Pyj¼0

ajak�jakþ1þj;

and aj ¼ 0 if j B f0; 1; . . . ; ng. Define also two non-linear operators on polynomials,Ua;Va : C½z� ! C½z�, by

Ua

�Pnk¼0

akzk

�¼Pnk¼0

bkðaÞzk and Va

�Pnk¼0

akzk

�¼Pnk¼0

ckðaÞzk:




We want to characterize the real sequences a for which Ua (or Va) send polynomials withonly real and non-positive zeros to polynomials of the same kind.

If PðzÞ ¼Pnk¼0

akzk, let

PEðzÞ ¼Pbn=2c

k¼0

a2kzk and POðzÞ ¼Pbðn�1Þ=2c

k¼0

a2kþ1zk:

The next theorem is a version of the classical Hermite–Biehler theorem, see e.g. [13], p. 197.

Theorem 4.1 (Hermite–Biehler). Let PðzÞ ¼ PEðz2Þ þ zPOðz2Þ A R½z�. Then PðzÞ is

weakly Hurwitz stable if and only if all non-zero coe‰cients of P have the same sign, and

� PEðzÞ1 0, and POðzÞ has only real and non-positive zeros, or

� POðzÞ1 0, and PEðzÞ has only real and non-positive zeros, or

� PEðzÞPOðzÞE 0, and PEðzÞ and POðzÞ have real and non-positive zeros which are

interlacing in the following sense. If z 0m e � � �e z 0

1 and zl e � � �e z1 are the zeros of PEðzÞand POðzÞ, respectively, then

� � �e z3 e z 02 e z2 e z 0

1 e z1:

Given a sequence m, we define two auxiliary operators, T Em ;T O

m : C½z� ! C½z�, by

T Em

�PðzÞ

�¼ Tm

�PðzÞ

�Eand T O

m

�PðzÞ

�¼ Tm

�PðzÞ

�O:

Let Pþn denote the set of all polynomials of degree at most n with only real and non-

positive zeros, and let Pþ ¼Sy

n¼0

Pþn .

Theorem 4.2. Let a ¼ fakgyk¼0 be a sequence of real numbers, and let n A N. The fol-

lowing are equivalent:

(i) UaðPþn ÞLPþ

n W f0g.

(ii) Ua

�ð1 þ zÞn

�A Pþ

n W f0g.

(iii)Pbn=2c

k¼0

�Pkj¼0

aj

ðk þ jÞ!ðk � jÞ!

�zk

ðn � 2kÞ! A Pþn W f0g:

Proof. Let m ¼ fa0; 0; a1; 0; a2; . . .g, and consider the operator Tm given by (7). ThenUa ¼ T E

m . By the Hermite–Biehler theorem, for each P A Pþn ,

TmðPÞ is weakly Hurwitz stable if and only if UaðPÞ A Pþn :

Now,

Pm;nðzÞ ¼ n!Pbn=2c

k¼0

�Pkj¼0

aj


�zn�2k

ðn � 2kÞ! ;




so Pm;nðzÞ is weakly Hurwitz stable or identically zero if and only if (iii) holds. The theoremfollows from Theorem 3.2. r

Example 4.3. Let us use Theorem 4.2 to give a second proof of Conjecture 1.1. Inthis situation a ¼ f1;�1; 0; 0; . . .g, and

Ua

�ð1 þ zÞn� ¼ Pn

k¼0

n

k

� �2

� n

k � 1

� �n

k þ 1

� � !zk ¼

Pnk¼0

1

n þ 1

n þ 1

k

� �n þ 1

k þ 1

� �zk:

These polynomials are known as the Narayana polynomials. There are numerous proofsthat the Narayana polynomials have only real zeros. The simplest is probably based onthe Malo theorem, see e.g. [6], Theorem 2.4.

The corresponding theorem for Va reads as follows.



(i) VaðPþn ÞLPþ

n W f0g.

(ii) Va

�ð1 þ zÞn� A Pþ

n W f0g.

(iii)Pbðn�1Þ=2c

k¼0

�Pkj¼0

aj

ðk þ 1 þ jÞ!ðk � jÞ!

�zk

ðn � 2k � 1Þ! A Pþn W f0g:

Proof. Consider m ¼ f0; a0; 0; a1; 0; . . .g. The proof proceeds just as the proof ofTheorem 4.2, since Va ¼ T O

m . r

5. Transcendental Polya–Schur characterizations of transformations

In this section, we provide transcendental characterizations of various transforma-tions. The following spaces of entire functions are relevant for our purposes:

� HðCÞ is the set of entire functions that are limits, uniformly on compact subsetsof C, of univariate polynomials that have zeros only in the closed left half-plane.

� HðRÞ is the space of entire functions in HðCÞ with real coe‰cients.

� The Laguerre–Polya class, L�P, of entire functions consists of all entire functionsthat are limits, uniformly on compact subsets of C, of real polynomials with only real zeros.A function f is in L�P if and only if it can be expressed in the form

fðzÞ ¼ Czne�az2þbzQyj¼0

ð1 þ rjzÞe�rj z;

where n A N, a; b; c A R, af 0, and frjgyj¼0 HR satisfies

Pyj¼0

r2j < y, see [9], Chapter VIII.




� L�Pþ consists of those functions in the Laguerre–Polya class that have non-negative Taylor coe‰cients. A function f is in L�Pþ if and only if it can be expressed as

fðzÞ ¼ CzMeazQyj¼0

ð1 þ rjzÞ;

where a;C f 0;M A N andPyj¼0

rj < y, see [9], Chapter VIII.

The following very useful lemma is due to Szasz [18].

Lemma 5.1 (Szasz). Let H HC be an open half-plane with boundary containing

the origin, and let f ðzÞ ¼ bMzM þ bMþ1zMþ1 þ � � � þ bNzN A C½z�, where bMbN 3 0. If

f ðzÞ3 0 for all z A H, then

j f ðzÞje jbM j jzjM expjbMþ1jjbM j jzj þ 3jzj2 jbMþ1j2

jbM j2þ 3jzj2 jbMþ2j

jbM j

!;ð8Þ

for all z A C.

Remark 5.2. The typical use of Lemma 5.1 is as follows. Suppose that fPnðzÞgyn¼0 isa sequence of polynomials that are non-vanishing in H, where H is as in Lemma 5.1. Write

PnðzÞ ¼PNn

k¼M

an;kzk;

and let fakgyk¼M be a sequence of complex numbers with aM 3 0. If limn!y

an;k ¼ ak for

each k fM, then there is a subsequence of fPnðzÞgyn¼0 converging, uniformly on compact

subsets of C, to the entire functionPy

k¼M

akzk. This follows from Montel’s theorem, since

fPnðzÞgyn¼0 is a locally uniformly bounded sequence by Lemma 5.1.

For a proof of the next lemma we refer to [9], Chapter VIII, or [1], Theorem 12.

Lemma 5.3. Let fðzÞ ¼Pyk¼0

akzk=k! be a formal power series with complex coe‰-

cients, and let H HC be an open half-plane with boundary containing the origin. Then fðzÞ is

an entire function which is the limit, uniformly on compact subsets of C, of polynomials that

are non-vanishing in H if and only if

fnðzÞ ¼Pnk¼0

n

k

� �akzk

is either identically zero or non-vanishing in H, for each n A N.

Theorem 5.4. Let m ¼ fmkgyk¼0 be a sequence of complex numbers and let fgkg

yk¼0 be

defined by (1). Define a formal power series by

TmðezÞ ¼Pyk¼0

gk

k!zk:




The following are equivalent:

(i) For all polynomials PðzÞ with only real and non-positive zeros, the polynomial

Tm

�PðzÞ

�is either identically zero or weakly Hurwitz stable.

(ii) TmðezÞ A HðCÞW f0g.

(iii) TmðL�PþÞLHðCÞW f0g.

Proof. Note that fnðzÞ is weakly Hurwitz stable if and only if znfnð1=zÞ is weaklyHurwitz stable. Combining Theorem 3.1 and Lemma 5.3 yields the equivalence of (i) and(ii). Clearly (iii) ) (ii). Assume (i) and let f A L�Pþ. Then, by Lemma 5.3, fnðzÞ is apolynomial with only real and non-positive zeros (unless identically zero) for each n A N.Thus Tm

�fnðz=nÞ

�is weakly Hurwitz stable or identically zero for each nf 1. Note that

limn!y

n

k

� �ak

nk¼ ak

k!;

for each k A N. By Remark 5.2, there is a subsequence fnjgyj¼0 such that

limj!y

Tm

�fnj

ðz=njÞ�¼ Tm

�fðzÞ

�;

where the convergence is uniform on each compact subset of C. Hence TmðfÞ AHðCÞWf0g.r

Remark 5.5. The characterization of the ‘‘good’’ sequences fmkgyk¼0 in Theorem 5.4

is in terms of the sequences fgkgyk¼0. How do we translate between the two sequences? The

answer is classical and is called the Chebyshev relation, see [15], p. 54:

gk ¼Pbk=2c

j¼0

k

j

� �mk�2j; for all k A N

if and only if

mk ¼Pbk=2c

j¼0

ð�1Þ j k

k � j

k � j

j

� �gk�2j; for all k A N:

The following lemma follows easily from Remark 5.2 and the Hermite–Biehlertheorem.

Lemma 5.6. Let fðzÞ be formal power series with real and non-negative coe‰cients.

Then fðzÞ A L�Pþ if and only if fðz2Þ A HðRÞ.



(i) UaðPþÞLPþW f0g.




(ii) UaðezÞ A L�PþW f0g, that is,

Pyk¼0

�Pkj¼0

aj


�zk A L�Pþ W f0g:

(iii) UaðL�PþÞLL�PþW f0g.

Proof. Let m ¼ fa0; 0; a1; 0; a2; . . .g. Then UaðPþÞLPþW f0g if and only ifTmðPþÞLHðRÞW f0g, by the Hermite–Biehler theorem. By Lemma 5.6, UaðezÞ A L�Pþ

if and only if TmðezÞ A HðRÞ, and UaðL�PþÞLL�PþW f0g if and only ifTmðL�PþÞLHðRÞW f0g. r

The proof of the next theorem is almost identical to that of Theorem 5.7.



(i) VaðPþÞLPþ W f0g.

(ii) VaðezÞ A L�PþW f0g, that is,

Pyk¼0

�Pkj¼0

aj

ðk þ 1 þ jÞ!ðk � jÞ!

�zk A L�Pþ W f0g:

(iii) VaðL�PþÞLL�PþW f0g.

6. Applications and examples

Let us apply Theorem 5.7 to a question posed by Fisk [7]. For r A N, let Sr ¼ Ua

where a0 ¼ 1, ar ¼ �1, and ai ¼ 0 for all i B f0; rg. In other words,

Sr

�Pni¼0

aizi

�¼Pni¼0

ða2i � ai�raiþrÞzi:

Fisk asked whether SrðPþÞLPþ for all r A N. We use Theorem 5.7 and the theory of mul-tiplier sequences to obtain partial results on Fisk’s question.

A sequence of real numbers flkgyk¼0 is a multiplier sequence if for each polynomialPnk¼0

akzk with only real zeros, the polynomialPnk¼0

lkakzk is either identically zero or has only

real zeros.

Multiplier sequences were characterized in a seminal paper by Polya and Schur [12].That multiplier sequences preserve the Laguerre–Polya class follows easily from Remark5.2 and Lemma 5.3, see also [9], Chapter VIII.

Theorem 6.1 (Polya and Schur). Let flkgyk¼0 be a sequence of real numbers, and let

T : R½z� ! R½z� be the corresponding (diagonal) linear operator defined by TðzkÞ ¼ lkzk, for




all k A N. Define FðzÞ ¼ TðezÞ to be the formal power series

FðzÞ ¼Pyk¼0

lk

k!zk:

The following are equivalent:

(i) flkgyk¼0 is a multiplier sequence.

(ii) TðL�PÞLL�PW f0g.

(iii) FðzÞ defines an entire function which is the limit, uniformly on compact sets, of

polynomials with only real zeros of the same sign.

(iv) Either FðzÞ or Fð�zÞ is an entire function that can be written as

CzneazQyk¼1

ð1 þ akzÞ;

where n A N, C A R, a; ak f 0 for all k A N andPyk¼1

ak < y.

(v) For all non-negative integers n the polynomial T ½ð1 þ zÞn� has only real zeros of the

same sign.

Proposition 6.2. Let r ¼ 0; 1; 2 or 3. Then SrðPþÞLPþ W f0g.

Proof. Fix r A N, and let SrðezÞ ¼Pyk¼0

ak; rzk. Then

ak; r ¼1

k!ðk þ rÞ!�ðk þ 1Þ � � � ðk þ rÞ � kðk � 1Þ � � � ðk � r þ 1Þ

�:

In particular,

S1ðezÞ ¼Pyk¼0

1

k!ðk þ 1Þ! zk:

For each m > 0, the sequence f1=Gðk þ mÞgyk¼0 is a multiplier sequence, see [6]. ThusF1ðzÞ A L�Pþ, by Theorem 6.1. This verifies the case when r ¼ 1 by Theorem 5.7.

Next, ak;2 ¼ ð2 þ 4kÞ=�k!ðk þ 2Þ!

�. Both sequences f1=ðk þ 2Þ!gyk¼0 and f2 þ 4kgyk¼0

are multiplier sequences. Hence, so is fð2 þ 4kÞ=ðk þ 2Þ!gyk¼0. The case r ¼ 2 now followsfrom Theorems 6.1 and 5.7.

Since ak;3 ¼ ð6 þ 9k þ 9k2Þ=�k!ðk þ 3Þ!

�, the case when r ¼ 3 follows from the fact

that f6 þ 9k þ 9k2gyk¼0 is a multiplier sequence. Indeed,

Pyk¼0

6 þ 9k þ 9k2

k!zk ¼ ð6 þ 18z þ 9z2Þez;




and the zeros of 6 þ 18z þ 9z2 are negative. By Theorem 6.1, f6 þ 9k þ 9k2gyk¼0 is a multi-plier sequence. r

Similarly, let S 0r ¼ Va, where a0 ¼ 1, ar ¼ �1, and ai ¼ 0 for all i B f0; rg.

Proposition 6.3. Let r ¼ 0; 1; 2 or 3. Then S 0rðPþÞLPþW f0g.

Proof. Fix r A N, and let S 0rðezÞ ¼

Pyk¼0

bk; rzk. Then

bk; r ¼1

k!ðk þ 1 þ rÞ!�ðk þ 2Þ � � � ðk þ 1 þ rÞ � kðk � 1Þ � � � ðk � r þ 1Þ

�:

The proof proceeds as the proof of Proposition 6.2. For example,

bk;3 ¼ 12

k!ðk þ 4Þ! ðk2 þ 2k þ 2Þ:

The sequence fk2 þ 2k þ 2gyk¼0 is a multiplier sequence since

Pyk¼0

k2 þ 2k þ 2

k!zk ¼ ð2 þ 3z þ z2Þez: r

We conjecture that S 0rðezÞ A L�Pþ for all r A N.

7. Refined results on the location of zeros

We provide here some general results on the e¤ect on the zeros of polynomials underthe transformations Tm, Ua and Va.

Theorem 7.1. Let m be a sequence of complex numbers, and let

PðzÞ ¼ 1 þ a1z þ � � � þ anzn ¼Qnj¼1

ð1 þ rjzÞ

be a complex polynomial of degree n. Suppose that K is a circular region containing no zeros

of Pm;nðzÞ. We further require K to be convex if m0 ¼ g0 ¼ 0. If z is a non-zero complex num-

ber for which

rizþ1

riz: 1e ie n

� �HK;

then TmðPÞðzÞ3 0.

Proof. Let z be as in the statement of the theorem, and suppose that TmðPÞðzÞ ¼ 0.Since

TmðPÞðzÞ ¼ Wm;nðr1z; . . . ; rnzÞ ¼ anznPn

k¼0

gken�k r1zþ1

r1z; . . . ; rnzþ

1

rnz

� �;




there is, by Theorem 3.1, a x A K such that

0 ¼Pnk¼0

gken�kðx; . . . ; xÞ ¼Pnk¼0

gk

n

k

� �xn�k ¼ Pm;nðxÞ;

which contradicts the assumptions on K . r

For 0 < y < 2p, let Sy ¼ freif : jp� fj < y and r > 0g be the sector centered on thenegative real axis, and that opens an angle 2y.

Theorem 7.2. Let a ¼ fakgyk¼0 be a sequence of real numbers such that

Ua

�ð1 þ zÞn� A Pþ

n :

Suppose that PðzÞ ¼Pnk¼0

akzk has zeros only in Sy, where 0e y < p=2. Then Ua

�PðzÞ

�1 0,

or all zeros of Ua

�PðzÞ

�are in S2y.

Proof. Suppose that PðzÞ ¼Pnk¼0

akzk has zeros only in Sy. Write

PðzÞ ¼ CQnj¼1

ð1 þ rjzÞ.

Then jargðrjÞj < y for all 1e j e n. If jargðzÞj < p=2 � y, then

rjzþ1

rjz: 1e j e n

( )H fz A C : ReðzÞ > 0g:

By Theorem 7.1, TmðPÞðzÞ3 0, where m ¼ fa0; 0; a1; 0; . . .g. Thus UaðzÞ ¼ Tmðffiffiffiz

pÞ3 0,

whenever jargðzÞj < p� 2y. r

The proof of the corresponding theorem for Va is almost identical.

Theorem 7.3. Let a ¼ fakgyk¼0 be a sequence of real numbers such that

Va

�ð1 þ zÞn

�A Pþ

n :

Suppose that PðzÞ ¼Pnk¼0

akzk has zeros only in Sy, where 0e y < p=2. Then Va

�PðzÞ

�1 0,

or all zeros of Va

�PðzÞ

�are in S2y.

8. Iterated Turan inequalities and the Boros–Moll conjecture

For Taylor coe‰cients of functions in L�Pþ, inequalities stronger than log-

concavity hold. Namely the Turan inequalities: IfPyk¼0

gkzk=k! A L�Pþ, then the sequence




fgkgyk¼0 is log-concave. Craven and Csordas [5] studied iterated Turan inequalities. De-

fine a transformation, T, on infinite sequences as follows. If fgkgyk¼0 is a sequence, let

TðfgkgÞ ¼ fmkgyk¼0, where mk ¼ g2

kþ1 � gkgkþ2. Note the shift of indices. Craven and Csor-das stated the following problem.

Problem 8.1. LetPyk¼0

gkzk=k! A L�Pþ. Is T iðfgkgÞ a non-negative sequence for all

i A N?

Craven and Csordas [5] proved that T2ðfgkgÞ is non-negative ifPyk¼0

gkzk=k! A L�Pþ,

and that T3ðfgkgÞ is non-negative ifPyk¼0

gkzk=k! A L�Pþ and g0 ¼ g1 ¼ 0. The second

result can be stated as follows: IfPyk¼0

gkzk=ðk þ 2Þ! A L�Pþ, then fgkgyk¼0 is 3-fold log-

concave. In [4], they posed the following problem.

Problem 8.2. Characterize the sequences fgkgyk¼0 such that

Pyk¼0

gk

k!zk A L�Pþ and

Pyk¼0

tk

k!zk A L�Pþ;

where ftkgyk¼0 ¼ Tðfgkgyk¼0Þ.

Theorem 5.7 provides a large class of entire functions for which Problem 8.2 holds.

Proposition 8.3. IfPyk¼0

gkzk A L�Pþ, then fgkgyk¼0 satisfies both conditions in Prob-

lem 8.2.

Proof. Since f1=k!gyk¼0 is a multiplier sequence, Theorem 6.1 implies

Pyk¼0

gkzk=k! A L�Pþ.

We claim that f1=ðk � 1Þ!gyk¼0, where 1=ð�1Þ! :¼ 0, is a multiplier sequence. Indeed

Pyk¼0

1

ðk � 1Þ!k! zk ¼ z

Pyk¼0

1

k!ðk þ 1Þ! zk A L�Pþ:

By Theorem 5.7,

Pyk¼0

ðg2k � gk�1gkþ1Þzk A L�Pþ;

and by Theorem 6.1,

Pyk¼0

g2k � gk�1gkþ1

ðk � 1Þ! zk ¼ zPyk¼0

tk

k!zk A L�Pþ: r




Let us describe the initial conjecture that motivated Boros and Moll to study infinitelylog-concave sequences. For l;m A N with lem, let

dlðmÞ ¼ 2�2mPmk¼l

2k 2m � 2k

m � k

� �m þ k

m

� �k

l

� �:ð9Þ

It is not trivial (at least without the use of computers) to prove that dlðmÞ is the lth Taylorcoe‰cient of the polynomial, defined for a > �1, by

PmðaÞ ¼2mþ3=2ða þ 1Þmþ1=2

p

Ðy0

1

ðx4 þ 2ax2 þ 1Þmþ1dx:

Based on computer experiments, Boros and Moll made the following conjecture,see [2].

Conjecture 8.4. For each m A N, the sequence fdlðmÞgml¼0 is infinitely log-concave.

Kauers and Paule [8] were able to prove log-concavity of fdlðmÞgml¼0, using computer

algebra. We make the following conjecture.

Conjecture 8.5. For each m A N, the polynomial

QmðzÞ ¼Pml¼0

dlðmÞl!

zl

has only real zeros.

We also make a stronger conjecture.

Conjecture 8.6. For each m A N, the polynomial

RmðzÞ ¼Pml¼0

dlðmÞðlþ 2Þ! z

l

has only real zeros.

Note that QmðzÞ ¼ ðd 2=dz2Þ�z2RmðzÞ

�, so Conjecture 8.5 is stronger than Conjec-

ture 8.6. The relevance of these conjectures stems from the results of Craven and Csordason Problem 8.1. If Conjecture 8.5 is true, then fdlðmÞgm

l¼0 is 2-fold log-concave. If Conjec-ture 8.6 is true, then fdlðmÞgm

l¼0 is 3-fold log-concave.

Acknowledgments. I thank the anonymous referee for carefully reading the paper.

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Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden

e-mail: [email protected]

Eingegangen 18. September 2009, in revidierter Fassung 26. Juli 2010




iterated sequences and the geometry of zeros

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