Parametric representation of a clas of multiply positive sequences

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Complex Variables, Theory and Application: An

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Parametric representation of a clas of multiply

positive sequences

I.V. Ostrovskii & N.A. Zheltukhina

To cite this article: I.V. Ostrovskii & N.A. Zheltukhina (1998) Parametric representation of a clas of multiply positive sequences, Complex Variables, Theory and Application: An International Journal, 37:1-4, 457-469, DOI: 10.1080/17476939808815144

To link to this article: https://doi.org/10.1080/17476939808815144

Published online: 29 May 2007.

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Parametric Representation

of

a Class

of

Multiply Positive

Sequences

I. V . O S T R O V S K I I ~ - ~ + * and N. A. Z H E L T ~ K H I N A ~ , ~

a Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey; b ~ . Verkin Institute for Low Temperature Physics and Engineering, 3 10 164 Kharkov, Ukraine

Communicated by A. Eremenko Dedicated to Professor A. A. Gol'dberg

(Received )

Keywords: Generating function; sequences; sequences of matrices

CIassiJicarion Categories: 30D99

1. INTRODUCTION

Multiply positive sequences were introduced by Fekete [ l ] in 1912 for study of zeros of real polynomials and entire functions. Since then these sequences have been studied by several mathematicians and have found several applications in Analysis (see, e.g. [2,3]).

Recall that the sequence

* E-mails: iossif@fen.bilkent.edu.tr, ostrovskii@ilt.kharkov.ua E-mail: zheltukhina@ilt.kharkov.ua

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45 8 I. V. OSTROVSKII AND N. A. ZHELTUKHINA

is said to be m-times positive for m E N U

{co),

if all minors of orders <

m

+

1 of the infinite matrix

ao- a1 a;! a3

- - .

0 a0 a1 a;!

...

0 0 a0 a ,

. . -

0

0 0 a0

...

.

. .

are non-negative. The class of all m-times positive sequences is denoted by PF,.

Evidently, the class P F I consists of all sequences (1.1). The class

P F 2 consists of all sequences of the form

where $ : N U {0) -t (-ca, + a ] , +(O) < ca, is a convex function.

In 1953, in joint works by Aissen, Edrei, Schoenberg, Whitney (see [2], p. 412) the complete solution of the problem of description of the class

P F , was found:

THEOREM AESW. The function f (z) = C ~ o a , z n is a generatingfirnc- tion of a sequence of P F , if and only if

where

Note that (1.3) gives the description of P F , in terms of independent

parameters ao, y , pi, qi (i = 1 , 2, 3,

. .

.). This means: (i) arbitrary values of the parameters can be chosen under the only conditions (1.4); (ii) there is a one-to-one correspondance between collections of these parameters and sequences of PF,.

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To the best of our knowledges, the problem of the description of the classes PF,, 3 5 m <

oo,

in, terms of independent parameters has not been solved until now. It is not clear even which kind of parameters could play the corresponding role (similar to that of ao, pi, qi and y in Theorem AESW).

This paper is devoted to description of a subclass Q3 C PF3 in terms of independent parameters. The subclass was for the first time considered by the authors [5] in connection with a generalization of P6lya's theorem on sections of power series.

DEFINITION We say that a sequence (1.1) of PF3 belongs to Q3 if all truncated sequences ₌_{{ao, a l , a2,}

.

. .

_,

_ak,0,O,

. .

.}, k E N,

belong to PF3. (Note that in [5] we denoted by Q3 the class of all corresponding generating functions.)

It turns out that the role of independent parameters describing Q3 play points of the set

(0,oo) x [O, co) x

U,

(1.5) where

U

is the set of all sequences { f f k } & , 0 I ak I 1, k = 2 , 3 ,

.

.,

such that if a , = 0 for some j , then ak = 0 for any k 2 j. To give a precise statement of our result, we define the numbers

Our main result is the following theorem.

THEOREM 1. A sequence (1.1) belongs to Q3

if

and only

if

Thus, the independent parameters are ao, a , ( ~ 2 , 0 1 3 ,

. . ..

The only res-

trictions on them are (1.8) i.e. belonging of the point (ao, a , ~ 2 ,

. .

.) to the set (1.5).

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460 I. V. OSTROVSKII AND N. A. ZHELTUKHINA

Since

Q3

is a subclass of PF2, any sequence { a k ] Z 2 of

Q3

admits the representation (1.2). The question arises what additional conditions on the convex function $ imply F a t

( U ~ ) ; P , ~

belongs to

Q3.

Denote N = min(n : $ ( n ) = S c o ] (N

=

+oo

if +(n) < +w, Vn E N ) .

(1.9)

Set A2$(n) := $ ( n ) - 2 $ ( n - l ) + + ( n - 2 ) , 2 5 n < N, n 2 N (if N <

+co).

Define a sequence ( w n ) Z 2 , putting

W2 = 1, Wn = [ a 2 ~ 3 ", f f n - ~ ] a ~ = ~ ~ =

...

mnnI=I. n 2 3.

Evidently, the sequence { w n } z 2 is increasing and, for any E

U,

we have

THEOREM 2. For a sequence (1.2) to belong to

Q3

it is necessary and suficient that

The equality holds in ( I . 1 1 ) for every n 2 2 for the sequence (1.2) given by (1.7) with cr2 = a3 = = 1.

Evidently, the bound in the right hand side of ( I . 1 1 ) has the following

properties: (i) it is contained in the half-closed interval (log c, log 21, (ii) it is equal to log 2 for n = 2, (iii) it tends to log c as n + oo. There- fore, using the second assertion of Theorem 2, we obtain the following corollary.

COROLLARY 1. For a sequence (1.2) to belong to

Q3

it is necessary to have

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and it is sufficient to have

Both above conditions are unimprovable in the following sense: thefirst one ceases to be necessary if one replaces Iog c by a larger constant; the second one ceases to be sufficient if one replaces log 2 by a smaller constant.

Note the following immediate corollary of Theorem 2.

COROLLARY 2. Let $ : [0, +w) -+ (-CG, foe) be a convex function

of c2[0, 00).

(i) Iffor all sufficiently large x we have $"(x) < log c, then the sequence (1.2) does not belong to

Q3.

(ii)

lf

for all x 2 0 we have $"(x) 2 log 2, then the sequence (1.2)

belongs to

Q3.

Theorem 2 and Corollaries 1, 2 can be used to determine whether a given sequence belongs to

Q3.

Example Consider the sequence

If ,9 1 2, then the function $(x) := dxp satisfies At$@) 2 A;$(2) =

d(2p

-

2) for n 2 2. By Corollary 1, (1.12), we have A(/?, d ) E

Qs

for d

2

(log2)/(2$

-

2). By Theorem 2, (1.1 1) with n = 2, we have A@, d )

g

Q3

for d < (log2)/(28

-

2). If

B

< 2, then $"(,K) -+ 0, as

x

+

00. By Corollary 2 we have A(,9, d) $

Q3.

Theorem 2 allows us to describe the possible character of the growth of $ for sequences ( 1.2) belonging to

Q3.

Define the lower order A[$] and the order p[$] of $ as log $(n A[$] = lim inf

log,

p[@] = lirn sup

-.

n - + ~ logn =+a? logn

Evidently, A[$] 5 p[$]. Since, by (1.12),

n2

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462 I. V. OSTROVSKII AND N. A. ZHELTUKHINA

we have A[$] 2 2. The question arises about description of the pairs

(A[+];p[$]), for which the sequence (1.2) belongs to

Qs.

Using Theo- rem 2, ,we prove that 2 5 A[$] 5 p[+] 5 oo is the only restriction. THEOREM 3. For any pair of numbers (a, b), 2

5

a

5

b

5 co,

there is a sequence (1.2) of

Q3

such that for the correspondingfunction

+

we have A[+]

=

a, p[+] = b.

2. PROOF OF THEOREM 1

Recall two lemmas and some notation from [5].

LEMMA 1. [5] Let ( 1 . 1 ) be a sequence of PF2. I f N <

co,

where N is defined by ( 1.9), then a, = 0 for any n 2 N.

For sequences ( 1 . 1 ) belonging to PF2, Lemma 1 allows us to intro- duce the numbers

% - I Pn Pn =

-

, 1 5 t z e N ;

a n = -

, 1 5 n < N . a, Pn- I Evidently, and where a = ( l j p l )

=

(allao).

LEMMA 2. [5] Let (ak}z=O = {ao, a , ,

. . .

, a,, 0, 0,

. .

.}, a0 > 0, a, > 0,

n 2 2 be a sequence of PF3. Then

I ) for n = 2, we have 62 2 2, 2 ) for n 2 3, we have 6, > 1 and

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We start the proof of Theorem 1.

(i) Let (1.1) be a sequence of

Q3.

Let us show that the representation (1.7) is valid. By Lemma 2 the numbers

1

y,, :=

-

2 4 n < ~ ,

6, - 1 ' (2.2)

are well-defined and the following inequalities hold:

Define the parameters

which will take part in (1.7). From (2.3), (2.4) it follows

{a,}z2

E

U.

From (2.4) we obtain

Hence, by (2.2),

Using (2.1), we have for 2

5

n < N

anan

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464 I. V. OSTROVSKII AND N. A. WELTUKHINA

(ii) Take any sequence E

U

and any numbers a0 > 0, a

>

0, and form the sequence ( a k ) g o according to (1.7). Let us prove that {ak)& E Q3.

Let i

:=

rnin{n : a, =

0}

(i

:= m if a,

#

0, Vn 2 2). Then we have

a, > 0 for 0

5

n < i, and a,

=

0 for n i. Form the sequence

Since 0 5 or, ( 1, n

>

2, the following inequalities hold:

Denote

=l+l/y,, 2 < n e i , n

>

i. Then we have

The proof of ( n , ) ~ O E

Q3

is very close to that of Lemma 5 of [5]. Nevertheless, we will present the proof for the reader's convenience.

As in [ 5 ] , we use the following theorem of Schoenberg.

THEOREM ([4]) Let {b,

~ k , , ~

be a finite sequence of numbers. Consider m matrices

where B, consists of n rows and k

+

n columns. Assrime the following condition is satisfied for n = 1,2,

. .

.

, m: all n x n -minors of B, , consisting of consec~itive columns are strictlypositive. Then the sequence (bo, bl ,

. .

.,

bk, 0, 0 .

.

.} is m-times positive.

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where E > 0 will be chosen sufficiently small later. Form three matrices

All minors of A1 are evidently positive for 0 < E < ak. Since

we have

for sufficiently small E > 0. Therefore all minors of A2 are positive for

such E .

Further, consider the determinants

P where 2 5 n < i

-

1. Moreover, (2.7) and (2.6) yield non-negativity of

the minor

, 2

5

n < i. N n =

In virtue (2.9), (2.7) and (2.6), we have

an an+l 0

an-1 an an+l

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466 I. V. OSTROVSKII AND N. A. ZHELTUKHINA

Now, consider 3 x 3-minors of A3 consisting of consecutive columns:

Positivity of Mo and Mk(&) for 0 < E < ak is evident. Since

we have Mn > 0, n = 1,2,

. . .

, k

-

3 , and M k - 2 ( ~ ) > 0. Further, since

we have Mk-l > 0 for sufficiently small E .

Applying Schoenberg's theorem, we conclude that the sequence (2.8) belongs to P F 3 for sufficiently small E > 0. Taking a limit as E -+ 0, we obtain that the truncated sequence (a,}k,=o

=

(ao, a , ,

. . .

, ak, 0, 0,

. .

.)

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Assume {ak}rd E

Q3.

Then the representation

(1.7)

of Theorem 1 yields the following expression of the function

I++

connected with { a k } g o

by

(1.2):

=

-

log "

)n-2...

(~+;)~-l

(I+=

This yields

Using the definition of the numbers w, and

(1.10),

we conclude that

i.e.

(1.1

1)

is true.

Now, assume that

(1.1

1) holds. Set

and show that the numbers ( ~ 2 , ( ~ 3 ,

. . .

E

[0,

I]

can be chosen such that the representation

(1.7)

is true.

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468 I. V. OSTROVSKII AND N. A. ZHELTUKHINA

Since by (1.11) we have A2$(2) ) log2, the number a2 defined

by the first equality (3.2) belongs to [0, 11. Further, we will define the numbers ~ 3 ~ 4 4 ,

. . .

inductively by means of the second equality (3.2). Since, for fixed a*, a; .a,-l (n 2 3), the function log(1

+

( l / a n J[cr2a3

-

as a function of a, E [O, 11 is decreasing and its range covers the closed half-ray [log 2, oo], the number a, is defined by (3.2) uniquely if a2, a j ,

. .

.

an- 1 have been defined. Having defined

{ ~ k } k m , ~ E

U,

we have (3.2) and (3.1) for all n 2 2. Inverting the calcula- tions in (3. I), we see that the representation (1.7) holds. By Theorem 1,

we conclude that ( a k ) g o E Q3.

For a = b = a, it suffices to consider any sequence (1.8) with a, = 0, n 2 N, where N is any integer

2

1. For such a sequence, we have $(n) = +oo for n 2 N.

Now, let us assume 2 < a

5

b < oo. Consider an auxiliary function

$OW

=

e x p ( W 1 , 1

O(x) = -(a

+

b

+

(b

-

a ) sin(1og log log x)} log x, 2

well-defined for x > xo = ee. Evidently, O(x) ) a l o g x ) 2logx for

x > xo, therefore qo(x) 2 x2, x > XO. Moreover,

Since

a

+

b

+

(b

-

a ) sin(1og log logx) at(x) =

2x

Ott(x) =

-

a

+

b

+

(b

-

a ) sin(1og log log x)

2x2 0 x + + m ,

we have

= fi'(x)h(x) 2

(:

+ o

(:))

XI, x + +m. !&'(x> = {O"(x)

+

Ot2(x)} $0 (x)

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a

+

b

+

(b

-

a ) sin(1og log logx) 2

a

+

b

+

(b

-

a)-sin(1og log logx) 2

Using the estimate

eo

2 x2, x > XO, we obtain

for sufficiently large x.

Choose a constant A so large that the function @(x) := $o(x + A )

satisfies the condition $"(x) 2 1 for all x 2 0. Hence,

Corollary 1 shows that the sequence ak = exp{-$(k)} belongs to

Q3.

Evidently,

U$I

=

U$oI

= a, p[$l = ~ E e o l = b.

The case 2 5 a c b = +m can be considered analogously by choosing

O(X) = { a

+

(log log logx) sin2(log log logx)) 10gx.

References

[I] Fekete, M. and P6Iyh G. (1912). ifber ein Problem von Laguerre. Rendiconti Circ.

Math Palenno.. 34. 89- 120.

[2] Karlin, S. (1968). Total Positivity, Stanford University Press, Stanford.

[3] Baker, G. A. and Graves-Moms, P. (1981). P d Appmximants, Addison-Wesley, London.

[4] Schoenberg, I. J. (1955). On the zeros of generating functions of multiply positive sequences and functions. Ann. Math., 62, 447-471.

[S] Ostrovskii, I. V. and Ueltukhina, N. A. On power series having sections with multiply positive coefficients and a theorem of Pdlya J, Lond Math Soc., (to appear).